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Reconstruction of Extended Environments from Image Sequences


Reconstruction of Extended Environments from Image Sequences

by

Stuart Butter?eld

Submitted in accordance with the requirements for the degree of Doctor of Philosophy

The University of Leeds School of Computer Studies May 1997

The candidate con?rms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others.

ii

Stuart Butter?eld School of Computer Studies University of Leeds

Doctor of Philosophy May 1997

Reconstruction of Extended Environments from Image Sequences
Abstract
The automatic recovery of the three-dimensional structure of a scene from a sequence of two-dimensional images has been the subject of considerable research in the ?eld of machine vision, with applications as wide-ranging as object recognition, virtual reality and robot navigation. Traditional attempts to solve this structure from motion (SFM) problem rely on calibrated cameras and involve the detection and tracking of features through successive images in the sequence. When considering long image sequences, taken with an ordinary hand-held video camera, the problem is signi?cantly harder, since both camera calibration parameters and matched feature information are dif?cult to obtain accurately. An additional complication is that small errors in the recovered structure will accumulate over long sequences, possibly resulting in a reconstruction which is internally inconsistent. To date, there has been no discussion in the SFM literature of attempts to tackle this important issue. Recently, a number of different techniques have been developed for scene reconstruction using uncalibrated cameras. In such cases the recovered structure is correct up to a projective transformation of the real structure. In this thesis, an original, incremental reconstruction system is described, based on this uncalibrated approach. A novel implementation for computing the fundamental matrix from a pair of images is presented, from which a projective reconstruction is obtained. For the ?rst image pair in the sequence, a small number of ground truth points are used to upgrade from projective to Euclidean structure. This structure is propagated through successive frames to obtain a complete Euclidean reconstruction for the entire scene. The inconsistency problem is addressed by attempting to detect when previously viewed sections of the scene are re-encountered. A solution method using the geometric hashing model-based object recognition paradigm is proposed.

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Acknowledgements
First of all, I would like to thank Professor David Hogg for his encouragement, guidance and patience throughout the duration of my research. Too numerous to mention are all those staff and students in the School of Computer Studies, who have helped make my time here so enjoyable. Particular, thanks must go to my colleagues in the Vision Group, both past and present, who created such a friendly atmosphere to work, and play, in. From demonstration classes to doughnuts, from research seminars to football, it’s been a pleasure. A special thank-you to Andy Bulpitt, proof-reader, agony uncle and a great friend. I know I wouldn’t have reached this point without him. My eternal gratitude goes to my parents and sister for their incredible support, especially during this last year, and thanks to Sarah, for everything, but especially for putting up with me for so long. I gratefully acknowledge the ?nancial support of the University of Leeds, for the ?rst three years of this research, and thank the School of Computer Studies for keeping me gainfully employed whilst writing up. Finally, for her unwavering belief, to Elizabeth, the usual.

Contents
1 Introduction 1.1 1.2 1.3 1 3 4 5 6

::::::::::::::::::::::::::::::::: Overview of the Thesis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Approach Taken

2 Background 2.1 2.2

2.3

:::::::::::::::::: The Fundamental Matrix : : : : : : : : : : : : 2.2.1 Fundamental v. Essential Matrix : : : : 2.2.2 Calculating the Fundamental Matrix : : Uncalibrated Reconstruction : : : : : : : : : :
Introduction

2.4 2.5

: : : : : 2.3.1 Other Applications of the Fundamental Matrix : Sequence-Based SFM : : : : : : : : : : : : : : : : : Model Based Recognition : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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3 Acquiring the Fundamental Matrix 3.1 3.2 3.3 3.4 3.5

::::::::::::: An Overview of Epipolar Geometry : Derivation of the Linear Criterion : : Other Properties of F : : : : : : : : : Implementing the 8-Point Algorithm :
Introduction iv

CONTENTS

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3.5.1 3.5.2 3.5.3 3.5.4

Solution Via Singular Value Decomposition - SVD Solution Via Eigenvector - EIG Enforcing the Rank Constraint Normalisation

3.6

:::: 3.5.5 Quantifying Success : Results and Conclusions : : : 3.6.1 Initial Experiments : : 3.6.2 Synthetic Data : : : :
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Real Data: Corridor Images Real Data: House Images

3.7

Initial Conclusions

3.8

:::::: 3.7.1 Further Experiments : 3.7.2 Further Results : : : : Conclusions : : : : : : : : :

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30 31 32 33 35 36 36 37 38 40 43 45 48 49 50 50 51 52 56 58 60 64 64 65 66 72

4 Reconstruction 4.1 4.2 4.3 4.4 4.5 4.6 4.7

:::::::::: Outline of the method : : : : : Image Sequence Processing : : Constructing Camera Matrices : Factorising F : : : : : : : : : : Projective Reconstruction : : :
Introduction

: : : : : : Upgrading to Euclidean Structure : 4.7.1 Method 1: A Direct Solution :

4.8

: : : : : : : : 4.7.2 Method 2: An Indirect Solution : 4.7.3 Initial Experiments : : : : : : : : Incremental Reconstruction : : : : : : :

CONTENTS

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4.8.1 4.8.2

A RANSAC Approach

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5 Structure Recognition and Matching 5.1 5.2 Introduction

5.3

::::::::::::: An Overview of Geometric Hashing : 5.2.1 2D Af?ne Invariance : : : : : 5.2.2 Model Representation : : : : 5.2.3 Matching : : : : : : : : : : : Hashing Euclidean Structure : : : : :
5.3.1

5.3.2 5.4

: : : : : : Computing Euclidean Invariants : Symmetry Considerations : : : :
Partitioning the Hash Table

Geometric Hashing and Image Sequences

5.5

5.6

::: 5.4.2 Algorithm Outline : : : : : : : : Results : : : : : : : : : : : : : : : : : : 5.5.1 Synthetic Data: Simple Models : 5.5.2 Real Data: Recovered Structure : Conclusions : : : : : : : : : : : : : : :
5.4.1

6 Conclusions 6.1 6.2 6.3

::::::::::::::::::::::::::::::::::::: Future Work : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Closing Comments : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Summary

A Essential Projective Geometry A.1 Homogeneous Coordinates A.2

:::::::::::::::::::::::::::: Some Simple Constructions : : : : : : : : : : : : : : : : : : : : : : : : : : :

CONTENTS

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A.3

::::::: A.2.2 The Intersection of Two Lines : : : : : : : : : : : : : A.2.3 Normalising Homogeneous Coordinates : : : : : : : : A.2.4 The Perpendicular Distance of a Point from a Line : : : Projective Transformations : : : : : : : : : : : : : : : : : : : A.3.1 P 3 to P 3 : : : : : : : : : : : : : : : : : : : : : : : : A.3.2 P 3 to P 2 : : : : : : : : : : : : : : : : : : : : : : : :
A.2.1 Computing The Line Through Two Points

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B RANSAC C The Quadrangle Image Sequence

List of Figures
1.1 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 4.1 4.2 Sample images Transfer

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2 13 15 17 25 26 37 37 39 39 41 42 42 42 44 45 46 52 54

::::::::::::::::::::::::::::::::::::: Example reconstruction using VSDF : : : : : : : : : : : : : : : : : : : : : : : Example data from the VANGUARD project : : : : : : : : : : : : : : : : : :
Epipolar geometry

::::::::::::::::: Pencil of epipolar planes and lines : : : : : : : : : Graphs of S against using perfect synthetic data : Graphs of S against QF using perfect synthetic data Graphs of S against for the corridor image pair : Graphs of S against QF for the corridor image pair Sample epipolar lines for the corridor image pair. : :: Graphs of S against QF for the house image pair : Sample epipolar lines for the house image pair. : : Instability of the SVDNORM method : : : : : : : Example Lowest QF Values : : : : : : : : : : : :
for the house image pair Overview of Euclidean Reconstruction : Graphs of S against

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Results of the INRIA matching algorithm for the house image pair

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::::::::::::::::::::: Poor performance of a local matching algorithm. : : : : : : : : : : : : : : : : :
viii

LIST OF FIGURES

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4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

::::::: Our manual feature detection and matching user interface. : Example of projective reconstruction. : : : : : : : : : : : Comparison of direct/indirect projectivity calculations : : : (a) Front view of the reconstructed points : : : : : : : : : (a) Side view of the reconstructed points : : : : : : : : : : (a) Top view of the reconstructed points : : : : : : : : : : Another top-down view of the reconstruction : : : : : : :
Effects of false matches on epipolar geometry

: : : : : : : : Plan view of the recovered quadrangle scene after 10 image pairs : Failed reconstruction of part of the quadrangle scene : : : : : : : : ::::::: The general scheme of the geometric hashing algorithm : Representing p using the af?ne basis triplet a00 a01a10 : : Hash table organisation : : : : : : : : : : : : : : : : : Constructing 3D Euclidean invariants : : : : : : : : : : Effect of basis point ordering on coordinate axes. : : : : Effect of basis point ordering on invariants : : : : : : : : Test models for the geometric hashing system : : : : : : Distributions of invariants : : : : : : : : : : : : : : : :
Example of internally inconsistent structure

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B.1 Effect of an outlier on the accuracy of least-squares C.1 Quadrangle Sequence: Images 0 to 11 C.2 C.3 C.4 C.5

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:: Quadrangle Sequence: Images 12 to 35 : Quadrangle Sequence: Images 36 to 59 : Quadrangle Sequence: Images 60 to 83 : Quadrangle Sequence: Images 84 to 104 :

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List of Tables
3.1 3.2 4.1 5.1 5.2 5.3 5.4 5.5 Comparison of the BEST-F and RANSAC algorithms Comparing RANSAC

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48 49 63 90 96 97 98 100

Reprojecting projective structure

:::::: Results of geometric hashing using perfect synthetic data : Results of geometric hashing using noisy synthetic data : : Grouping of features into structure patches : : : : : : : : : Results of geometric hashing using real data : : : : : : : :
Effect of basis ordering on invariant coordinates

x

Chapter 1

Introduction
The work described in this thesis is motivated by a seemingly simple task - to determine the structure of a scene as viewed in a segment of video footage, typically obtained using an ordinary handheld video camera. Our particular concern is with ‘extended environments’, for which a small part of the scene is visible in each frame. Reconstruction of such environments introduces associated problems in ensuring the internal consistency of the recovered structure. In particular, structures that are seen more than once during the image sequence must be correctly identi?ed with each other. In terms of machine vision, the task is a combination of the fundamental problems of structure from motion (SFM) and structure matching, which have, and continue to be, extensively researched. The aim of this work is to bring together ideas and techniques from these two areas in an attempt to solve the overall task at hand. A successful system would have many, varied uses, for example in constructing a virtual reality ‘walk-through’ of a real building, navigating a moving robot, automatic acquisition of CAD models etc. Ideally, it should not be dependent on the type of camera motion, or the nature of the scene being viewed. Sample images from the kinds of sequences such a system might be expected to deal with are shown in ?gure 1.1.

1

CHAPTER 1. INTRODUCTION

2

Figure 1.1: Sample images.

CHAPTER 1. INTRODUCTION

3

1.1 Approach Taken
The choice of approach is dependent on the nature of the input image sequence. In our case the large disparity between successive images called for the use of feature-based SFM, rather than optical ?ow [1]. A manual corner detection and matching system is used, to avoid the additional dif?culties caused by poor localisation and false matches inherent in automatic methods. Even if all the images in the sequence are taken with the same camera (which is likely, but not certain), there is no guarantee that the internal camera parameters do not change during the course of the sequence, for example due to zooming. Thus it is necessary to employ an uncalibrated reconstruction technique. Note that even camera self-calibration [13] is not possible, since this depends on unchanging intrinsic parameters. The procedure is based around the calculation of the fundamental matrix, which embodies the epipolar geometry of a pair of images. The calculation is performed via a novel combination of the well-known 8-point algorithm [39], a recently developed normalisation technique [26] and the RANSAC parameter estimation paradigm [15]. The fundamental matrix is factorised to obtain a representation for the camera matrices, and the structure of the scene in the two images is recovered by back-projection. The reconstruction so obtained is only correct up to a projective transform of the real structure. Knowledge of a small number of ground truth points in the ?rst image pair is used to compute a projectivity, which transforms the projective structure to Euclidean. For subsequent image pairs, the projectivity calculation is performed using previously estimated Euclidean structure instead of ground truth. Thus, as the sequence is processed, a complete reconstruction of the scene is incrementally acquired. The second major part of this work is motivated by the acceptance of the fact that the reconstruction system will not produce perfect results, and as the sequence is processed, errors in the recovered structure will accumulate. If part of the scene is re-encountered, it will have two Euclidean structure estimates, at some displacement and orientation in the world coordinate system. In order to be able to update the structure so that it is internally consistent, it is necessary to obtain the mapping between the points in the two structure estimates. This, in turn, depends upon the ability of the system to recognise when structure that has been seen previously has come back

CHAPTER 1. INTRODUCTION

4

into view. These two problems are addressed by incorporating a model-based object recognition system into the reconstruction process. A variation of the standard geometric hashing algorithm is employed, whereby model acquisition and recognition are performed concurrently, the usual offline preprocessing step being omitted. At each step of the reconstruction process, a new segment of structure is recovered. This is combined with other recently acquired structure to obtain a local structure patch. The hash table is updated using Euclidean invariants computed from this set of 3D points. Thus, recognition should occur if any of these patches of structure are re-encountered later in the sequence. In this case the information stored in the hash table permits the mapping between old and new scene structure to be immediately obtained.

1.2 Overview of the Thesis
The remainder of the thesis is organised as follows:

Chapter 2: A review of the relevant background material and related research. Chapter 3: A description of a novel method used to estimate the fundamental matrix, based on a normalised version of the 8-point algorithm. Chapter 4: A description of an original incremental reconstruction system for recovering the Euclidean structure of a scene from a long image sequence. Chapter 5: A description of a version of the geometric hashing algorithm which has been incorporated into the reconstruction system in an attempt to allow previously encountered scene structure to be recognised. Chapter 6: Conclusions and a discussion of future work. Appendix A: An overview of some of the essential principles of projective geometry. Appendix B: A detailed description of the RANSAC parameter estimation paradigm. Appendix C: The quadrangle image sequence, which is used in experiments throughout this work.

CHAPTER 1. INTRODUCTION

5

1.3 Notation
We will use boldface letters to denote vectors and matrices e.g. element at the i’th row and j ’th column of matrix

M. Transposition of vectors and matrices is indicated by T , e.g. MT , the inverse of a matrix M by M? and the inverse transpose by M?T . The determinant of a matrix is denoted det(M). The cross product of two 3D vectors, x and y is represented by x y. Given some vector t =(tx ; ty ; tz ), it is useful to consider the anti-symmetric
1

M. The notation Mij denotes the

matrix

1 0 0 ?tz ty C B C B t] = B tz 0 ?tx C C B A @ ?ty tx 0
This is the matrix representation of the cross product. For any vectors and ,

(1.1)

s

t

sT t] = s t
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(1.2)

t] s = t s
On a related note, given the square matrix
+ ( )

(1.3)

M, we use the notation M to represent the matrix of cofactors of M, that is, the matrix de?ned by Mij = (?1)i j det(M ij ) where M ij is the matrix derived from M by removing the i’th row and j ’th column. If M is non-singular then M = det(M):(M?T ). In other words M M?T , where indicates equality up to a scale factor. If a and b are 3 1 vectors and M a 3 3 matrix then Ma Mb M (a b).
( )

Uppercase letters will be used to denote 3D points and lines, whereas lowercase letters indicate 2D features. We differentiate between the geometric objects themselves and their represenvector would be . The line between two points P1 and P2 is represented by hP1 ; P2i. tations. For example, a point in the image plane would be denoted by p, whereas its coordinate

p

Chapter 2

Background
2.1 Introduction
The recovery of the three-dimensional information lost when projecting a scene onto an image plane is a central problem in machine vision. At least two images are required, either taken by a pair of cameras in a stereo con?guration, or a single moving camera. Reconstruction via the latter of these is termed structure from motion (SFM). An early structure from motion theorem, due to Ullman [66], states that:

Given three distinct orthographic projections of four non-coplanar points in a rigid con?guration, the structure and motion compatible with the three views are uniquely determined, up to a re?ection about the image plane.

Over the years there have been innumerable structure from motion algorithms described in the vision literature. Due to the assumption of orthographic projection1, which is not an accurate model of the real image formation process, a general purpose structure recovery system based on this theorem is impractical. However, it does serve to illustrate a number of the ways in which structure from motion algorithms can be characterised. 1 Where rays are projected from an object point along a direction parallel to the camera’s optical axis, so that they
strike the image plane orthogonally.

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CHAPTER 2. BACKGROUND

7

First of all, there is the question of how many projections (images) are required for the algorithm to function. Some systems work with just two or three images, others have these as a minimum number for structure recovery, but are able to deal with long sequences of images. With image sequences, there is the additional consideration of whether the images are actually processed sequentially or as a batch. The number and type of image features used are also varied. Possibly the two most important characteristics are the projection model and whether or not the camera is calibrated. The most general method of projection from 3D space onto a 2D image is perspective projection (see Appendix A). Other methods approximate this under certain imaging conditions, for example scaled orthographic (weak perspective), paraperspective [48] and af?ne [47]. A description of the most appropriate camera model for a given situation is provided in [69], along with a method for switching models during processing to ensure the best model is used at all times. In our case, we make no assumptions about the nature of the scenes being viewed and so use the full perspective model. If camera calibration2 is known, scene structure can be recovered up to a scale factor [11]. However, the calibration process usually relies upon accurately measuring calibration objects and is very sensitive to errors [64]. Secondly, calibration has to be recomputed if the internal parameters change, for example, the camera zooms. These dif?culties have led to the development of the projective approach to reconstruction, also known as uncalibrated stereo [52]. This method models the geometric relationship between the cameras, matched points and corresponding 3D positions, which is encapsulated in the fundamental matrix. The non-metric nature of this projective approach means there is no dependency on camera parameters, but recovered structure differs from the real structure by a projective transformation. The acquisition of the fundamental matrix and a projective reconstruction technique form a signi?cant part of this work. These are fully detailed in Chapters 3 and 4. We continue in this chapter with a more general background description of the two subjects. This is followed by a review of current research on reconstruction from image sequences, most relevant to this thesis. The chapter concludes with an overview of some of the techniques of model-based recognition and background material on the method we use: geometric hashing. 2 Focal length, aspect ratio and principle point.

CHAPTER 2. BACKGROUND

8

This section opened with Ullman’s structure from motion theorem. In truth, the theorem was originally due to Kruppa, sixty-four years earlier, and rediscovered by Ullman. This is just one of several cases in which results that have only recently been obtained in the vision community, have been known for many years by photogrammetrists. A fascinating potted history of photogrammetry, from a projective geometry viewpoint, is given in [9], whilst [27] explores the relationship between photogrammetry and machine vision.

2.2 The Fundamental Matrix
In 1981 Longuet-Higgins published a seminal paper [39], in which he described a method for the recovery of the structure of a scene from eight point correspondences. By establishing constraints on the relationship between the two sets of image coordinates, their corresponding 3D points, and the optical centres of the cameras, a 3 3 matrix is de?ned, the essential matrix

E, which con-

veniently encapsulates the epipolar geometry (see section 3.2) of the imaging arrangement. Using homogeneous coordinates (see Appendix A) the essential matrix and a pair of corresponding points p and p0 in the two images are related as follows:

p0TEp = 0
homogeneous in the elements of

(2.1)

This equation, known as the linear criterion or the Longuet-Higgins relation, is linear and

E.

Thus, given a set of at least eight point correspondences

(hence the name), the essential matrix can be solved for, up to a scale factor. In the original paper, it is shown how

E can be factorised to obtain a pair of camera matrices, and hence the structure

of the scene recovered. There is the implicit assumption in Longuet-Higgins’ description that the cameras are calibrated - at least the focal length and principal point are known.

2.2.1

Fundamental v. Essential Matrix

There appears to have been some confusion amongst researchers about the distinction between the essential and fundamental matrices. The 8-point algorithm can be used to compute either of

CHAPTER 2. BACKGROUND

9

them, and equation 2.1 applies equally to essential and fundamental matrices i.e. if fundamental matrix, then

F is a 3

3

p0TFp = 0

(2.2)

The question is therefore: when does the 8-point algorithm produce an essential matrix and when does it produce a fundamental matrix? The answer to this question lies in the camera calibration. If the images are formed by projection onto the unit sphere, then the matrix is the product of an orthogonal matrix and an anti-symmetric matrix and it is therefore an essential matrix [13]. In the case of general projection, the matrix

A of intrinsic camera parameters transforms the im-

age into the image that which would have been produced by projection onto the unit sphere. This gives the relation:

F = A? T EA?
1

1

(2.3)

Thus, when we are dealing with uncalibrated images, the 8-point algorithm is used to recover the fundamental matrix. The essential matrix can be obtained if the images are taken with a calibrated camera. The two matrices can also be characterised as follows: the essential matrix has zero determinant and its two non-zero singular values are equal, thus it depends on ?ve independent parameters. The fundamental matrix is singular and has rank two. As such, it depends on seven independent parameters. Other properties of the essential matrix have been investigated in [31] and [42].

2.2.2

Calculating the Fundamental Matrix

The main attraction of the 8-point algorithm is its conceptual and numerical simplicity, the fundamental matrix being obtained from a simple set of linear equations. Unfortunately, the 8-point algorithm is susceptible to noise in the matched feature data; poor localisation and false matches causing the algorithm to fail. Until recently, it was generally held that this problem rendered the 8-point algorithm useless for any practical application, and research has concentrated on devel-

CHAPTER 2. BACKGROUND

10

oping other alternatives of computing the fundamental matrix, all of them more complex than the 8-point method. However, in [26] a normalising technique was presented, that claimed to improve the performance of the 8-point algorithm to a level as good as, and even surpassing, that of these alternatives. The rationale was that the poor performance of the eight-point algorithm is wholly due to implementations which do not take account of the numerical considerations involved, particularly the conditioning of the set of linear equations being solved. The proposed solution to this problem is a simple normalisation (scale and translation) of the matched point coordinates prior to processing. Numerical analysis demonstrates that this transformation leads to a signi?cant improvement in the conditioning of the system [26]. A detailed description of the normalisation method is given in Chapter 3. Here we give a brief overview of some of the alternative methods that have been proposed. Implementations of the 8-point algorithm attempt to minimise minF

P (p0TFp ), which
i i
i

is based on the linear criterion. Typically this is solved either using a closed form solution, setting one of the coef?cients of

F to 1, or via a least-eigenvector method, both described in section 3.5.

In [40] these linear methods are compared against a number of non-linear criteria, for example minimising the distance of a point from its corresponding epipolar line. In [62] it is shown that the fundamental matrix can be estimated from the image correspondences of only seven 3D points in general position. First, the correspondences are used to obtain a non-unique solution to the standard system of equations formed from the linear criterion. A further solution of a cubic equation is then necessary to determine the fundamental matrix that corresponds to the given point con?guration. The idea here is to use the least possible number of points in determining the fundamental matrix, to reduce the likelihood of including outliers. An in-depth discussion of the application of robust parameter estimators to calculate the fundamental matrix, is given in [60]. Methods considered here include, RANSAC (see appendix B), the Hough transform [33] and M-estimators. Standard least squares attempts to minimise the sum of the squares of the residuals

P r2 i.e. the difference between the observed and ?tted data. P (r ), where is a symmetric M-estimators replace r2 by other functions of the residuals min
i
positive-de?nite function.

CHAPTER 2. BACKGROUND

11

In [8] Boufama describes a completely new approach for computing the fundamental matrix based on virtual parallax. This method still requires at least eight point matches, but instead of computing the fundamental matrix directly, it relies on estimating the position of an epipole and a 2D homography. One advantage is that this method implicitly constructs a fundamental matrix that is of rank 2. Unfortunately, errors in feature localisation and matching are not the only possible causes of problems in fundamental matrix calculations. In the original paper on the 8-point algorithm [39], it is remarked that certain con?gurations of the eight points will cause the algorithm to fail, due to linear dependencies entering the computation. Examples of such con?gurations are; four of the points being in a straight line, seven points in a plane, or eight points at the vertices of a cube. A more complete analysis of the problem of degeneracy is given in [62]. Torr de?nes degenerate con?gurations of 3D points as those whose resulting image correspondences fail to de?ne a unique epipolar transform (section 3.2). Thus there exist two or more linearly independent fundamental matrices which encapsulate the epipolar geometry, and scene structure cannot be recovered unambiguously. An algorithm for detecting such degenerate con?gurations, which is robust to the presence of outliers, is described. A complete review of the issues involved in fundamental matrix theory was recently published in [41].

2.3 Uncalibrated Reconstruction
In 1992 Faugeras published a ground-breaking paper [12] that proposed a technique for recovering projective structure of a scene, from a set of matched points in a pair of uncalibrated images. The idea is to constrain the form of the two camera matrices. This is achieved by ?xing the coordinates of ?ve of the 3D points to be the standard projective basis, and making appropriate coordinate assignments for their corresponding image points. As a result the camera matrices can be represented as functions of just two arbitrary parameters. The exact determination of these parameters requires the coordinates of the epipoles and hence, computation of the fundamental matrix. With this done, the camera matrices are fully spec-

CHAPTER 2. BACKGROUND

12

i?ed and the locations of the 3D points can be recovered, relative to the coordinate system de?ned by the ?ve points. Thus the reconstruction is correct up to a projective transform of the real structure. The problem with this method is in the reliance on the accuracy of the chosen basis features. Mismatched or poorly localised basis features will impair the quality of the projective reconstruction. Just one month later, Hartley published his paper [23], which achieved the same results, in a different manner and without the reliance on basis points. In this case, the projective reconstruction is achieved through the analysis of the fundamental matrix. Just as the essential matrix can be factorised to obtain camera matrices, so too can the fundamental matrix. In the latter case however, it is shown that the factorisation is not unique, and that the camera matrices can be transformed by an arbitrary projectivity and still be a valid factorisation. Hence, once again, the recovered structure and camera locations differ from the real ones by a projective transform. The projective reconstruction method employed in Chapter 4, is based around this technique. The standard basis method above is one of several projective reconstruction techniques compared in a recent paper by Rothwell [52]. Rothwell places the basis method in a group of what he calls explicit reconstruction algorithms. The goal for each of these is to compute a pair of camera matrices from a set of image correspondences. Two other explicit methods are described. One estimates structure by computing the intersection of camera rays on which the 3D points must lie. Another uses singular value decomposition to obtain estimates for the camera matrices which are consistent with the epipolar geometry. Two implicit techniques are described, which compute structure using three-dimensional invariants of the camera con?guration and point sets. All the methods described in this paper assume knowledge of the weak calibration between the two cameras; i.e. the fundamental matrix. Rothwell concludes that the most reliable reconstructions are obtained using camera matrices derived via singular value decomposition, which is the method we use in Chapter 4.

CHAPTER 2. BACKGROUND

13

P

image 1 p1

image 2 p2

C1

C2

image 3 epipolar line for P from image 2 l 23

epipolar line for P from image 1

p3 l 13

C3

Figure 2.1: Transfer.

2.3.1

Other Applications of the Fundamental Matrix

Transfer

One simple application which follows naturally from the de?nition of the fundamental matrix is in the area of automatic feature matching. The fundamental matrix maps a point in one image to its corresponding epipolar line in the other. This constraint can be used to reduce the search in image 1 can be used to generate an epipolar line l13 for P in image 3; similarly, observation of space for the matching point. This idea extends to three images. The observation of the point P

P in image 2 can be used to generate a second epipolar line l23 for P in image 3. As shown in ?gure 2.1 the intersection of the epipolar lines l13 and l23 uniquely de?nes where P must appear
in image 3. This concept is known as transfer.

CHAPTER 2. BACKGROUND

14

Self-Calibration

A new approach to calibrating cameras, called self-calibration was presented by Faugeras et al in [13]. In contrast to existing techniques, which rely on calibration objects [64], all that is required is a set of point matches tracked over three or more camera displacements. The displacements are used to calculate the fundamental matrix, using the non-linear minimisation of the image plane distance of a point from its corresponding epipolar line, mentioned earlier. The epipolar transform (section 3.2) is derived from the fundamental matrix and used to solve for the coef?cients of the absolute conic, from which the camera intrinsic parameters are obtained[9]. The original authors [13] note that the precision of feature localisation required to obtain reasonable calibration results is at the limit of even the best feature detectors. Nevertheless, the method has been used in attempts to calibrate cameras automatically, and thus to recover Euclidean structure [25, 2].

2.4 Sequence-Based SFM
In this section we present a brief review of some current related research into reconstruction from image sequences.

The VSDF
The Variable State-Dimension Filter (VSDF) is not really a reconstruction system. Rather, it is a recursive estimation algorithm which has been applied to the problem of structure from motion [43]. In fact, one of the bene?ts of the VSDF is that it can be applied to any problem that can be formulated as a suitable measurement equation, for example [44]. The algorithm was designed speci?cally for real-time applications. Having computed optimal estimates for the structure and motion over a small number of initial images, the recursive part of the algorithm takes over and recomputes sub-optimal estimates, using new image data. An

CHAPTER 2. BACKGROUND

15

example reconstruction3 is shown in ?gure 2.2.

Figure 2.2: Example reconstruction using VSDF.

Vanguard
Vanguard[4] is a project being developed by a European Union consortium, including the Robotics Research Group at Oxford University. The goal is automatic 3D model building from long uncalibrated monocular image sequences, and the use of these models for rendering scenes in telepresence applications. This entails extracting both geometry and surface descriptions (re?ectance) at a level suitable for high quality graphical rendering. The system uses a robust tracking algorithm for corner and line segment features, based on the trifocal tensor [61]. The trifocal tensor performs a similar role for three views as the fundamental matrix does for two: it encapsulates all geometric constraints between the three views, that are independent of scene structure. Given point correspondences in two images, the trifocal 3 This reconstruction was obtained using software kindly supplied by the VSDF author, Phil McLauchlan.

CHAPTER 2. BACKGROUND

16

tensor determines the position of the point in the third. This is similar to transfer, described earlier in this chapter, but more robust. The trifocal tensor is computed from an initial set of feature matches in three images but for future images does all the matching itself. Camera matrices are generated from the tensor for these images, and used to instantiate 3D point and line structure. As each new image is processed, matches between it and the previous image provide a correspondence between existing 3D structure and new features, enabling a camera matrix for the new image to be obtained, thus determining the new camera position relative to the existing world coordinate system. Existing structure estimates are updated using an Extended Kalman Filter. Figure 2.3 shows some sample data and results obtained [59].

The Factorisation Method
The factorisation method, originally due to Tomasi and Kanade [58], is a batch method for recovering the structure and motion of an object from an image sequence. Point features are tracked through the sequence and used to construct a measurement matrix. Structure and motion are obtained by factorising the measurement matrix, using singular value decomposition [50]. It is implicitly assumed that camera intrinsic parameters are known. The method has been through many stages of development, starting with simple planar motion, then moving on to arbitrary motion in 3D with 2D images obtained under the orthographic camera model. The orthographic model was too simplistic to be of any practical use, being unable to model even the effect of distance on image size, the scaling effect. Further updates followed through the scaled orthographic and paraperspective models [49] to the projective model [48]. While still in batch form the algorithm was unsuitable for use in real-time applications, and as a result a sequential version was developed [46], but this needed most of the feature points to be visible in each frame, which is not feasible in an extended environment. The latest addition to the factorisation approach was in a recent paper by Held [28], which described an incremental windowed factorisation method. At each step Held applies a version of Morita’s sequential factorisation algorithm, to obtain a new shape matrix. This is then related

CHAPTER 2. BACKGROUND

17

Sample Image

Recovered Structure

Two novel rendered views. Figure 2.3: Example data from the VANGUARD project.

CHAPTER 2. BACKGROUND

18

to the shape matrix at the previous step by computing an af?ne transform between their shared shape points, then applying it to the points which have just been lost. Thus the scene structure is recovered incrementally as the ‘factorisation window’ moves through the image sequence. The major differences between our method and Held’s are that he has gone back to using the orthographic projection model, whereas we use full perspective, and he makes no attempt to register between structure recovered at different times.

2.5 Model Based Recognition
The structure matching required within our framework is most closely analogous to model-based approaches to object recognition. As background to our choice of matching strategy, this ?nal part of the chapter reviews the major approaches in this area. Object recognition is one of the most fundamental of all problems studied in machine vision. The aims of an object recognition system are twofold: to detect the presence of a given object, and to establish its location and orientation. The latter of these is known as pose determination. The pose of an object can be expressed in terms of the rigid transformation required to rotate and translate to the object position from the origin of the given coordinate system. The most predominantly used approach to object recognition is model-based, which relies upon the construction of explicit representations (models) of the geometric shape of the objects which are to be recognised. In this section we provide a brief overview of a number of different model-based recognition schemes, ?nishing with the geometric hashing paradigm, which is the method used in this work. For more detail the reader is referred to the general surveys on modelbased recognition in [5] and [10]. In general, model-based recognition is a two-step process: hypothesis generation and veri?cation, for example [51]. In the ?rst stage, a number of likely candidate objects and poses are singled out for further investigation. These are examined more closely in the second stage and incorrect candidates are eliminated. Hypothesis generation involves matching subsets of model and scene features, for example corner points and edges. Veri?cation requires the computation

CHAPTER 2. BACKGROUND

19

of the transformation between the model and scene, in an attempt to induce more feature correspondences and thus accept or reject the candidate match. In model-based recognition, pose determination involves computing the transformation between the model and the scene. Note that we use the terms model and scene without referring to their dimensionality, or the nature of the transformation between them. For example, the model and scene could both be image features, related by a 2D transform [37]. Equally, they could both be sets of structure points, related by a 3D transform, for example this work and [14]. The model and scene could even be of different dimensionality, as in the case of recognising 3D objects from 2D images [16, 6].

Alignment
In an alignment scheme [32], candidate poses are hypothesised, based on the correspondence of a minimal number of model and scene features. Each minimal set of correspondences is just enough to determine a unique pose, which is then veri?ed.

Pose Clustering (Hough Transform)
Pose clustering is similar to alignment but uses a more intelligent, rather than exhaustive, approach to selecting candidate poses for veri?cation. The idea is each pairing of model/scene features tallies a vote for the pose they determine. Correct poses should be voted for many times by different pairings of features and only high-scoring poses are considered for veri?cation. In practice, things are not quite as simple as that, since correct feature pairings will generally determine poses that are similar, but not identical, due, for example, to measurement error. The solution is to look for clusters of poses (hence the name). The usual way of doing this is by creating a parameterised pose space in which to tally the votes. For example, a 2D af?ne transform can be represented by by six independent parameters. Thus voting is performed in a six-dimensional pose space, represented by a multi-dimensional array, with all elements, or bins, initially zero. Although each parameter may take on continuous values, each dimension of the pose space is quantised, resulting in a set of six-dimensional volumes of admissible poses. The parameters of each candidate pose are calculated and used to increment the vote count at

CHAPTER 2. BACKGROUND

20

the corresponding bin. When all votes have been cast, the pose clusters correspond to those bins containing more than some given number of votes. A pose at the centre of each cluster is passed on to the veri?cation stage. Pose clustering methods are generalisations of the Hough transform [3], which was originally used for the detection of straight lines in images [30]. A survey of the Hough transform is given in [33].

Interpretation Tree Search
Alignment and pose clustering techniques produce candidate poses from correspondences between model and scene features, eliminating incorrect candidates by direct veri?cation. In contrast to this, interpretation tree search methods [21, 19] attempt to assign model features to all scene features, in all feasible combinations. Such assignments are called interpretations and the method involves performing a tree search to generate all of the combinations of interpretations as required. Nodes in the ?rst level of the tree contain assignments for the ?rst scene feature. Those at the second level contain explicit assignments for the second scene feature, and an implicit assignment for the ?rst, and so on for each level of the tree. Likewise, each branch corresponds to a different model feature. Each node in the tree represents a partial match between model and scene features and the path to the node from the root of the tree gives all the correspondences in the partial match. Finally, each leaf node de?nes a complete candidate match (interpretation) between the model and the scene. As described, the method is infeasible, since there is a vast number of interpretations for even a moderately complex model. The idea is to eliminate the need to verify many false matches by pruning the tree. This is possible because false matches can be identi?ed without needing to assign all the pairings. Each time a new node is to be added to the tree a set of fast consistency checks are performed against all nodes on the path back to the tree root. For example, if attempting to match a set of planar surface patches against a model in 3D, the angle between the surface patch normals could be examined to ensure it falls between some range of values, this range having been

CHAPTER 2. BACKGROUND

21

obtained by preprocessing the model before the search begins. Thus, the tree is pruned below any partial interpretation that is found to be inconsistent.

Indexing and Geometric Hashing
An alternative model-based recognition paradigm has been proposed, known as indexing. All indexing schemes share a common approach to the recognition problem. They compute invariants [47] from scene features and use them to index a look-up table containing references to model objects. The look-up table returns a weighted set of candidate matches, for veri?cation. Whereas the methods described previously must be applied separately for each of the models they should recognise, indexing schemes attempt to recognise all models simultaneously. This has obvious ef?ciency bene?ts when the size of the model database is large. Indexing techniques have been applied to a number of problems. For example, in [45] Mohan describes a method for recognising 3D objects from a 2D image sequence, assuming the weak perspective camera model. Object models are acquired automatically from the image sequence by tracking features through at least three frames, and extracting Euclidean (similarity) invariants [68]. In [54, 56, 55] Rothwell et al. describe the development of a complete model-based recognition system (LEWIS). Their system also acquires its models directly from images, this time using projective invariants to index the look-up table. Perhaps the best-known indexing method is geometric hashing, originally developed for the task of recognising ?at rigid objects [35, 34, 36] from images, using an af?ne approximation to the full perspective camera. However, the same approach can be used for many recognition problems involving a variety of transformations in two and three dimensions [37]. A full description of the method is given in Chapter 5, where it has been applied to the problem of recognising familiar segments of 3D structure, as they are recovered by the reconstruction system. Geometric hashing has an additional property which makes it ideal for our purposes; the nature of the information stored in the hash table means that the pose of an object is known as soon as it is recognised. There has been some discussion about the susceptibility of geometric hashing to sensor

CHAPTER 2. BACKGROUND

22

error [20]. A simple technique for taking errors into account is shown in [70], which involves tallying votes in a region of the hash table, rather than at a single, indexed location. Other additions to the original geometric hashing method have been developed. For example, in [63] Tsai presents a system which uses invariants computed from line features, under the assumption that these can be acquired from images with greater accuracy then points. In [14] an interesting application of geometric hashing is described, that of matching protein molecules. The system uses a two-atom basis and indexes the hash table using the lengths of the sides of the triangle the basis forms with each of the remaining atoms. This approach reduces the complexity of the process from

O(n4) to O(n3) at the expense of a non-unique representa-

tion of the atoms4. Constraints on the allowable lengths of the triangle sides mean that not all basis/atom pairs need to be considered. Finally, a new technique has recently been proposed, called enhanced geometric hashing [38]. This extends the basic method in two ways. First of all, the use of quasi-invariants [7] extracted from connected segments, reduces the number of invariants that have to be computed. Secondly, the voting scheme is augmented through the use of pose clustering. Each candidate pose, obtained via the normal voting procedure, de?nes a geometric transform, which is parameterised and used to vote in the pose space, as described above. Clusters form around coherent poses and the matched model is taken to be the one whose pose space contains the highest density cluster.

4 Where n is the number of points used to de?ne a basis for the transformation being considered.

Chapter 3

Acquiring the Fundamental Matrix
3.1 Introduction
Since the pioneering work of Faugeras [12] and Hartley [24], there has been a great deal of research into developing methods for recovering the structure of a scene from images taken with uncalibrated cameras. A recurring theme in this work, and in the method we describe in the next chapter, is the requirement for the accurate estimation of the fundamental matrix. The fundamental matrix is of vital importance in the the analysis of pairs of uncalibrated images, because it encapsulates all the information about camera motion, camera parameters and epipolar geometry that can be obtained from a set of point correspondences. The most simple method of computing the fundamental matrix is the 8-point algorithm, due to Longuet-Higgins [39]. This involves the solution of a set of linear equations derived from the linear criterion, which relates the fundamental matrix, image points and corresponding epipolar lines. A common criticism of the 8-point algorithm is that it is very sensitive to noise in the measured image features. In fact, the prevailing view is that this ?aw renders the 8-point algorithm useless for any practical application [40]. This has led to the development of a number of alternative methods of solution, as discussed in Chapter 2. These are, without exception, more complicated than the 8-point algorithm, involving iterative schemes [25] or non-linear minimisation [13, 40]. However, in a recent paper [26], Hartley described a novel normalisation technique,

23

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

24

which claimed to improve the performance of the 8-point algorithm to a level as good as (and even surpassing) that of the more complex methods. This technique was based on numerical analysis of a particular method of implementing the 8-point algorithm. One of the aims of this chapter is to determine what effect, if any, the normalisation process has when a different implementation method is used. Hartley described a series of experiments, in which the fundamental matrix was computed using different sized subsets of the matched image points. For each subset size, several trials were performed, each time randomly selecting the points used in the calculation. Given a measure of the accuracy of a fundamental matrix, results were presented in terms of subset size versus the median accuracy value obtained over all the trials. We have carried out similar experiments, but in contrast, our results show the effects of normalisation and subset size on the best accuracy values. There is the additional problem of determining the best subset of matched points to use to estimate the fundamental matrix. Rather than simply choosing the one that gives highest accuracy value over all the trials, we employ a more ef?cient, elegant solution based on the RANSAC parameter estimation paradigm [15]. This chapter describes our implementation and experiences in using the normalised 8-point algorithm to compute the fundamental matrix. We begin with an overview of epipolar geometry and derive the linear criterion on which the 8-point algorithm is based. This is followed by a brief description of some of the important properties of the fundamental matrix. Next we describe two different methods of solving the linear criterion equations. We then outline the normalisation process itself and explain how it can help improve the accuracy of the solution of these equations. Finally, we present the results of a series of experiments to compute the fundamental matrix, using a variety of real and synthetic data, and show how we can embed the normalised 8-point algorithm into the RANSAC paradigm. Appendices A and B provide essential background material on projective geometry and the RANSAC paradigm, respectively.

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

25

P

L

L’

p l C e e’ l’

p’

C’

plane intersects the images along the two epipolar lines l and l0.

optical centre is at positions C and C 0. The plane through CPC 0 is the epipolar plane for P . This

Figure 3.1: Epipolar Geometry: Two images of a point P are taken by a moving camera, whose

3.2 An Overview of Epipolar Geometry
Figure 3.1 illustrates the epipolar geometry. Suppose we have a pair of images of a scene, taken with either a single moving camera or a stereo pair. The 3D locations of the camera optical centres are at C and C 0 . The 3D scene point P is projected along rays L and L0 onto the image planes at

p and p0, respectively. Projection of C along the line joining the two optical centres, gives a point e0 in the second image, known as the epipole. Similarly, projecting C 0 gives the epipole e in the

?rst image. Note that, for reasons of clarity, the imaging arrangement we have shown results in the epipoles lying within the bounds of the two images. In practice this is not necessarily the case, indeed the line between the two optical centres may only intersect the image planes at in?nity. The 3D point P and the two optical centres de?ne the epipolar plane for P . This intersects

L0 onto the ?rst image and l0 is the projection of the ray L onto the second image. This property
has important implications when attempting to match points in the two images. If the 3D point

the image planes along the epipolar lines l and l0. It can be seen that l is the projection of the ray

P has been observed at position p in the ?rst image, then P must lie somewhere on the ray L. This, in turn, constrains the position of p0 : it must lie somewhere along the projection of L in the

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

26

second image i.e. on the epipolar line l0. Consequently, given a mechanism for calculating the epipolar line corresponding to a point in one image, the search space for the matching point can be dramatically reduced from the whole image to just a line. The fundamental matrix provides such a mechanism.
Image C2

Epipole

C1 Epipolar Lines

Figure 3.2: Each 3D point de?nes a plane with the camera optical centres. These planes intersect the images, forming two pencils of epipolar lines, with intersections at the epipoles Consider ?gure 3.2 and imagine a set of 3D points at general positions in the scene. Each point forms a different epipolar plane with the optical centres. As we have seen, each plane forms an epipolar line at its intersection with the image planes, thus multiple planes lead to the formation of a pencil of lines, which intersect at the epipole in either image. Let be any such plane, then projects to an epipolar line l in the ?rst image and l0 in
1

the second. The correspondences

^l and ^l0 are homographies between the two pencils of

epipolar lines and the pencil of planes through the optical centres.

It follows that the correspon-

dence l^l0 is also a homography. This homography is the epipolar transform. It is determined by the coordinates of the two epipoles and three

l^l0 correspondences. It follows that the epipolar

transform depends on seven independent parameters. In fact, it can be shown ([13, 41]) that the epipolar transform determines, and is itself determined by, the fundamental matrix. 1 We use the symbol ^ to denote homographic correspondence.

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

27

3.3 Derivation of the Linear Criterion
In this section we describe an alternative derivation of the linear criterion, due to Hartley [24]. This projective geometry version ?ts better with the uncalibrated/projective nature of the fundamental matrix than the calibrated/Cartesian method used by Longuet-Higgins in the original paper. Given the same imaging arrangement as in ?gure 3.1, we ?x the ?rst camera position, C , at the origin of our object space coordinate system. The second camera,

C 0, is located at some

displacement from this. We can represent the two cameras by the 3 4 transformation matrices they use to project from 3D object space coordinates to the 2D image planes (see Appendix A). We can assign the two camera matrices as follows:

C = (Ij0)

and

C0 = (Rj ? Rt)

(3.1)

where we have partitioned the 3 4 matrices into a 3 3 left sub-matrix and a 3 1 column vector,

I is the identity matrix, R is a 3D rotation matrix and t = (tx; ty ; tz ) is a 3D translation matrix. These two cameras matrices project a 3D image point P = (X; Y; Z; 1)T as follows: p = CP , (u; v; w)T = (IjO)(X; Y; Z; 1)T
and (3.2)

p0 = C0P , (u0; v0; w0)T = (Rj ? Rt)(X; Y; Z; 1)T

(3.3)

L by C 0. Another way of looking at L is as the set of all 3D points which project onto p under C . We now choose two points from this set; the camera origin (0; 0; 0; 1)T and the point at in?nity (u; v; w; 0)T . The coordinates of the projections of these points in the second image are ?Rt and R(u; v; w)T respectively. The homogeneous equation of the epipolar line, l0 , passing through
these two points can now be recovered using the cross product:

point, p0 in the second image is constrained to lie on the epipolar line, l0, which is the projection of

Now, as we saw in the previous section, given a point p in the ?rst image, the corresponding

(a; b; c)T = Rt R(u; v; w)T = R(t (u; v; w)T )

(3.4)

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

28

where 0

l = (a; b; c) represents the line au + bv + cw = 0. If we now de?ne the anti-symmetric matrix S = t] as: 1 0 B 0 ?tz ty C B C B tz 0 ?tx C (3.5) C B A @ ?ty tx 0
(a; b; c)T = RS(u; v; w)T
(3.6)

the properties of the cross product allow us to rewrite equation 3.4 as:

We set

F = RS, giving F as the 3

3 fundamental matrix and can now write:

l0 = Fp

(3.7)

Thus, given a point in one image, the fundamental matrix allows us to compute the corresponding epipolar line in the other. It is also clear that, since, by de?nition, the corresponding image point, p0 belongs to epipolar line l0, we obtain the linear criterion:

0 = p0 TFp

(3.8)

It is worth noting that by reversing the roles of the two images in this derivation, the fundamental matrix is switched to its transpose. That is, the following relations also hold:

l = FTp0
3.4 Other Properties of F
It has already been mentioned that

and

0 = pT FTp0

(3.9)

F determines and is determined by the epipolar transform,

which has only seven independent parameters. Although the fundamental matrix has nine elements, it too has only seven independent parameters. We can account for one of the degrees of freedom by noting that to

F will have no effect on the results of equations 3.7and 3.8.

F is only de?ned up to a scale factor. That is, applying an arbitrary scale

The other degree of freedom is taken care of by the fact that the fundamental matrix must have rank 2. To see why this is so it may be necessary to consider ?gure 3.1 once again. As we

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

29

know, application of the fundamental matrix to a point in one image, yields the corresponding epipolar line in the other, but what if the point in question is one of the epipoles? For example,

Fe = le. Geometrically, le is the projection of the ray hC; ei onto the second image, but by conproperty:

struction this line is reduced to a point, the corresponding epipole e0 . Thus, we have the following

Fe = FTe0 = 0
Consequently,

(3.10)

F is singular, has zero determinant and its rank must be less than or equal to

2. In general it is of rank 2. It is not possible for the rank to be 1, since this implies that the line between the optical centres belongs to the intersection of the image planes [41].

3.5 Implementing the 8-Point Algorithm
Equation 3.8 is linear and homogeneous in the nine unknown coef?cients of the fundamental ma-

F, up to a scale factor. Taking one such pair of points p = (u; v; 1), p0 = (u0; v0; 1) and labelling the coef?cients of F, gives an expanded linear criterion equation: 1 0 f f f C B 0 ; v 0; 1)T B f f f C (u; v; 1) = 0 C B (u (3.11) C B A @
be able to obtain a unique solution for
11 12 13 21 22 23

trix. Thus, in general, given a set of eight point matches pi

$ pi0 in the two images, we will

f31 f32 f33

Multiplying out and rearranging in terms of the known coordinates of

p and p0 gives:
(3.12)

uu0 f11 + vu0f12 + u0 f13 + uv 0f21 + vv 0f22 + v 0f23 + uf31 + vf32 + f33 = 0

Each pair of matched points gives one such equation. With more matches, we can build up

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

30

a set of homogeneous linear equations

0 Bf B B 0 1B f B B B uu0 vu0 u0 uv0 vv0 v0 u v 1 C B f B B B ::::::::::::::::::::::::::::::::::: CB f CB B CB B B ::::::::::::::::::::::::::::::::::: CB f CB B CB B CB B B ::::::::::::::::::::::::::::::::::: CB f CB B CB @ AB ::::::::::::::::::::::::::::::::::: B f B B Bf B @

Af = 0:

11 12 13 21 22 23 31 32

f33

1 C C C C 0 1 C C C B 0 C C B C C B C C B C C B C C B C C=B C C B C C B C C B C C B C C @ A C C C C C C A

(3.13)

The process of generating and solving the above system of equations is the 8-point algorithm. In practice, we are given many more than just eight matches, resulting in an over-determined set of equations. In order for there to exist a non-trivial solution to de?cient, that is, although

A has nine columns, its rank will be at most eight. This is generally

Af = 0, the matrix A must be rank-

true if we are dealing with perfect data, but in practice inaccuracies in the matched points lead to

A having full rank. In this case we seek a least-squares solution to the linear criterion equations.
We will now look at two different methods of obtaining this solution.

3.5.1
Since

Solution Via Singular Value Decomposition - SVD

F is only de?ned up to a scale factor, we can ?x one of its coef?cients to a known value
= 1 we obtain the following modi?ed system

and then compute a closed-form solution to the linear criterion equations. Numerical analysis of non-linear solutions has shown that the choice of this normalising coef?cient is not arbitrary: it should be one of the ?rst six [40]. By setting f13

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

31

of linear equations

0 Bf B 0 1B f 0 vu0 uv 0 vv 0 v 0 u v 1 B B uu B CB f B B ::::::::::::::::::::::::::::::: CB CB B CB B B ::::::::::::::::::::::::::::::: CB f CB B CB B CB B B ::::::::::::::::::::::::::::::: CB f CB B CB f @ AB B ::::::::::::::::::::::::::::::: B Bf B @

^f A^ = ?u0:

11 12 21 22 23 31 32

f33

1 C C 0 C 1 C C B ?u0 C C B C B C C B C C B C C B C C=B C C B C C B C C B C C B C C @ C C A C C C C A

(3.14)

This system can then be solved, using Singular Value Decomposition [50], which is known to give the best estimate for the remaining coef?cients of , in a least-squares sense. This is perhaps the simplest method for computing the fundamental matrix, and possibly the one which is most prone to the effects of noise and outliers in the matched features. Although the normalisation transform, discussed in the next section, was not designed speci?cally with this approach in mind, it will be interesting to see what effects it has. In the sequel we will refer to this method of solution as SVD.

F

3.5.2

Solution Via Eigenvector - EIG

Another, more popular, method of computing the fundamental matrix is to formulate the linear criterion equations as a classic minimisation problem and seek the vector which minimises:

f

min kAf k f
unit norm eigenvector of

subject to

kf k = 1

(3.15)

where k:k indicates the Frobenius norm 2 . It is well known that the solution to this problem is the

ATA, corresponding to the smallest eigenvalue. In our implementation we solve this problem by ?rst reducing AT A to tri-diagonal form, via the Householder method.

The eigenvalues and eigenvectors can then be recovered using the QL algorithm. See [50] for the details. From now on we will refer to this solution method as EIG. 2 The square root of the sum of squares of the coef?cients. For a vector, the Frobenius norm is equivalent to the two
norm.

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

32

3.5.3

Enforcing the Rank Constraint

As noted in section 3.4, important properties of the fundamental matrix are that it is singular, has rank 2 and

det(F) = 0. In fact, most applications of F, depend on this singularity or rank conF, com-

straint. However, in general, the fundamental matrix obtained using the methods based on the linear criterion will not have these properties, or, to put it another way, the linear criterion cannot express the rank constraint. To see the effect this has, consider a fundamental matrix, puted via a linear criterion method, a point

epipolar line l0. Furthermore, we denote the coordinate vector of the epipole in the ?rst image by

p = (u; v; 1) in the ?rst image and its corresponding

e = (eu; ev ; 1) and ?x the horizontal and vertical offsets of p from e as x and y, respectively. We can now express p as: 1 0 1 0 B u C B eu ? x C C B C B p = B v C = B ev ? y C (3.16) C B C B A @ A @
1 1
Now, applying the fundamental matrix, gives:

0 1 1 0 eu ? x C BxC B C B B C l0 = Fp = F B ev ? y C = Fe? F B y C B C C B @ A A @ 0 1 | {z }
loffset

(3.17)

F is singular, Fe = 0, exactly, and the right hand side of the equation simpli?es to lo set, which is an epipolar line, as required. However, when F is non-singular, l0 is the sum of a constant p residual vector r = Fe and the vector lo set , whose norm is bounded by x + y kFk. It can be seen that as p ! e so (x; y ) ! (0; 0) and thus l0 ! r, which is inconsistent with the principles
If
2 2

of epipolar geometry. In fact, the closer p gets to the epipole, the greater will be the error in its

corresponding epipolar line. Thus, if the epipole lies within the bounds of the image, then the epipolar geometry encompassed in a fundamental matrix obtained using the linear criterion will be inaccurate. Fortunately, this problem is not insurmountable. The solution is to correct the estimated fundamental matrix to ensure it has the required rank. To do this we employ a technique developed by Tsai and Huang [65], whereby

^ F is replaced by the matrix F that minimises:

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

33

^ kF ? Fk

subject to

^ det(F) = 0

(3.18)

F = UDVT , where ^ D is a diagonal matrix D =diag(r; s; t) such that r s t. Setting F = Udiag(r; s; 0)VT has ^ been shown to give F as the closest singular matrix to F under Frobenius norm.
This is achieved by ?rst computing the singular value decomposition,

3.5.4

Normalisation

Here we give a brief description of a recently developed technique [26] for improving the accuracy of the fundamental matrix computed using the 8-point algorithm. This approach is based on numerical analysis of the EIG method of solving the linear criterion equations. It is theorised that the poor performance of the 8-point algorithm can be attributed to methods of implementation that do not take suf?cient account of the conditioning of the set of equations being solved. The condition number, of a matrix

M is given by
= kMk:kM?1k
(3.19) is large, a small change in as small

It plays an important role in the analysis of linear problems. When

the data can lead to large variations in the computed solution. Thus, we aim to make

as possible, improving the conditioning of the system of equations and leading to a more stable and accurate solution. The EIG method of solving of

ATA. As such, the result of Hartley’s analysis was the development of a normalising transform to reduce the condition number of the AT A matrix, allowing us to obtain a better estimate for F.
However, as we will show, the bene?cial effect of normalisation is not just limited to the EIG implementation, it also improves the results of the SVD method. The normalisation process is straightforward. Prior to generating the linear constraint equations, the coordinates of the matched image points undergo a combined translation and scale transformation such that the centroid of the points is at the origin and the average distance of a point

Af = 0 requires the computation of the unit norm eigenvector

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

34

from the origin is

p

2. 101; 102; 103. Translating by 100 results in 1; 2; 3. Thus in the

The effect of the translation is intuitively obvious. Consider a set of points in a 200 200 image, whose u-coordinates are

untranslated coordinates, the important values are not found until the third signi?cant ?gure, being obscured by the offset of 100. This has a detrimental effect on the conditioning of

ATA.

However, this problem can be solved by a simple translation which promotes the coordinates’ signi?cant ?gures. The theory behind the scaling effect is quite complex and we will not discuss it in any great depth here. In essence, a lower bound for the condition number of the

ATA is derived, based on

the interlacing property3 for the eigenvalues of a symmetric matrix and their relationship with the values of the diagonal elements of

ATA. Clearly, the magnitudes of these diagonal elements ATA.

are themselves related to the coef?cients in equation 3.12, which, in turn, are determined by the matched point coordinates. Thus, it can be shown that scaling so that the average homogeneous point coordinate is unity will improve the conditioning of

Note that the translation and scale transforms are computed separately for each image. Each

T and T0, for the ?rst and second image respectively. Thus two matched image coordinates p and p0 are replaced by
pair of transforms is then combined to give a single transformation matrix their normalised versions as follows:

p = Tp ^
thus

and

^ p0 = T 0 p0 ^ p0TT0?T = p0T

(3.20)

T?1 p = p ^
which, by the linear criterion (3.8), gives:

and

(3.21)

r principal sub-matrix of an n n symmetric matrix A, and i (A) represents the i-th largest eigenvalue of A, then for r = 1; 2; : : : ; n ? 1 the following interlacing property holds [18]:
r+1 (Ar+1 ) r (Ar ) r (Ar+1 )

3 If Ar denotes the leading r

^ p0TT0?T FT?1 p = 0 ^
:::

(3.22)

2 (Ar+1 )

1 (Ar )

1 (Ar+1 )

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

35

This implies that , the fundamental matrix for the normalised point correspondences is:

^ F

^ F = T0?T FT?1
and we can recover

(3.23)

F by:

^ F = T0TFT

(3.24)

Thus we can outline the normalised 8-point algorithm as follows: 1. Given a set of pairs of matched image points pi

$ p0i, apply the normalising transforma^ ^ tions T and T0 to obtain pi = Tpi and p0i = T0 p0 . i

2. Compute the fundamental matrix 3. Recover the fundamental matrix

^ F corresponding to these normalised points.

^ F corresponding to the ‘un-normalised’ points as F = T0T FT.

In the sequel we will refer to the normalised implementations as SVDNORM and EIGNORM.

3.5.5

Quantifying Success

Our overall aim is to obtain the best possible estimate for the fundamental matrix, based on the linear criterion, for later use in our structure recovery sy stem. In doing so, we can evaluate which combination of normalisation and method of solution (SVD/EIG) provides the best results. The question is, how to quantify ‘best’? The measure we use is based upon the relationship between matched points and their epipo-

l = Fp, we know that p0 should lie somewhere on l0. In fact, this is only exactly true when the computed F is perfect and in practice the point will actually lie some distance from its epipolar line. We can use this as a measure of the quality of F. Formally, we
lar lines. Speci?cally, given 0 tance of each point from its corresponding epipolar line. Thus we would like QF to be as low as possible. A method for computing the perpendicular distance of a homogeneous point from a homogeneous line is given in Appendix A. de?ne the quality of a fundamental matrix, denoted by QF , to be the average perpendicular dis-

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

36

The aim of the normalisation process is to obtain a better estimate for

F by improving the

^ ^f A in A^ = ?u0 (section 3.5.1). For the EIG method of solution, we are interested in the condition number of matrix AT A in Af = 0 (section 3.5.2).
SVD method we keep track of the condition number, , of the matrix

conditioning of a linear system. In this respect, its success or failure is easy to quantify. For the

3.6 Results and Conclusions
3.6.1 Initial Experiments

In this section we describe the results of our experiments to determine the effects of the normalising transform on our two implementations of the 8-point algorithm. Experiments were performed on a variety of synthetic data and real images, using manual and automatic feature detection/matching methods. Here we give a representative sample of the results obtained. In order to be able to make a valid comparison, our experiments follow along similar lines to those carried out in Hartley’s original paper on normalisation [26], in that we show the effect of varying the number of matched points used in the computation of the fundamental matrix. Thus, given a dataset containing M matched points, we begin by selecting a random 8-point subset,

S , which is used to obtain an estimate for F. The condition number, , of the solved system of equations is recorded, as is the quality measure, QF . This process is repeated for a number of

trials, each time selecting a different random subset of matched points. Once all these trials are completed, we then do exactly the same thing using 9-point subsets, then 10-point subsets, etc., until we have computed

F using all subset sizes S 2 8 : : :M ].
and QF change as S is varied. Here we

For each dataset we plot graphs which show how

use the median values obtained over all the trials for each size subset. Note that, regardless of the subset size, all M matches are used in the determination of QF . In addition, we vary the number of trials used per subset as a fraction of the total number of S -point subsets in M . A pseudo-code overview of the experiments is as follows:

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

37

Given an M-point dataset:

for subset size S = 8 to M { for some variable number of trials T { randomly select an S-point subset from M compute F using this subset and each of the 4 methods enforce the rank constraint record the kappa and Qf values for this particular subset } record the median and best kappa/Qf values for this size subset }

3.6.2
1e+30

Synthetic Data
1e+06 EIG EIGNORM 1e+05 SVD SVDNORM

Median Condition Number K

1e+20

Median Condition Number K 0 10 20 30 40 50 Number of matched points used to compute F 60

1e+04

1e+03

1e+10

1e+02

1

1e+01 0 10 20 30 40 50 Number of matched points used to compute F 60

(a) Figure 3.3: Graphs of S against
3 EIG EIGNORM 2.5 2.5

(b) using perfect synthetic data
3 SVD SVDNORM

Median Qf (pixels)

1.5

Median Qf (pixels)

2

2

1.5

1

1

0.5

0.5

0 0 10 20 30 40 50 Number of matched points used to compute F 60

0 0 10 20 30 40 50 Number of matched points used to compute F 60

(a)

(b)

Figure 3.4: Graphs of S against QF using perfect synthetic data

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

38

We begin by applying the algorithms to a perfect dataset. The idea was to obtain a frame of reference for future experiments and to see whether any differences could be detected when the algorithms were used under ideal conditions. We simulated a scene consisting of 60 randomly generated points within a constrained 3D volume. Two arbitrarily placed cameras project the 3D scene points into the perfect 2D matched data. Figures 3.3 and 3.4 show how varying the number of points used in the fundamental matrix calculation affects QF and . Looking at the graphs of , two things are immediately obvious. Firstly, it is clear that normalisation has resulted in a dramatic reduction in the condition number for both the SVDNORM and EIGNORM methods, the improvement being of the order of 106 and 103 respectively. Secondly, it would seem that as the subset size increases, so is reduced. It is also particularly no-

ticeable that the EIG implementation, in both its basic and normalised forms, is extremely illconditioned when 8-point subsets are used to compute . As one would expect for this kind of dataset, the values of QF are very low, on the whole to sub-pixel accuracy. However, it does look as though the EIG method is producing marginally more accurate results. One thing that is also worth mentioning at this point is that for any given 8-point subset, both methods determine an exact solution for the elements of the fundamental matrix. Thus they generate identical values for QF .

F

F matrices that are the same, to within a scale factor, and hence, give

3.6.3

Real Data: Corridor Images
4

The two images used here are part of a larger, widely-used sequence of a corridor. detection and matching was done by hand, to produce 40 matched point pairs.

Feature

Figure 3.5 shows that, as with perfect data, normalisation leads to a reduction in the condition number of the system, for both methods of solution, but, again, we note the high values for EIG and EIGNORM when using exactly eight matched points. However, overall the improvements were of the order of 108 for EIGNORM and 103 for SVDNORM. The impact this has on QF is easier to see than with the previous dataset (?gure 3.6). Both

4 This sequence was provided by the Robotics Research Group at the University of Oxford.

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

39

1e+30 Standard EIG Normalised EIG

1e+07 Standard SVD Normalised SVD 1e+06

Median Condition Number K

1e+20

Median Condition Number K 5 10 15 20 25 30 35 Number of matched points used to compute F 40

1e+05

1e+04

1e+10

1e+03

1

1e+02 5 10 15 20 25 30 35 Number of matched points used to compute F 40

(a) Figure 3.5: Graphs of S against

(b) for the corridor image pair

4 Standard EIG Normalised EIG 3.5

4 Standard SVD Normalised SVD 3.5

3 Median Qf (pixels) Median Qf (pixels)

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5 5 10 15 20 25 30 35 Number of matched points used to compute F 40

0.5 5 10 15 20 25 30 35 Number of matched points used to compute F 40

(a)

(b)

Figure 3.6: Graphs of S against QF for the corridor image pair

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

40

normalised methods result in a small, but consistent, decrease in perpendicular pixel error, relative to their basic counterparts. The sub-pixel accuracy can be attributed to the fact that this manually generated dataset contains only small errors in localisation and no false matches. Overall, EIGNORM is the best performer, but there is very little to choose between any of the four methods. This is emphasised by ?gure 3.7, which shows the image pair overlaid by epipolar lines generated by the best5 fundamental matrices, obtained using each of the four methods. As can be seen, the locations of the epipoles and lowest QF values are very similar.

3.6.4

Real Data: House Images

These images of a toy house were obtained from the VASC image database6. They are part of a large image sequence (over 180 images) which has been used extensively for machine vision research, particularly structure from motion, for example [48]. The images were processed using the INRIA epipolar geometry server7 , which in this case found a set of 116, rather noisy, point matches (see ?gure 3.8). Figure 3.9 shows that, once again, the normalisation process has the desired effect of improving the conditioning of the systems, a reduction in of the order of 1010 for EIGNORM and

106 for SVDNORM. As before, this leads to a decrease in the median values of QF , as shown in
?gure 3.10. For this noisy dataset, the bene?cial effect is more dramatic than we have seen previ-

noticeable that, for the basic EIG and SVD methods, QF begins to increase as the subset size ap-

ously, reducing QF by more than 3 pixels, when the computation involves large subsets. It is also

proaches the total number of matched points. This is another feature of our noisy dataset. When large numbers of points are used, there is a greater likelihood that the subset will include false point matches, thus degrading the least-squares solution. However, the monotonically decreas5 The fundamental matrix with the lowest QF over all trials and all subset sizes. 6 Available on the World Wide Web at http://www.ius.cs.cmu.edu/IUS/ppt usr0/yx/idbm/image html 7 This enables users to have a pair of their own images processed remotely on the INRIA computer, via the World
Wide Web. Matched point correspondences are obtained [71] and fed into an algorithm which computes the epipolar geometry [8]. All resulting information is available, for example detected/matched features, the fundamental matrix, images overlaid with matches, epipolar lines etc. This provided an alternative method of obtaining point matches and a useful check on the accuracy of our own results. The URL is: http://www.inria.fr/robotvis/demo/f-http/html/

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

41

(a) EIG QF

= 0:54

(b) EIGNORM QF

= 0:54

(c) SVD QF

= 0:56

(d) SVDNORM QF Figure 3.7: Sample epipolar lines using the four best

= 0:55

F matrices from the corridor image pair.

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

42

Figure 3.8: Results of the INRIA matching algorithm for the house image pair.
1e+30 EIG EIGNORM 1e+07 Median Condition Number K Median Condition Number K 1e+08 SVD SVDNORM

1e+06

1e+20

1e+05

1e+04

1e+10

1e+03

1e+02

1 0 20 40 60 80 100 Number of matched points used to compute F 120

1e+01 0 20 40 60 80 100 Number of matched points used to compute F 120

(a) Figure 3.9: Graphs of S against
11 10 9 Median Qf (pixels) Median Qf (pixels) 8 7 6 5 4 3 2 0 20 40 60 80 100 Number of matched points used to compute F 120 EIG EIGNORM

(b) for the house image pair
11 10 9 8 7 6 5 4 3 2 0 20 40 60 80 100 Number of matched points used to compute F 120 SVD SVDNORM

(a)

(b)

Figure 3.10: Graphs of S against QF for the house image pair

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

43

ing curves for the EIGNORM and SVDNORM methods suggest that the normalisation process goes some way towards counteracting this effect. The two normalised methods are clearly the best performers on this dataset, with EIGNORM doing slightly better than SVDNORM. Although the basic methods give relatively poor results for median QF values, ?gure 3.11 shows that the best fundamental matrices obtained over all their trials are very similar to those of their normalised counterparts.

3.7 Initial Conclusions
In every experiment we performed, the normalisation process had the desired effect of improving the conditioning of the system of linear equations. For the least eigenvector method, the reduction in was of the same order of magnitude as reported by Hartley, at approximately 108. The theory behind normalisation is speci?cally linked to this method of solution and one of the things we were interested in was whether it could also be used to enhance the conditioning of the SVD method. Indications are that this is the case, with our results showing a reduction of about 104. The magnitude of the improvement is clearly dependent on the quality of the dataset, the most bene?cial effects occurring with noisy data, containing may false matches and localisation errors. The same applies with respect to the perpendicular pixel error; normalisation gave the biggest decreases in

QF on the noisiest datasets.
In every one of our EIGNORM experiments the enhanced conditioning resulted in lower values for QF . In the vast majority of cases, SVDNORM gave only slightly poorer results, but occasionally it was unstable (see ?gure 3.12). Here, in spite of the lower condition number, SVDNORM actually gives worse results for

QF than basic SVD. While the SVD method itself gives quite erratic values, it at least shows the
familiar decreasing curve as the subset size increases, which is not the case for SVDNORM. One possible cause of these anomalies could be the composition of the dataset, which in this case contained a large proportion of projections of planar 3D points. Hence, many of the random matched point subsets would have contained degenerate con?gurations [62]. We have not yet carried out a more in depth investigation of this phenomenon, since we are primarily interested in a high-level

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

44

(a) EIG QF

= 1:83

(b) EIGNORM QF

= 1:81

(c) SVD QF

= 1:82

(d) SVDNORM QF

= 1:82

Figure 3.11: Sample epipolar lines using the four best

F matrices from the house image pair.

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

45

4 3.5 3 Median Qf (pixels) 2.5 2 1.5 1 0.5 0 5 10 15 20 25 30 35 Number of matched points used to compute F 40 45 SVD SVDNORM EIG EIGNORM Median Condition Number K

1e+30 SVD SVDNORM EIG EIGNORM

1e+20

1e+10

1 5 10 15 20 25 30 35 Number of matched points used to compute F 40 45

Figure 3.12: Instability of the SVDNORM method comparison of the methods of solution. Here we merely note its, albeit infrequent, occurrence and the fact that the EIG and EIGNORM methods seem unaffected. In short, for this series of tests, the EIGNORM method produced consistently lower perpendicular pixel errors and better estimates for the coordinates of the epipoles. The algorithm copes very well with noisy data, and its performance degrades gracefully as the level of noise is increased. Therefore, we have concluded that of the various implementations tested, the normalised eigensystem method is the clear winner, and from now on this is the method of implementation we have used.

3.7.1

Further Experiments

The experiments we have performed so far have con?rmed Hartley’s ?ndings, that the normalisation process does lead to more accurate estimation of the fundamental matrix via the 8-point algorithm. However, the way these experiments were carried out is not a very practicable solution to the problem. Rather than doing a laborious series of trials for different sized subsets we would prefer a more ef?cient way of ?nding the best fundamental matrix for a given dataset. Looking at the graphs of subset size against perpendicular pixel error that we have presented, one might conclude that the best way to calculate it must be remembered that these graphs plot the median values of QF . Consider instead, ?gure 3.13. This is an example plot of the lowest values of

F would be to use the largest possible subset size. However,
QF
for each subset size, obtained during

the experiment on the house images described previously, but it is typical of the results for other

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

46

datasets.
2.05 EIGNORM 2

Lowest Qf (pixels)

1.95

1.9

1.85

1.8 0 20 40 60 80 100 Number of matched points used to compute F 120

Figure 3.13: Example Lowest QF Values. Looked at in isolation, without the swamping effect of being compared with the high error, basic method, it is clear that the EIGNORM performance also degrades as the subset size approaches the total number of matches. As has already been mentioned, this is due to the fact that large subsets are more likely to include outliers in the data; false matches which taint the least-

F causes other problems. Clearly, as the size of our subset approaches that of the complete matched point set (S ! M ), the number of
squares result. The use of large subsets when computing permutations of points in that subset increases exponentially, as does the number of trials required to ?nd the best S -point subset. In addition larger subsets incur a computational overhead, because they lead to bigger systems of equations which take longer to solve. However, it is equally apparent from this, and many other experiments on real data, that the answer is not to do the computation using the smallest possible (8-point) subset. It may well be the case that a fundamental matrix estimated from an 8-point subset is very accurate in terms of those 8 points, but is not so good when applied to all the matched points. It seems that the optimum subset size i.e. the one that gives the lowest QF , can be anywhere except at the extremes of the range S

2 8 : : :M ]. The question is, how to ?nd it? A brute force and ignorance solution, which we will refer to as BEST-F, is simply to choose the best F matrix over all trials and subset sizes.

However, we would prefer a more intelligent and ef?cient approach.

CHAPTER 3. ACQUIRING THE FUNDAMENTAL MATRIX

47

The solution we employ is a variant of the RANSAC parameter estimation paradigm, which we denote by RANSAC .8 As before a series of trials are performed using randomly selected subsets of the total number of matched points, but here 8-point subsets are chosen. At each trial, we take the computed some threshold value. These matches form the consensus set. The ( set are then used to compute another estimate for

F and ?nd all the point matches whose perpendicular pixel error is beneath F for which we calculate QF , in the usual man-

8) points in the consensus

ner. The old consensus set is replaced if the new consensus set is of equal or larger size and has a lower QF . The process terminates either once all trials have been completed or the consensus set size or QF reach a speci?ed target. In detail, the RANSAC version of our experiments is as follows: Given a dataset, M , of point matches, then for some ?xed number of trials: 1. Randomly select an 8-point subset from M . 2. Compute

F for this subset using EIGNORM.

3. Enforce the rank constraint. 4. Determine the consensus set C of matches whose perpendicular pixel error is within some threshold. 5. Compute a new 0 based on C . 6. Calculate QF for 0. 7. Replace the old consensus set if C is the same size or bigger, and has lower QF value. 8. Terminate if the size of C or QF reach speci?ed targets. At the end of this procedure, we hope to have obtained a fundamental matrix, based on many of the matched points, with a low QF value, and to have done so in a relatively small number of trials. 8 For a full description, refer to appendix B.

F

F

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3.7.2

Further Results

There is little point in attempting to quantify the increase in speed of RANSAC relative to the BEST-F approach, since, by altering the number of trials per subset size in the latter case, we could come up with whatever ?gures we liked. Instead, we note that for a given dataset, however many trials per subset size are needed in order to obtain a good estimate for F using BEST-F, invariably, RANSAC requires fewer trials to give a fundamental matrix which is almost as accurate. Some example results on different datasets9 are given in table 3.1. Dataset Matched Points Perfect Corridor Quadrangle House 60 40 43 116 BEST-F RANSAC Trials 4413 23674 32842 42839

QF
0.30 0.54 0.43 1.81

Subset Size 52 31 23 50

QF
0.31 0.62 0.55 1.90

Consensus Size 45 23 32 43

Trials 2340 6166 4570 5666 algorithms

Table 3.1: Comparison of results of obtaining

F using the BEST-F and RANSAC

The RANSAC method results in slightly inferior QF values than can be obtained using BEST-F. However, this small decrease in accuracy is outweighed by big improvements in ef?ciency. To put the performance of RANSAC into perspective, table 3.2 shows how it compares against some of the more complicated alternative methods for estimating the fundamental matrix. NONLIN is a non-linear least-squares method, LMEDS is the least median of squares and M-EST uses the M-estimators technique, as discussed in Chapter 2. Clearly, there is very little to choose between the various implementations. With perfect data, as one would expect, they all give virtually the same results, especially when applied to our noisiest dataset, the house images. There is more of a differential when using hand-matched data, containing only small localisation errors, but we are still only talking in terms of a few hundredths of a pixel accuracy. The performance of RANSAC is even more impressive with our automatically processed house dataset, which 9 We have already seen the perfect dataset and the corridor and house images. The quadrangle dataset is a pair
of hand-matched images from an outdoor image sequence of part of the Leeds University campus. The entire image sequence is shown in Appendix C.

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Dataset RANSAC Perfect Corridor Quadrangle House 0.31 0.62 0.55 1.90

QF
NONLIN 0.31 0.56 0.49 1.92 LMEDS 0.31 0.56 0.47 1.84 M-EST 0.31 0.55 0.45 1.90

Table 3.2: Comparing RANSAC with more complicated algorithms

includes many mismatches. Here it is only bettered by the least-median of squares algorithm.

3.8 Conclusions
In this chapter we have investigated Hartley’s normalisation technique for improving the performance of the 8-point algorithm. Although this technique is based on numerical analysis of one particular method of solution (EIG), we have shown that its bene?cial effects are quite general in that its application to another solution method (SVD) also leads to more accurate results. We have demonstrated that, when using the normalised 8-point algorithm (EIGNORM) to estimate the fundamental matrix, the size of the matched point subset used in the computation is of great importance. We can obtain an excellent solution simply by carrying out thousands of trials for each possible subset size and then picking the best solution overall, however, this is very time-consuming. To this end, we have embedded EIGNORM within a RANSAC-style procedure which attempts to obtain a quick, accurate estimate for the fundamental matrix, based on a medium-sized point subset (RANSAC ). We do not claim that this will produce an optimum solution, but have shown that results of our simple, linear method are similar to those obtained with other, more complex alternatives. In the next chapter we will use the fundamental matrices acquired using RANSAC , as the starting point for our 3D scene reconstruction.

Chapter 4

Reconstruction
4.1 Introduction
In his seminal paper [12] Faugeras showed that it is possible to obtain the projective structure of a scene, given just a set of matched points in a pair of images taken with uncalibrated cameras. This generated a great deal of interest in the development of algorithms for performing uncalibrated stereo, that is, recovering the three-dimensional structure of a scene, without explicit knowledge of the intrinsic or extrinsic camera parameters. Descriptions of some of these algorithms have been given in Chapter 2. In the absence of any other information or simplifying assumptions, the reconstruction will be correct up to a projective ambiguity i.e. it will differ from the real Euclidean structure by a projective transformation. Projective structure can be very useful in its own right, for example, objects may be recognised via projective invariants [56]. However, although geometrically related, it is unlikely that the projective and Euclidean structures will look anything like each other. If the intention is to use the recovered structure for graphical reconstruction and subsequent viewing by a person, this clearly poses a problem. A method of converting from a projective to Euclidean reconstruction was presented in [25]. This uses the constraint of unchanging intrinsic camera parameters, which is the basis of camera self-calibration theory[13], requiring at least three views of the scene, taken with the same 50

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camera. The method is highly complex and relies heavily on the Levenberg-Marquardt iterative parameter estimation algorithm [50]. Although it appears to give good results using synthetic data, the method’s performance on images of real scenes is less impressive. For this reason, we have adopted a simpler approach, using ground truth points. If the Euclidean positions of ?ve or more of the points in the projective structure are known, it is possible to compute the projectivity that maps between them. This can then be applied to all the projective points to achieve a full Euclidean reconstruction. In this chapter we describe our implementation of a Euclidean reconstruction method, based on the properties of the fundamental matrix. Rather than restricting ourselves to a single pair of images, we present a novel incremental version of the basic algorithm which allows us to recover the Euclidean structure of an extended environment viewed over a long image sequence.

4.2 Outline of the method
Before embarking on a detailed description of each of the stages in our implementation, we ?rst provide a brief overview of the entire system. A diagrammatic representation is given in ?gure 4.1. We begin by estimating the fundamental matrix for the ?rst pair of images in the sequence, using the algorithm described in the previous chapter. This is then factorised in a manner which allows us to construct estimates for the pair of camera matrices used in the formation of the two images. Next, we recover the projective structure of each of the matched points by computing the intersection of the two rays, projecting back from the two cameras through the corresponding image features. For the ?rst image pair only, we transform our projective structure into a Euclidean frame, using a projectivity derived from a small number of user-supplied ground truth points. With subsequent images, we proceed in a similar manner. A new fundamental matrix is estimated, using matched points in the current and previous images. As before, this is factorised to obtain a pair of camera matrices and hence, another batch of projective structure. Once again, we upgrade the projective structure using a projective transform, but here, the ground truth information is not

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Real World Scene

Camera

Image Pairs

Feature Detection & Matching

Point Correspondences

Camera Matrix Construction M and M’ Projective Reconstruction

R and [e] x

Factorisation

F

Fundamental Matrix Calculation

P P

Structure Upgrade Transformation

P E

Euclidean Structure

Figure 4.1: Overview of the system supplied directly by the user. Instead, we use common points from our previously computed Euclidean structure. The entire process is repeated, generating new Euclidean structure as we step through each pair of images in the sequence. Note that, as the sequence is processed, old points will disappear and new ones come into view: there is no requirement for points to be visible in all the images.

4.3 Image Sequence Processing
Before embarking on a description of the reconstruction algorithm we ?rst consider some of the inherent problems of image sequence processing and the effects these have on the overall design of our system. Any system which involves the processing of long image sequences must address a number of issues. First of all, unless the system is capable of processing live video directly from a camera, or some other input source such as a VCR, then the sequence must be digitised onto disk, and thus

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we need to look at the space requirements. Take, for example, our quadrangle sequence. In this case, six and a half minutes of original video footage, digitised at full frame rate (25 frames per second), took up over a gigabyte of disk space! We simply did not have enough resources to keep this amount of data on our system for any length of time, so had to ?nd a way to reduce the size. One obvious solution was simply to reduce the digitisingframe rate. When attempting to do so, it was important to remember that each pair of images in the resulting sequence must contain enough overlap to allow us to obtain a set of matched features, for use in our fundamental matrix algorithm. Now, our footage was ?lmed by walking around with an ordinary palm-corder and the route we followed involved a number of twists and turns, some of them quite sharp. As such, it was extremely dif?cult to keep the camera moving at a steady pace and so some parts of the scene passed in and out of view much more quickly than others. Where this was so, it was essential to digitise at full frame rate, to ensure enough inter-frame overlap, but elsewhere this resulted in lots of almost identical, redundant frames. Ideally, we would have liked to have been able to vary the digitisation rate, based on the speed of camera motion. Without this facility, the best solution we could manage was to digitise at full frame rate and then manually extract representative frames, discarding the rest. This process reduced over 10000 digitised frames to 104, resulting in the image sequence shown in Appendix C, which only takes up 10 megabytes of disk space. Unfortunately, this space saving comes at a cost. Suppose we have processed a pair of images from our sequence, with a corner detector, for example [22, 57, 67]. The next task is to solve the correspondence problem i.e. ?nd the set of matching corners between the two images. The fact that there is such a relatively large baseline between our images, causes problems for local strategies, such as nearest-neighbour, which search for a match in the area of the second image corresponding to the location of the other corner in the ?rst. For example, consider ?gure 4.2. This shows the result of applying a simple local matching algorithm, based on cross-correlation, to a set of corner features, found via the Plessey detector [22], in a pair of the corridor sequence images. The usual match search area parameter of about ten pixels is of no use for this image pair, as the disparity is much greater than that. However, it can be seen that using a larger value is not the answer. The image on the right is overlaid with matched corners, in red, and yellow lines

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Figure 4.2: Poor performance of a local matching algorithm. indicating the corner trajectories. Clearly the matching process has not been very successful. In order to have a chance of ?nding the correct match, the search area must be increased. However, this not only increases the computational cost of the process, it also makes it more prone to false matches. This is particularly so for scenes containing repeated structure (windows etc), where it is easy to match one instance incorrectly against another. The effect of false matches can be seen in Figure 4.3. Figure 4.3(a) shows the results of processing the ?rst pair of quadrangle images with the INRIA point detection and matching algorithm [71]. Evidently, a signi?cant number of false matches have been detected. The fundamental matrix estimated from these noisy matches does not encapsulate the correct epipolar geometry for this pair of images. This can be seen by comparing the epipolar lines in ?gure 4.3(a), with those in 4.3(c), which were generated by a fundamental matrix computed from hand-matched data. We have already mentioned one possible solution in the previous chapter. The fundamental matrix constrains the location of a matched point to lie on a given epipolar line, which can reduce the search space dramatically. Unfortunately, this is like the ‘chicken and the egg’ situation, in that we need to ?nd at least eight point matches in order to compute the fundamental matrix! However, if an initial set of matches is obtained, by some other means, the fundamental matrix approach can then be used to discard false matches in the set, as described in [71]. A variation on this theme is

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(a) Noisy matched data.

(b) Epipolar lines from noisy matches.

(c) Epipolar lines from hand-matched data.. Figure 4.3: Effects of false matches on epipolar geometry

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discussed in [4], which matches over three images using the trifocal tensor. It was not the objective of this work to investigate automatic feature matching, which is a research topic in itself. For this reason, a simple, manual feature detection and matching system was developed, using an X11-based graphical user interface (see ?gure 4.4). The user can process each image of the sequence in turn, matching existing features or specifying new ones, by clicking the mouse button at the desired image location. Matched features are highlighted in red, unmatched features are either old (yellow) or new (orange).

4.4 Constructing Camera Matrices
In the previous chapter a method was described for estimating the fundamental matrix, , from a pair of views of a scene, taken with an uncalibrated camera. Acquiring

F is the key to recovering

F

the projective structure of the scene, since it encapsulates all the geometric information relating the scene and the cameras that can be extracted from a set of matched image features. In particular, knowing

F allows us to construct camera matrices for our two cameras.

The reason for this is directly related to the derivation of the linear criterion in section 3.3. There it was shown that, by choosing

As before, it is possible to determine the epipolar line corresponding to a point (u; v; w)T in the ?rst image. Two points which must lie on this line are the images under optical centre and the point at in?nity, given by

C = (Ij0) and C0 = (Rj ? Rt) as our partitioned camera matrices, the fundamental matrix could be factorised as F = RS, where S = t] . Now consider the more general case, where the two camera matrices are C = (Rj? Rt) and C0 = (R0j? R0 t0 ). C0 of the ?rst camera’s

0 B @
These project to

1

1 tC A

and
1

1 0 ? (u; v; w) C BR A @
1

0

(4.1) The epipolar line, 0

(q; r; s), through these points is given by the cross product

R0(t ? t0) and R0R? (u; v; w) in the second image.
(q; r; s) = R0(t ? t0 ) R0 R?1 (u; v; w)

l =

(4.2)

Figure 4.4: Our manual feature detection and matching user interface.

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Which, by the properties of cofactor matrices, can be rewritten

(q; r; s) = R0 ((t ? t0 ) R?1 (u; v; w))
and

(4.3)

F and thus we obtain an expression for F in terms of the components of our camera matrices.

(q; r; s) = R0 t ?{z0 ] R?1 (u; v; w) t | }

(4.4)

F = R0 t ? t0] R?

1

(4.5)

It has been shown [23] that this relationship does not determine the two camera matrices

C1 and C01 are two camera transforms satisfying equation 4.5, then so are C2 = HC1 and C0 = HC0 , where the matrix H is a 4 4 projectivity. Consequently, 2 1
uniquely. In particular, if the 3D structure of a scene derived from any such pair of camera matrices is de?ned only up to an arbitrary projective transformation. The task now is to construct a pair of camera matrices for which equation 4.5 holds. A number of different ways of doing this have been proposed, for example [23, 24, 4, 53]. A hybrid of some of these methods is described, which we believe leads to a more intuitive solution.

4.5 Factorising F
We follow convention and choose the origin of the world coordinate system as the optical centre of the ?rst camera, its axes aligned with the camera axes. The second camera is displaced from the ?rst by some translation and rotation, giving the two familiar camera matrices,

C0 = (Rj ? R0t0). Substituting into equation 4.5 gives: F = R0 ?t0]
The problem now is to ?nd a way to factorise

C = (Ij0) and
(4.6)

F in the above form. F = UDVT , where U and
Note that one of the singular
1

The ?rst step is to compute the singular value decomposition

VT are orthogonal matrices and D is the diagonal matrix (r; s; 0).

1 Some swapping of matrix rows and columns may be necessary so that r > s > 0.

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values will always be zero, due to the enforcement of the rank constraint, discussed in section 3.5.3. De?ning the two matrices:

1 0 0:0 ?1:0 0:0 C B C B E = B 1:0 0:0 0:0 C C B A @
0:0 0:0 1:0

and

1 0 0:0 ?1:0 0:0 C B C B Z = B 1:0 0:0 0:0 C C B A @
0:0 0:0 0:0

(4.7)

The fundamental matrix can now be factorised as:

F = M e]
where

(4.8)

M = Udiag(r; s; )EVT
with any non-zero number, best chosen to lie between r and s, to make as possible2, and

(4.9)

M as well-conditioned
(4.10)

e] = VZVT
where

e is the coordinate vector for the epipole in the ?rst image.
This works because the process of singular value decomposition explicitly constructs or-

thonormal bases for the nullspace and range of a matrix. Speci?cally, the rows of

ViT for i = 1; 2; 3) whose corresponding singular values are zero are an orthonormal basis for the nullspace. Given the ordering of the singular values and the fact that Fe = 0, it is clear that VT VT = VT = e, which accounts for e] = VZVT . Furthermore, since F = M e] , it follows that eT MT F eT MT M e] = eT e] = 0. Hence, FT (Me) = 0 and so Me e0.
1 2 3

VT (denoted

The end result of this factorisation, is the following pair of camera matrices

C = (Ij0)

and

C0 = (Mjp0)

(4.11)

Once again, it must be stressed that these are in no way intended to be the true camera matrices, but they are related to them by a projective transformation, which is enough to allow us to recover projective structure. One ?nal point of note with which to end this section: the matrix

M

is the epipolar trans-

form discussed in section 3.2. It provides the mapping between epipolar lines in the two images. 2 We set = (r + s)=2.

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l = Fp. The epipolar line through p in the ?rst image is l = p e = e] p. Hence, M l = M e] p = Fp = l0 , as required. Thus the matrix M maps epipoles to epipoles and the matrix M maps epipolar lines to epipolar lines.
second image, hence 0

To see this, let p be a point in the ?rst image and let l0 be the corresponding epipolar line in the

4.6 Projective Reconstruction
Suppose

C and C0 are camera matrices, consistent with the fundamental matrix obtained from a p = (u; v; 1) , p0 = (u0; v0; 1). The point PP = (X; Y; Z; T ) in P

set of point matches in a pair of images. Armed with this information, it is a relatively straightforward process to recover the corresponding projective structure of the scene. Consider one such point match
3

that projects

onto p and p0 is located at the intersection of the two rays which originate from the optical centres of the cameras and pass through the matched points. This places constraints on standard perspective projection equations:

PP , based on the

0 1 0 B uC BC B C=B B vC BC B C B @ A @
and

11

21

C31

C12 C13 14 C22 C23 24 C32 C33 C34

0 1B X C CB CB Y C CB CB Z AB B @
T

1 C C C C C C C A

(4.12)

0 1 1B X C 1 0 0 B u0 C B C 0 C 0 C 0 C 0 C B Y C B 0 C B 0 0 0 0 CB C B v C = B C C C C CB C B C C B B AB C A @ 0 0 0 0 CB Z C @ @ A C C C C
11 12 13 14 21 22 23 24 31 32 33 34

(4.13)

T

Multiplying out we get

u = C11X + C12Y + C13Z + C14T v = C21X + C22Y + C23Z + C24T = C31X + C32Y + C33Z + C34T

(4.14)

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0 0 0 0 u0 = C11X + C12Y + C13Z + C14T 0 0 0 0 (4.15) v 0 = C21X + C22Y + C23Z + C24T 0 0 0 0 = C31X + C32Y + C33Z + C34T All the various camera matrix elements are known, as are u; v and u0; v 0, so, cancelling out the
scale factors, and , then rearranging, gives a set of four linear equations in the four unknowns

and

X; Y , Z and T . 0 B C11 ? uC31 B B C21 ? vC31 B B 0 0 0 B C11 ? u C31 B @ 0 0 C21 ? v 0C31

With perfect data, the solution to this homogeneous system of equations is PP , the point of inter-

C12 ? uC32 C22 ? vC32 0 0 C12 ? u0C32 0 0 C22 ? v 0C32

C13 ? uC33 C23 ? vC33 0 0 C13 ? u0C33 0 0 C23 ? v 0 C33

C14 ? uC34 C24 ? vC34 0 0 C14 ? u0C34 0 0 C24 ? v 0C34

10 CB X CB CB Y CB CB CB Z CB A@
T

1 0 1 C B0C C B C C B0C C=B C C B C C B0C C B C A @ A
0

(4.16)

section of the two rays. In practice, measurement errors mean the rays will not intersect and we solve for the point of their closest approach. We generate and solve such a system for each pair of matched points to build a complete projective reconstruction. A variation on the above is to constrain the form of the points in P 3 to

PP = (X; Y; Z; 1).

This leads to a system of four linear equations in three unknowns, which can be solved by standard least-squares techniques. However, it has been pointed out [52] that this formulation makes the, possibly invalid, assumption that the projective points do not lie on the ideal plane. As such, some choices of camera matrices may give poor results. The two methods produce identical projective structure estimates, so although none of our experiments have reproduced the aforementioned problem, we will stay with the original formulation. In fact, our efforts to compare the two variations raised an interesting question: how to check the correctness of the resulting structure? A visual inspection of projective structure is of little use, a point made clear in ?gure 4.5. This shows the ?rst pair of images from our quadrangle sequence, along with two views of the projective structure recovered by our system. It is quite obvious that the projective structure bears no resemblance whatsoever to the real scene. In general, this will be the case for any projective reconstruction. In the absence of any other information there is no way to quantify the accuracy of the projective structure. The best we can hope for is to show that it is consistent, i.e. that it is actually a

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Figure 4.5: The ?rst pair of images in the quadrangle sequence and two views of their projective reconstruction.

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correct solution to the reconstruction problem. The simplest way to do this is to compute residuals for each system of linear equations. In other words, use the two camera matrices to reproject the 3D points and compare their reprojected image coordinates with those of the original matched features. We compute the ratio of the ratios of the original and reprojected x and y coordinates for each point pair. Ideally these values should all be 1, which provides an easy way to tell, at a glance, if the projective structure is correct. Also measured is the overall average distance, in pixels, between the original and reprojected image points, which one would hope to be close to zero. Image 1 Feature Label 0 8 16 24 34 60 76 85 95 109 . . . Original Reprojected Original Ratio 0.998 1.002 0.998 1.000 1.000 0.999 1.018 1.004 1.025 0.997 . . . Image 2 Reprojected

x
159 218 32 100 77 255 142 70 46 17 . . .

y
236 167 135 226 139 60 61 78 5 74 . . .

x0
159.0 218.0 32.0 100.0 77.0 255.0 142.1 70.0 46.0 17.0 . . .

y0
235.5 167.4 134.8 226.1 139.0 59.9 62.1 78.3 5.1 73.7 . . .

x
246 307 108 174 132 343 222 146 117 90 . . .

y
256 184 155 247 159 71 77 96 22 93 . . .

x0
246.0 307.0 108.0 174.0 132.0 343.0 221.0 145.0 116.0 90.0 . . .

y0
256.0 184.0 155.0 247.0 159.0 71.0 77.0 96.0 22.0 93.0 . . .

Ratio 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 . . .

Average Absolute Image Coordinate Differences 0.02 0.37 0.00 0.00

Table 4.1: Results of reprojecting projective structure

Table 4.1 shows some example results of these computations for the image pair and projective structure of ?gure 4.5, using a representative sample of matched points having a range of coordinate values. The original, manually-detected, image points are shown to the nearest pixel, while the reprojected coordinates are rounded to the nearest tenth of a pixel. The reason for this

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sub-pixel accuracy is to illustrate that the original and reprojected coordinates are not necessarily identical. The ratio of ratios of the coordinates is given to three decimal places. Again, this is primarily to indicate that the ratio is not always exactly 1, but it also serves to show that the ratio of ratios is furthest from 1 when the x-coordinate is signi?cantly larger than the y -coordinate, as with feature 95. As can be seen, in this case all the coordinate ratios are equal, or very close, to 1, and the overall average coordinate differences are extremely small. Thus we can conclude that, regardless of what it looks like, the projective structure recovered by our system is indeed consistent with the given point matches and fundamental matrix. It is interesting to note the relative inaccuracy of the Image 1 reprojections. A possible explanation for this is that while the Image 2 camera was explicitly computed, the Image 1 camera matrix was ?xed to would be required to determine why the average Image 1 x and y -coordinates differences vary by more than a factor of ten.

C = (Ij0). Further investigation

4.7 Upgrading to Euclidean Structure
4.7.1 Method 1: A Direct Solution

The recovered projective structure is related to the real scene structure by a 3D projective transform. So, in order to obtain a full Euclidean reconstruction we need to determine the 4 4 transformation matrix,

PEi . For one such pair of corresponding structure points, the mapping is

H, which maps the set of projective points PPi to their Euclidean counterparts
1 0 0 XE C B H B B Y C BH B E C B C B B B C B B ZE C = B H C B B A @ @ 10 C B XP CB Y CB P CB CB C B ZP CB A@
TP

11 21 31

H41

H12 H22 H32 H42

H13 H23 H33 H43

H14 H24 H34 H44

1 C C C C C C C A

(4.17)

Multiplying out, cancelling the scale factor, i , and rearranging, gives the following set of three linear equations in the sixteen unknown elements of

H:

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0 1 0 1 H11 ?XP ?YP ?ZP ?TP 0 0 0 0 0 0 0 0 XE XP XE YP XE ZP XE TP C B .. C B C B B 0 0 0 0 ?XP ?YP ?ZP ?TP 0 0 0 0 YE XP YE YP YE ZP YE TP C B .. C = 0 B C @ A B .. C B C A 0 0 0 0 0 0 0 0 ?XP ?YP ?ZP ?TP ZE XP ZE YP ZE ZP ZE TP @
H33
(4.18)

H is only de?ned up to a scale factor (see Appendix A), and therefore has just ?fteen degrees of freedom. Since each correspondence PPi , PEi gives rise to three such equations, we only need ?ve ground truth points to solve for the elements of H. This transformaThe projectivity tion can then be applied to all the projective points to obtain Euclidean structure.

4.7.2

Method 2: An Indirect Solution

An alternative method for obtaining the projectivity has been suggested which does not rely di-

H maps from projective to Euclidean structure, its inverse can be used to perform the reverse transformation i.e. PPi = H? PEi. Thus we can formulate a set of constraints on the elements of H? , based on the perspective projection equations
rectly on the projective structure [23]. Just as
1 1

1 0 ui C B i C B B ivi C = CH? C B A @
i
and

1

0 B XEi B B YEi B B B ZEi B @
TEi

1 C C C C C C C A 1 C C C C C C C A

(4.19)

1 0 0 B iui C B 0 C = C0H? B ivi C C B A @
i

1

0 B XEi B B YEi B B B ZEi B @
TEi

(4.20)

Multiplying out, cancelling the scale factors, i and i, then rearranging, gives a set of four, rather involved, linear equations for each combination of point matches and corresponding ground truth. However, only three of these are linearly independent, so we need at least ?ve such

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combinations to solve for

H , up to a scale factor. We can then invert, to obtain H and upgrade

our projective structure as before.

4.7.3

Initial Experiments

In order to compare the performance of the two methods, described above, we carried out a series of simple experiments. Although only ?ve ground truth points are actually needed to solve for

H,

it helps to have more available, to be able to measure the accuracy of the projectivity with respect to how it transforms points which were not used in its computation. For these experiments a set of twelve3 ground truth points PEi was used, corresponding to matched points in the quadrangle images, and chosen to give a good distribution in the 3D world space. First of all, a random subset of ?ve of the ground truth points was selected. This was used to estimate projective transform matrices,
4

H

1

and

H , for each of the two methods. Next, the pro2

jectivities were applied to all of the projective points, to obtain new estimates for their Euclidean locations tivities, QH1 and QH2 , were computed as the average Euclidean distance between the twelve esti-

^ ^ H PPi = PE i and H PPi = PE i . Finally, measures of the quality of the projec1 1 2 2

mated Euclidean points and the corresponding ground truth dist( ^ E 1i ; Ei ) and dist( ^ E 2i ; Ei ).

P

P

P

P

The experiment described above was performed 10000 times and the QH values for the two methods were examined. There was little correlation between the two sets of values. That is, a subset of correspondences that generated an accurate projective transform using the direct method would not necessarily do so for the indirect method. We have not performed a full statistical analysis of these results and so can only state that, on a trial for trial basis, the direct method outperformed the indirect method on approximately 68% of occasions. The sets of QH values were sorted and Figure 4.6 shows a graph of the 100 lowest values obtained for each method. The ground truth points for this pair of images were spread over a world space volume of approximately 50 10 50 metres, so both methods exhibit good best-case 3 This is in fact just a subset of over a hundred ground truth points for the initial images of the quadrangle sequence.
These were obtained by the author, with the aid of a trusty tape-measure, one cold, dark, miserable Sunday evening, when the campus was, thankfully, deserted! 4 In these experiments a total of 43 projective/Euclidean correspondences were available.

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400 350 Euclidean Structure Error QH (cm) 300 250 200 150 100 50 0 0 20 40 60 Sorted Results of the 100 Best Trials 80 100 Method 1: Direct Solution Method 2: Indirect Solution

Figure 4.6: Comparison of the accuracy of the direct and indirect methods for computing the transform from projective to Euclidean structure. performances, getting the average Euclidean distance error down to well under a metre. However, it is clear that, on average, the direct method gives the most accurate results. In the sequel, all our discussions and experiments will be based on this method of solving for the projective transform. Figures 4.7–4.10 show the Euclidean structure recovered by this method, using the trial above which had the lowest QH value, for the lecture theatre building in the image pair of ?gure 4.5. As it is rather dif?cult to portray three-dimensional structure on a two-dimensional page, especially using just point features, several different views of the reconstruction are shown. A partial triangulation of some of the points into 3D facets is performed, and subsequent texturemapping5 makes it easier to see the relationship between the real and recovered structure. These results demonstrate that it is possible to obtain a good estimate for the Euclidean structure of a projective scene, if just ?ve ground truth points are known. 5 The texture-mapping process is very basic, simply interpolating the pixel values from the original image between
the triangle vertices. Since camera calibration is unknown, there is no attempt to correct for camera motion, and thus the texture-mapping only looks realistic when the structure is viewed from close to the original camera position.

Figure 4.7: (a) Front view of the reconstructed points

(b) Partial triangulation with texture-mapping

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(c) Texture triangles in the original image This view of the reconstruction is from a position close to the original camera location, looking directly towards the lecture theatre building. The ’staircase’ of points in the top half of (a) corresponds to the corners of windows on the front face of the building, while the 12 points at the bottom right belong to the middle and right supporting pillars. In (b), the front face of the building has been modelled using 3 texture-mapped triangles. The original image locations of these triangles are shown in (c). The thin triangles at the extreme left and bottom of (b) correspond to the left side of the building and the ground plane, respectively. These are easier to see in ?gures 4.8 and 4.9, which show the reconstruction from different viewpoints. 68

Figure 4.8: (a) Side view of the reconstructed points

(b) Partial triangulation with texture-mapping

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(c) Texture triangles in the original image This view of the recovered structure is from a position at the right-hand side of the lecture theatre building, looking directly along the plane of its front wall. The features on this wall (corners of windows etc.) form a vertical band of points in (a), while features on or near the ground plane form a perpendicular group. The triangular facet in (b) is part of the lefthand wall of the building, but the texture-mapping is not consistent with the viewpoint due to the signi?cant change in camera position. The original image location of the texture triangle is shown in (c). 69

Figure 4.9: (a) Top view of the reconstructed points

(b) Partial triangulation with texture-mapping

CHAPTER 4. RECONSTRUCTION

(c) Texture triangles in the original image This is a top-down view of the recovered structure of the lecture theatre. The vertical band of points in (a) represents features on the front wall of the building, while the group of points at the top-right belongs to the pillars and bollards at the left side of the building. Clearly visible in (b) is the right-angled structure formed by the triangulations of some of these points (as shown in ?gures 4.7 and 4.8). The kite-shaped polygon, consisting of two triangles, depicts the ground plane. The original 70 image locations of the texture triangles are shown in (c).

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Figure 4.10: This is another top-down view of the lecture theatre reconstruction. The viewpoint is slightly different to that in ?gure 4.9 to show more of the texture-mapping on the sides of the building and give a better sense of the recovered structure.

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4.8 Incremental Reconstruction
Once a Euclidean structure estimate has been acquired for the scene in the ?rst image pair, it can be extended incrementally by processing each remaining image in the sequence. The initial steps of this processing are identical to the ones used for the ?rst image pair i.e.

1. Load the next image in the sequence and call it the current image. 2. Obtain a set of matched features in the current and previous images. 3. Use the matched features to estimate a fundamental matrix. 4. Factorise the fundamental matrix to obtain a pair of camera matrices. 5. Use the camera matrices to determine a projective reconstruction for the matched points.

At this point we calculate the projective transform to upgrade to Euclidean structure, as before. However, there is a difference: we no longer rely upon the user to supply the system with ground truth data. If the newly-computed projective structure includes at least ?ve points for which there exist previously estimated Euclidean coordinates, then these correspondences are used to obtain a projective transform. This is then be applied to the remaining6 projective structure to map it into the existing Euclidean frame. The whole process is repeated for each subsequent image in the sequence, at each step ‘stitching’ new and old structure together via the projectivities. Thus a Euclidean model of the entire scene is gradually built, from a basis of as few as ?ve ground truth points. This algorithm places some restrictions on the visibility of features between images in the sequence. Our incremental Euclidean reconstruction process requires ?ve or more projective points which have already been assigned Euclidean 3D coordinates. In order for this to be true, these points must previously have taken part in the process of fundamental matrix calculation, projective reconstruction and upgrade to Euclidean structure. Hence, at some point they must have been included in a set of eight or more matches between two successive images. Thus, each extension 6 Those points without Euclidean correspondences.

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to our overall Euclidean structure requires ?ve points which have been viewed in at least three images. These constraints have been built into our GUI-based feature detection/matching system, so at each step the user knows exactly how many image matches or 3D correspondences are required to proceed. There is one other implementation detail we have not yet discussed. So far, we have been happy to assume that the projectivity calculation could be performed simply by choosing the best result from a small number of trials. This approach is perfectly valid for the ?rst pair of images in the sequence, when we know the correspondences between projective points and accurate ground truth data. This accuracy also means that there is little to be gained from using more than ?ve of the ground truth points. However, when processing the remainder of the sequence, the projectivity is computed using a set of previously estimated Euclidean points. Obviously, some of these will be more accurate than others and, hopefully, there will be many more than ?ve correspondences. We would like to devise an ef?cient way to choose the best subset of these with which to calculate the projective transform. The is reminiscent of the problem faced in section 3.7.1, when attempting to estimate the fundamental matrix from a noisy set of matched image points. Here, we employ a similar solution.

4.8.1

A RANSAC Approach

We have embedded the direct method of computing the projective transformation matrix,

H, into

a RANSAC algorithm. Given a set of ?ve or more correspondences between projective and existing Euclidean structure, we perform a series of trials. In each trial, the ?rst step is to select, at random, a 5-element subset, from all the correspondences. This is used to compute the projectivity which is then applied to all the projective points to obtain new estimates for their Euclidean positions. We then compute the distance between each newly estimated and existing Euclidean points. Those correspondences for which this distance is below a given threshold form the consensus set. The (

5) correspondences in the consensus set are then used to calculate another projectivity, for

which we compute the average Euclidean distance error, over all correspondences, as in section 4.7.3. The old consensus set is replaced if the new one is of equal or larger size and has a lower

QH value. We then move on to the next trial and the process continues until either all the trials

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have been completed or the consensus set size or QH value reach a speci?ed target. In detail, the RANSAC projectivity calculation is as follows: Given ?ve or more projective,Euclidean structure correspondences fPPi ; PEi g, then for some ?xed number of trials: 1. Randomly select a subset of 5 correspondences, P5C 2. Use P5C to compute the projectivity 3. Apply

^ fPPi ; PEi g.

H.

^ H to obtain new Euclidean structure estimates HPPi = PEi .

4. Determine the consensus set C of correspondences whose Euclidean distance error is below some threshold. 5. Calculate a new projectivity, 6. Compute QH for

H0, based on C .

H0 .

7. Replace the old consensus set if C is the same size or larger, and has a lower QH . 8. Terminate if the size of C or QH reach speci?ed targets. At the end of this procedure, we hope to have obtained a projective transform, based on many of the correspondences, with a low number of trials.

QH value, and to have done so in a relatively small

4.8.2

Results

Figure 4.11 shows the results of the reconstruction algorithm after processing the initial ten frames of the quadrangle sequence. Although the two-dimensional constraints of the page make it dif?cult to get a feel for the recovered three-dimensional structure, this plan view indicates how the structure has been extended from that shown in ?gure 4.9. Clearly visible at the middle right of the diagram, is the right-angled corner of the building, which is in view over the ?rst few frames. The set of points leading away from this, to the south, corresponds to a series of bollards, while the long straight, east-west con?guration represents the struts of a continuous concrete bench. The

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set of points at the very top of the ?gure consists of features on the vertical surface of a distant building. The remaining points in the centre of the plan belong to various prominent scene features. Although these do not provide much in terms of readily identi?able scene structure, they are a valuable source of extra data for satisfying the constraints of section 4.8.

Figure 4.11: Plan view of the recovered quadrangle scene after 10 image pairs. Although the algorithm performs well at recovering the structure of objects which are close to the camera, objects at long range cause problems. The reason for this is the image resolution, which is too low to allow feature points on distant objects to be detected and matched with suf?cient accuracy. Take, for example, the point at the top left of ?gure 4.11, which is approximately ?fty metres from the camera. Changing the coordinates of one of its corresponding image points by just a single pixel resulted in a seven metre shift in its recovered Euclidean structure. These erroneous structure points can have a detrimental effect on the overall reconstruction, if they are used in the calculation of the projective transform at later steps. For example, ?gure 4.12 shows a plan view of another reconstruction of the same section of the quadrangle scene, which has failed

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for this reason; the points at the upper left of the ?gure being completely wrong. At the moment, this problem is avoided by only considering foreground structure when performing manual feature detection and matching.

Figure 4.12: Failed reconstruction of part of the quadrangle scene Results indicate that at each step in the reconstruction process, the error in the recovered Euclidean structure increases, even that of objects which are close to the camera. This is to be expected, due to the knock-on effect of calculating the projective transform at each step using previously estimated Euclidean structure. In the case of a long image sequence, the accumulated error could become very large, leading to inconsistencies in the recovered structure. In the next chapter we will describe our attempts to deal with this problem.

Chapter 5

Structure Recognition and Matching
5.1 Introduction
Our experiments of the previous chapter have shown that errors in the recovered Euclidean structure can accumulate rapidly, over the course of a few frames, and even when the reconstruction process is initialised with ground truth data. We must consider the effect this would have on a long sequence of images, especially one in which the camera eventually returns to view a part of the scene that has been seen previously. For example, ?gure 5.1 shows a simple diagrammatic representation of such an occurrence: a complete walk-around of a rectangular building. In this example, the reconstruction process starts by recovering structure for the front of the building. With our reconstruction system, this structure is extended incrementally as the viewer walks clockwise around the building, until eventually, the camera returns to somewhere close to its starting position. At this point it will be viewing the front of the building for the second time and thus we will have obtained two different sets of Euclidean structure for this section of the building. The errors in the structure recovery process will mean that the two structure segments are not aligned correctly in the world coordinate frame. If our reconstruction system is to be of any practical use, we need to ?nd a way to turn this initial incorrect structure into one which is internally consistent. In order to do this, two problems must be solved. First of all, we must be able to recognise 77

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S T A R T

F I N I S H

Figure 5.1: Errors in the reconstruction process can lead to internally inconsistent structure. when two recovered structure segments actually correspond to the same part of the real-world scene. Once this has been accomplished, it is necessary to ?nd the transformation which brings the two structure segments into alignment. In terms of model-based recognition, these are the problems of indexing and matching. The task is further complicated by the fact that occlusion and/or a change of viewpoint could mean that the two segments do not contain exactly the same points. We show that these problems can be solved using the geometric hashing paradigm, a general method for model-based object recognition. It was originally developed for the task of identifying ?at rigid objects [35, 34] from images, using an af?ne approximation to the full perspective camera, but the same approach can be used for many recognition problems involving a variety of transformations in two and three dimensions [37]. Classical geometric hashing is a two-stage process, object representation and matching. Brie?y, the ?rst stage, which is usually done off-line, involves the construction of a hash table, to provide an invariant representation for each of a series of model objects that are to be recognised. In the second stage, features extracted from images of the scene are used to compute invariants, with which to index the hash table and tally votes for candidate models i.e. those that could feasibly be present in the scene. A high vote count not only indicates the likely presence of an instance

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of a particular model object in a view of the scene, but the corresponding invariants determine the model!view transformation. The same basic algorithm can be used to solve our problem of matching two segments of Euclidean structure, with one important difference. Rather than having a priori knowledge of a set of model objects to be recognised, our hash table starts off completely empty. As the incremental reconstruction system recovers new estimated Euclidean points, these are hashed into the table. In a sense, the system ’learns’ its own models of segments of scene structure as it goes along. Later in the image sequence, if we come across the same structure segment again, the geometric hashing system should recognise this, and provide the transform between the old and new segments, thus solving both our indexing and matching problems. As with all indexing techniques, geometric hashing allows for the recognition of many models simultaneously. This is important for our application, in which an extended environment could be represented by hundreds of structure segment models. We begin this chapter with an overview of the original geometric hashing paradigm, followed by a discussion of the details of our implementation. We describe our method for representing 3D structure in a form which is invariant to Euclidean transformations, and present a simple, symmetry-based approach for improving the ef?ciency of the hashing process. Finally, we describe the way we have embedded the geometric hashing algorithm into our structure recovery system and discuss the initial results of experiments on real image sequences.

5.2 An Overview of Geometric Hashing
In this section we review the geometric hashing paradigm in the context of its original application: the recognition of ?at objects under the af?ne transformation (rotation, translation, scale and shear). This allows for a very simple explanation of the algorithm’s main concepts and will help us to show, later in the chapter, how our implementation differs from that of the standard method. The general scheme of the geometric hashing paradigm is shown in ?gure 5.2. When viewing ?at objects which are relatively distant from the camera, orthographic projection (with a scale factor), provides a good approximation to the full perspective camera [29].

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P R E P R O C E S S I N G

Model Acquisition Interest Feature Extraction Transformation Invariant Coordinate System

HASH TABLE h(coordinate) = (Model,Basis)

R E C O G N I T I O N

Computation of Interest Feature Coordinates In Given Basis

VOTE For (Model,Basis) Pairs

NO Basis Choice

(Model,Basis) With High Vote? YES Find Best

Interest Feature Extraction BAD SCENE

Least-Squares Match Verify Object Against Scene GOOD Eliminate Model PROCEED

Figure 5.2: The general scheme of the geometric hashing algorithm

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Under this assumption, two different images of the same ?at object are related by a 2D af?ne trans-

A and 2D vector b, such that a non-homogeneous point p in the ?rst image is transformed to the corresponding point Ap + b
formation 2 matrix in the second. The transformation has six degrees of freedom and can thus be fully determined by three point correspondences. Assuming that sets of interest points have been extracted from images of the model objects and the scene, the problem becomes that of determining if some transformed subset of scene points matches a subset of any of the model point-sets. The key to this lies in representing the point sets in a manner which is invariant to the af?ne transform.

T. In other words, there exist a non-singular 2

5.2.1

2D Af?ne Invariance

Suppose we have extracted a set of

m points from an image of one of our model objects.

We

can pick any ordered, non-collinear triplet of points from this set and use them to represent all the other points. Consider ?gure 5.3. Let a00 , a10 and a01 be three, non-collinear points in the image. The vectors
10 00 01 00

linear basis. Any point p in the image can be represented as a linear combination of these two basis vectors. In other words, there is a pair of scalars ( , ) such that:

i = (a ? a ) and j = (a ? a ) are linearly independent, hence they are a 2D

p = i + j + a = (a ? a ) + (a ? a ) + a Application of the linear af?ne transform T to this representation for p gives
00 10 00 01 00

00

(5.1)

Tp = (Ta ? Ta ) + (Ta ? Ta ) + Ta
10 00 01 00

00

(5.2)

Tp has the same coordinates ( , ) with respect to the af?ne basis Ta Ta Ta as has p with respect to a a a . For example. suppose we make the following coordinate assignments for the points in ?gure 5.3: a = (5; 1), a = (10; 1), a = (7; 5) and p = (24; 9). Writing these in terms of equation 5.1 gives:
Hence the transformed point
00 10 01 00 10 01 00 10 01

0 1 0 1 0 1 0 1 B 24 C = B 5 C + B 2 C + B 5 C @ A @ A @ A @ A 0 9 4 1 | {z } | {z }
i j

(5.3)

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βj

p

a01 j

a00

i

a10

αi

Figure 5.3: Representing p using the af?ne basis triplet a00a01 a10. which means, by inspection

Ax + b where

4 Applying this to our four points gives: Ta00 = (10; 15), Ta10 = (15; 25), Ta01 = (20; 23) and Tp = (45; 61). Once again, in terms of equation 5.1 we have:

1 0 1 2C A=B A @
2 1

= 3 and = 2. Now, suppose we have an af?ne transform T =
and

0 1 3 b=B C @ A

(5.4)

0 1 0 1 B 45 C = B 5 C + @ A @ A 10 61 | {z }
i
Clearly

0 1 0 1 B 10 C + B 10 C @ A @ A 8 15 | {z }
j

(5.5)

obtained a representation for the point p which is invariant to the af?ne transform

= 3 and = 2 is a solution to this simple pair of linear equations. Thus we have

T. Armed with

the mechanism for computing such invariants, we proceed with a description of the two steps that make up the geometric hashing method: model representation and matching.

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5.2.2

Model Representation

Classical geometric hashing usually begins with a preprocessing step, to generate a hash table containing invariant descriptions of the model objects that the system needs to be able to recognise. In the 2D af?ne case, this implies we have a database of images of our model objects, and that from each of these we have extracted sets of interest points corresponding to model features. The model representation is constructed by considering every possible three point subset of interest points as an af?ne basis and, in each case, hashing the invariants computed for all the remaining points. The outline for this preprocessing stage is as follows: For each model: 1. Extract a set of m interest points. 2. For each ordered, non-collinear triplet of interest points (af?ne basis): Compute the invariant coordinates of the remaining m ? 3 interest points in the af?ne frame de?ned by the basis triplet. Pass each set of invariant coordinates to a hash function which generates indices into the hash table. At each given hash table location, known as a bucket, store a record of the model and af?ne basis from which the invariants were obtained. Note that the ?nite size of the hash table will often lead to collisions, whereby more than one (model,basis) pair needs to be stored in each bucket. In such cases the bucket holds a list of these pairs (see ?gure 5.4).

The preparation of the hash table can be looked upon as a kind of learning process, in which various different representations of the models are memorised. Since it requires no knowledge about the scenes in which the models are to be recognised, it is usually performed off-line. Once preprocessing is completed, the geometric hashing system is ready for the matching stage.

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Hash Table B U C K E T S 0 1

List Of Elements In Each Bucket

B-1

Figure 5.4: Hash table organisation

5.2.3

Matching

In geometric hashing the object representation and matching stages follow along very similar lines. When matching, we are presented with an image of a scene, and wish to determine which, if any, of our model objects are currently in view. To do this we: 1. Extract a set of n interest points from the image. 2. Choose an arbitrary ordered triplet of non-collinear points and use them as a basis with which to compute the af?ne invariant coordinates of the remaining n ? 3 points. 3. Pass each set of invariant coordinates to the hash function which generates indices into the hash table. 4. Check each indexed hash table bucket and tally a vote for every (model,basis) pair stored there. 5. Look for a (model,basis) pair which scores a high number of votes. Each such pair implies that an instance of the given model is present in the scene. Furthermore the uniquely de?ned af?ne transformation between the candidate model and image basis triplets is assumed to be the transformation that maps between the model and the scene. 6. Apply the transformation obtained in step 5 to all the image points that voted for the candidate model to induce additional model-image point correspondences. Find the best transformation between all the correspondences, in a least-squares sense.

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7. Transform the entire low-level representation of the model (which may include additional information, such as edge features, colours etc.) via the af?ne transformation obtained in step 6 and verify it against the scene. If the veri?cation con?rms the existence and orientation of the model, the matching process is complete. If this candidate solution is rejected, another one from step 5 is examined. If there are no more candidate solutions, go back to step 2.

It should be noted that, in general, the voting scheme will not result in just one candidate solution. In fact, that is not really the aim. Rather, the intention is to reduce signi?cantly the number of candidates which make it through to the veri?cation step. Also, since votes are cast for all models simultaneously, the complexity of the recognition process is independent of the size of the model database. One of the nice features of geometric hashing is its ability to recognise partially occluded objects. This is made possible by the preprocessing stage, which constructs model representations using all the basis triplets from the points of interest. Thus, for matching to succeed it is enough to pick three points in the scene which belong to some model, in which case the appropriate (model,basis) pair will score highly in the voting procedure.

5.3 Hashing Euclidean Structure
The description of geometric hashing in the previous sections was based around the problem of recognising models and scenes related by a 2D af?ne transform. As we have seen, computing invariants under such a transform requires a three-point basis. Other transformations have different basis requirements. For example, recognition of objects which have undergone a 2D or 3D translation needs only a one-point basis, and a four-point basis for a projectivity between two planes [37]. A projectivity from P 3 to P 3 requires a ?ve-point basis [12]. In our case, we wish to match two sets of 3D Euclidean structure, which differ by some rigid transformation. It will be shown that this problem can be solved using a three-point1 basis. 1 In fact, this is also the case for the 3D similarity transform, so we can solve for this more general problem with no
additional complexity.

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worst case complexity of the geometric hashing algorithm is O(nk+1 ) [37]. Thus, by upgrading from projective to Euclidean structure as part of the reconstruction process, the complexity of the recognition task has been reduced from O(n6 ) to O(n4).

In passing, we note that given a set of n model/scene interest points and a k-point basis, the

5.3.1

Computing Euclidean Invariants

Suppose we have a set of four 3D Euclidean structure points. Three of these can be used to de?ne a new coordinate system (basis) in which the position of the fourth point is invariant to a 3D rigid or similarity transform. Consider ?gure 5.5. The three points, E1,

E2 and E3, de?ne the unit length and the xy -plane of our new coordinate system, XE YE ZE , with XE as the origin. The normal to this plane de?nes the new z -axis. The task is to compute the position P with respect to
this new coordinate frame.

Y E3 Z

V3 P YE ZE E1 VP XE E2 X
Figure 5.5: Points E1, E2 and E3 de?ne a 3-point basis for computing the new coordinates of P , which are invariant under 3D rigid and similarity transformations. We begin by constructing orthogonal vectors corresponding to the direction of each of our new coordinate axes, as follows:

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1. The new x-axis vector is simply

XE = E ? E .
2 1

2. To compute the new z -axis vector we ?rst need to de?ne the vector gether, the vectors

XE and V

3

de?ne our new xy -plane. We can now obtain our new z -axis

V = E ? E . To3 3 1 3

as the normal to this plane, given by the cross-product E

Z = XE V .

3. Finally, our new y -axis vector is given by the cross-product The invariant coordinates of the point P , denoted inv lating the component of the vector
1

Y E = ZE X E .

VP = P ? E , in the direction of each of our new coordinate

P = ( ; ; ), are obtained by calcu-

axis vectors, and dividing by the unit length, as follows:

1. 2. 3.

= (VP XE =jXE j)=jXE j = (VP YE =jYE j)=jXE j = (VP ZE =jZE j)=jXE j
As a quick check that this method actually works, consider computing invariant coordinates

from a set of four 3D points under some simple transformations. In each case, the ?rst three points are used to compute the invariant coordinates of the fourth. Firstly, the results for the original four points.

25 These points give the invariant coordinates of P4 as ( ; ; ) = (0:76; 0:51; 1:22) and a unit length of 10:39. 10 12 15
After scaling the points by a factor of two:

0 1 0 1 0 1 0 1 15 14 C 20 C 10 C B C B C B C B C B C;P = B C;P = B C;P = B C B 11 C B 20 C B 12 C P = B 10 C B C B A B A B A @ A @ @ @
1 2 3 4

(5.6)

0 1 0 1 0 1 0 1 B 30 C B 28 C B 40 C B 20 C B C B C B C B C P = B 20 C ; P = B 24 C ; P = B 40 C ; P = B 22 C B C B C B C B C @ A @ A @ A @ A
1 2 3 4

(5.7)

20

24

30

50

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As one would expect, this time the unit length has increased to 20:78, but the invariant coordinates of

P

4

remain as (0:76; 0:51; 1:22).

This time a translation of (5; 10; 20)T , followed by the same scaling as above, transforms the points to:

0 1 0 1 0 1 0 1 38 C 50 C 30 C B 40 C B C B C B C B C B C B C B C P = B 40 C ; P = B 44 C ; P = B 60 C ; P = B 42 C B C B A B A B A @ A @ @ @
1 2 3 4

(5.8)

60

64

70

90

Which gives identical results to the previous test of unit length 20:78 and invariant coordinates

(0:76; 0:51; 1:22).

For one quick ?nal test, the points were transformed by a rotation of 180 degrees about the

z-axis, followed by a rotation of 90 degrees about the x-axis, giving:

11 20 Once again, these give the invariant coordinates of P4 as (0:76; 0:51; 1:22) and a unit length of 10:39. 10 12
5.3.2 Symmetry Considerations

1 0 1 0 1 0 1 0 ?15 C ?14 C ?20 C B ?10 C B C;P = B B C;P = B B C;P = B B C B ?25 C B ?15 C B ?12 C P = B ?10 C C B C B C B C B A @ A @ A @ A @
1 2 3 4

(5.9)

For any given set of three points, A, B and C , there are six different ways of ordering them to construct a Euclidean basis:

ABC , ACB, BAC , BCA, CAB and CBA. Each of these bases can be used to compute the invariant coordinates of a fourth point D = ( i; i; i), where i = 1 : : : 6. An interesting question is whether or not one would expect to obtain different invariant
coordinates with each basis, or if it is possible to extrapolate the result of one computation from another and thus speed up the hashing process. Let A1 =B1 and A2 =B2 be two different labellings for the points A and B corresponding

to the normal ordering of points (ABC ) and when the positions of A and B have been swapped

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C Y2 Y1 U1 U2

Z1 X1 X2 A1 B 2

B 1 A2

Z2

Figure 5.6: Effect of basis point ordering on coordinate axes. (BAC ), respectively. Figure 5.6 shows the two bases A1 B1 C and B2 A2 C . As described in section 5.3.1, the ?rst stage in the computation of invariant coordinates is the construction of a new set of coordinate axes. The x-axis vectors for the two bases are which differ only in their signs.
2 2

U = C ? A . Due to the change of point ordering, these two vectors will be different. The z-axes are computed as the cross-products Z = X U and Z = X U . Thus Z and Z
1 1 1 2 2 2 1 2

X = B ?A , The next step is to compute the vectors U = C ? A and
1 1 1

X = B ?A

and
1

2

2

2

1

computed as the cross-products of the x and z -axes vectors, will be the same for both bases.

are both normal to the plane ABC , but have different signs. It follows, that the y -axes, which are
2

of D are obtained by projecting the

Now, consider ?gure 5.7 and the addition of a fourth point D. The invariant coordinates

V

1

and

V

2

vectors onto the coordinates axes de?ned by the

two bases. First of all, it can be seen that the two vectors

V

1

and

V

2

are related as follows:

V ?V = X
1 2

1

(5.10)

Thus, projecting in the direction of the for the point D in the two bases:

X

1

, gives a relation between the invariant -coordinates

2 Since i

j = k ?! ?i ?j = k.

1

+

2

=1

(5.11)

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D

V2 V1 Y1 Z1 Z2 X1 X2 Y2

Figure 5.7: Effect of basis point ordering on invariants. The y -axis vectors for both bases are identical, therefore the invariant -coordinates are

identical. Similarly, the z -axis vectors differ by only by their signs, therefore so do the invariant coordinates. As a result, if the three invariant coordinates are computed for one of these bases, then the invariants for the other are trivially de?ned. This result holds for the basis pairs ABC=BAC , ACB=BCA and CAB=CBA. Thus, out of a possible total of eighteen different invariant coordinates3, there are only nine independent values, which can be obtained from just three of the basis orderings. Table 5.1 summarises the relationships between the ordering of the basis points and the invariant coordinates. ABC BAC ACB BCA CAB CBA
1 1 1 2 1 2 2 3 2 3 3 3 1

1? 1? 1?

?
2

1

?
3

2

?

3

Table 5.1: Effect of basis ordering on invariant coordinates.

3 Three for each of the six different bases.

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This symmetry property can be used to reduce the computation time of the hashing process, since only half of all the possible basis choices for a given point set need to be considered. The important point is that, in each of the related basis pairs, only the positions of the A

and B points have been swapped and therefore only the signs of the x-axis vectors differ. Other for which the positions of points B and C have been exchanged. This effectively swaps the xaxis vector cross-product of these, only changes sign4, but the y -axis vector, which is the cross-product of positional changes have different effects. For example, consider the two bases ABC and ACB ,

X and the U vector, and changes the unit length. The z-axis vector, obtained as the V onto each of

the x and z -axes vectors, will be different. As a result, projecting the point vector

these axes and dividing by the unit length, will give different invariant coordinates for D in the two bases.

5.4 Geometric Hashing and Image Sequences
A geometric hashing system has been developed, based on the Euclidean invariants discussed in the previous section. In this section we describe how it has been incorporated into our incremental reconstruction system. The most important difference between our method and that of standard geometric hashing, is that we do not perform a preprocessing step, at least, not in the usual sense. The reason for this is simple: there is nothing to preprocess, no prede?ned database of models to be recognised, no a priori knowledge of the scene. Rather, the intention is that the system should acquire its own models automatically and do so concurrently with the matching stage. Each step of the reconstruction system processes the next pair of images in the sequence and, starting with a set of matched points, eventually recovers a 3D Euclidean structure segment for the part of the scene currently being viewed. These segments are stitched together, incrementally, to build up a reconstruction for the entire scene. 4 In fact, since no matter which ordering of points is chosen, the resultant X and U vectors always lie in the plane,

ABC , the z -axis vector is the same (up to a change of sign) for all these basis permutations.

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The model acquisition part of the geometric hashing system follows along the same lines. As the reconstruction system generates new structure segments, their 3D points are used to compute Euclidean invariants (section 5.3.1) with which to index and update the, initially empty, hash table (section 5.2.2). Thus, a complete hash table representation for the entire scene is gradually obtained. Similarly, in the recognition stage, recovered structure segments are used to compute invariants with which to index the hash table and tally votes (section 5.2.3). Hence, if a structure segment containing points that have previously been hashed comes back into view later in the sequence, the system should recognise this fact, in addition to providing the transformation between the corresponding points.

5.4.1

Partitioning the Hash Table

One approach to updating the hash table with new structure information would be to hash the new 3D points together with all of the old ones, which have previously been estimated. This would create one large model of the scene viewed over the whole image sequence, but it would be a highly redundant representation, since the majority of combinations of points used in computing the invariants could never actually be viewed together in the scene. The hashing strategy actually used in the implementation of this scheme is based around a simple observation: both the model acquisition and recognition stages depend entirely upon the recovered structure segments which contain only local information. In other words, for a long image sequence, each image only gives a view of a small section of the entire scene. Thus, a more ef?cient approach to updating the hash table would be to compute invariants using only the currently viewed structure segment. In fact, the method used is to compute invariants based on points in the current and previous d structure segments5. This creates some overlap in the hash table, but allows for recognition from a wider range of viewpoints. In essence, when a segment of scene structure is viewed for the ?rst time in the image sequence, it is used, along with neighbouring segments, to construct an invariant representation for 5 Where d is determined by the disparity between pairs of images in the sequence.

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93

the local patch of structure. Thus, when updating the hash table, the information recorded is not (basis,model), but (basis,patch).

5.4.2

Algorithm Outline

Each step in the reconstruction algorithm results in a structure segment containing a set of newly estimated Euclidean points. At the same time, a set of previously estimated old structure points will have just gone out of view. The old points are used to update the hash table, while the new ones are used for recognition. The bene?t of this approach is to ensure that the new points are not used for model representation and matching at the same step. Suppose, at some step i of the reconstruction process, a structure segment has been recov-

ered and sets of old and new points, denoted SOLDi and SNEWi , have been determined. Model acquisition and matching proceed as follows:

Model Acquisition Combine the points in SOLDi with those obtained at d previous stages (SOLDi?1 : : :SOLDi?d ), to obtain a local structure patch. Let this patch contain a total of non-collinear triplet of these points: (Euclidean basis):

m points.

For each ordered,

1. Compute the invariant coordinates

( ; ; ) of the remaining m ? 3 points points in the

Euclidean coordinate frame de?ned by the basis triplet. 2. Use the invariants to generate indices into a 3D hash table. 3. At each given hash table location, store a record of the patch and Euclidean basis from which the invariants were obtained.

Matching 1. Suppose SNEWi contains a total of n points.

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94

2. Choose an arbitrary ordered triplet of non-collinear points as a Euclidean basis and compute the invariant coordinates of the remaining n ? 3 points. 3. Use the invariants to generate indices into a 3D hash table. 4. Check each indexed hash table bucket and tally a vote for every (patch,basis) pair stored there. 5. Look for a (patch,basis) pair which scores a high number of votes. Each such pair implies that at least part of the given patch is present in the scene. The uniquely de?ned Euclidean transformation between the candidate patch and scene segment bases is assumed to be the transformation that maps between the patch and the scene. 6. Apply the transformation obtained in step 5 to all the SNEWi points that voted for the candidate patch, to induce additional point correspondences. Find the best transformation between all the correspondences, in a least-squares sense. 7. Compute the average Euclidean distance between the candidate patch points and those obtained by applying the above transform to SNEWi . 8. If the transformation results in a low average Euclidean distance between points in SNEWi and the candidate patch, then the existence and orientation of the patch has been veri?ed and matching is complete. If not, this candidate solution is rejected and another one from step 5 is examined. If there are no more candidate solutions, go back to step 2.

5.5 Results
5.5.1 Synthetic Data: Simple Models

We begin with a very basic test of the geometric hashing system, to ensure that the implementation is algorithmically correct and bug-free. Figure 5.8 shows a set of simple 3D models, the ?rst ?ve of these are quite distinct, while the sixth is a slightly skewed (noisy) version of the ?rst. The corner points of models 1–5 were used

CHAPTER 5. STRUCTURE RECOGNITION AND MATCHING

95

1

2

3

4

5

6

Figure 5.8: The set of simple models used in initial tests on the geometric hashing system. Models 1–5 are structure segments used to build the hash table structure. Model 6 is a skewed version of model 1, used to see if the system copes with noisy data.

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96

as structure patches, with which to initialise the hash table, as described in the previous section. Subsets of the same points were then fed into the matching process, to see if the system could recognise each of the models using perfect data. For this test, an exhaustive6 series of trials were performed, with the best match taken to be the model which received the most votes in any single trial. Results are shown in Table 5.2. Subset Model 1 2 3 4 5 Matched Model 1 2 3 4 5 Most Votes 5 5 2 3 1 Trial 1 1 1 1 1

Table 5.2: Results of geometric hashing using perfect synthetic data.

As one would expect, the system had no dif?culty matching the point data to the original models. In every case the correct match, with the highest number of votes, was achieved with the ?rst choice of basis points. It is worth noting that each of the ‘Most Votes’ values is actually the maximum number of votes possible, since an n-point structure patch will cast basis7. We mention, in passing, that exactly the same results as above were obtained from a supplementary test, which attempted to match using model points that had undergone arbitrary Euclidean transformations. This is additional con?rmation of the invariance property discussed in section 5.3.1. The next test of the system was an attempt matching with imperfect data. The sensitivity of the geometric hashing paradigm to such errors was discussed in [20], while methods for dealing with noisy data were presented in [70] and [16]. The suggested approach is to modify the voting part of the matching scheme. Rather than just indexing a single hash table bucket and tallying a 6 Using all possible basis/point combinations. 7 Assuming, as in this case, an even distribution of basis/model pairs over the hash table, such that no pair appears
more than once in a given bucket.

n ? 3 votes per

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97

vote for each of the elements it contains, all buckets within a region of interest around the indexed location take part in the voting process. The implementation of this idea required only a minor extension to our 3D geometric hashing system, whereby votes are cast in a spherical region around the indexed location. The imperfect data we hoped to match was based on the sixth model of ?gure 5.8. In fact, several variations of this model were created by adding different levels of random noise 8 to the original model 1. Results from a series of trials, as before, are shown in Table 5.3. % Noise 0 5 10 15 20 30 40 Matched Model 1 1 1 1 1 1 3 Most Votes 5 4 4 3 2 1 1 % Max Correct 100 100 100 100 90 66 46

Table 5.3: Results of geometric hashing using noisy synthetic data

The ?rst thing to note from the table is that up to and including the 30% noise level the geometric hashing system successfully matched the input data with model 1. However, as the noise level increased, so the largest number of votes scored in any single trial decreased. The ?nal column of the table gives the percentage of the trials with the highest vote count that proposed the correct match. In the 10–15% noise range this was all of them. At the 20% noise level this dropped to 90%, and with 30% noise only 66% of trials with the given highest vote count (1) matched correctly. This means that although the matching process was successful eventually, many more candidate matches had to go through the veri?cation procedure. Unsurprisingly, with 40% noise the matching system proposed the wrong match the majority of the time. 8 The noise level was calculated as a percentage of the size of the original model. Model 1 was a cube of 10 10 10
units, so, for example, a noise level of 10% means a random coordinate shift of 1 unit.

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98

5.5.2

Real Data: Recovered Structure

For this series of tests we attempted to match the structure recovered from the ?rst pair of quadrangle images, against the original ground truth. The structure correspondences were grouped into ?ve models/patches according to 3D location, as shown in Table 5.4. Model Number 1 2 3 4 5 Number of Points 8 9 6 14 13 Description of features Supporting pillars at the front of the lecture theatre Supporting pillars at the side of the lecture theatre Features on the distant building Windows on the left of the lecture theatre Windows on the right of the lecture theatre

Table 5.4: Grouping of features into structure patches

Each of the recovered structure patches was used as input to the matching process. Initial experiments with this data highlighted a problem. The system was proposing (incorrect) candidate matches with a ‘Most Votes’ value that was three or four times the expected maximum. This was caused by an uneven distribution of invariant coordinate values, as shown in Figure 5.9. As a result, a given basis/model pair could be stored many times in the same hash table bucket. Thus, each time that bucket was indexed for voting, multiple votes were cast for that pair. We attempted to solve this problem using a logarithm-based hash function. The results of subsequent experiments are shown in Table 5.5. Unfortunately, none of the recovered structure patches were successfully matched against the corresponding ground truth data. The ‘Most Votes’ column still shows a larger number of votes for the proposed match which is bigger than the maximum we would expect from an evenly distributed hash table, but there has been a marked improvement. The ?nal column gives the largest number of votes cast in a single trial for the correct model. Only the trials with patches 4 and 5 result in a high number of votes for the correct model. In the other trials the proposed best match seems almost arbitrary. This is not really surprising if we take a closer look at the data the geometric hashing system

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99

700 Ground Truth Alpha 600 20 Reconstructed Alpha

500 Occurences Occurences

15

400

300

10

200 5 100

0 -10

-8

-6

-4

-2 0 2 Invariant Coordinate

4

6

8

10

0 -10

-8

-6

-4

-2 0 2 Invariant Coordinate

4

6

8

10

700 Ground Truth Beta 600

3 Reconstructed Beta 2.5

500 2 Occurences 400 Occurences

1.5

300

1 200 0.5

100

0 -10

-8

-6

-4

-2 0 2 Invariant Coordinate

4

6

8

10

0 -10

-8

-6

-4

-2 0 2 Invariant Coordinate

4

6

8

10

2 Ground Truth Gamma 1.8 1.6 1.4 Occurences Occurences 1.2 1 0.8 0.6 0.4 0.2 0 -10

3 Reconstructed Gamma 2.5

2

1.5

1

0.5

-8

-6

-4

-2 0 2 Invariant Coordinate

4

6

8

10

0 -10

-8

-6

-4

-2 0 2 Invariant Coordinate

4

6

8

10

Figure 5.9: Distributions of invariants. The left column of graphs shows the distribution of , and invariants for ground truth data. The right hand column shows the distributions for the

corresponding recovered structure. The ground truth graph for has had the zero entries removed, as they were swamping the other values. This was caused by many of the basis/point pairs being coplanar.

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100

Recovered Patch 1 2 3 4 5

Matched Model 4 5 4 1 4

Most Votes 12 13 5 17 16

Correct Votes 0 7 0 14 15

Table 5.5: Results of geometric hashing using real data.

has to work with. For example, four points chosen at random from set of correspondences number 2. The ground truth values for these points are:

1 0 1 0 1 0 1 0 ?587:00 C ?662:00 C 0:00 C B 20:00 C B C;P = B B C;P = B B C;P = B B C B 328:00 C B 108:00 C B 39:00 C P = B 39:00 C C B C B C B C B A @ A @ A @ A @ 1050:00 244:00 ?743:00 ?350:00
1 2 3 4

(5.12) The invariant coordinates of

P

4

are (

; ; ) = (?3:47; ?1:78; 0:56), using the ?rst three points

as a basis. If we now examine the corresponding recovered structure points:

1 0 1 0 1 0 1 0 ?535:36 C ?731:63 C ?75:79 C ?33:46 C B B B B B C B C B C B C B 26:53 C ; P = B 28:32 C ; P = B 99:23 C ; P = B 325:74 C P =B C B C B C B C A @ A @ A @ A @ 935:04 331:68 ?688:49 ?304:85
1 2 3 4

(5.13) These give the invariant coordinates of

P

4

as

( ; ; ) = (?3:05; ?1:71; 0:62).

The

and

values are quite similar, but in the context of the overall range, the

value has changed quite dra-

matically. Such differences between the computed invariants cause the wrong hash table location to be indexed, resulting in spurious votes being cast and incorrect candidate matches.

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101

5.6 Conclusions
Although the results of experiments with synthetic data have been encouraging, so far we have been unable to get the geometric hashing system to recognise real structure patches successfully. It is apparent that the cause of this problem is not the geometric hashing system itself, but rather, inaccuracies in the recovered Euclidean structure, which forms the input to the system. A common approach to dealing with such noisy data is to use a region-based voting algorithm, but this has also proved inadequate. The dif?culty lies in determining the size of the region of interest. Too small a region achieves nothing, but if it is too large then many candidate matches are found, which defeats the object of performing geometric hashing in the ?rst place. So far we have been unable to develop a sensible value for the size of the voting region which is large enough to allow recognition to take place but small enough to generate a manageable number of candidate matches. It may be that performance can be improved slightly by careful choice of parameters such as the size of the voting region or modifying the hash function, but, in conclusion, it would seem that unless a way can be found to produce more accurate structure estimates, the geometric hashing method of matching will not be successful.

Chapter 6

Conclusions
6.1 Summary
The work presented in this thesis addresses the problem of automatically reconstructing a model of an extended environment, from a long image sequence taken with an ordinary hand-held video camera. An uncalibrated approach to structure recovery has been taken, based on the calculation of the epipolar geometry of successive pairs of images in the sequence. Knowledge of ground truth data, for the ?rst image pair only, is used to propagate an estimate for the Euclidean structure of the entire scene. Over a long image sequence, it is anticipated that errors will accrue, and as a result, the recovered structure will be internally inconsistent. A method which attempts to detect such anomalies has been developed, using the geometric hashing paradigm. In Chapter 3 a method is described for acquiring the fundamental matrix from a set of matched points in a pair of uncalibrated images. It is based on a novel implementation of the wellknown 8-point algorithm, which combines a recently developed normalisation technique with a variation on the RANSAC parameter estimation algorithm. This linear method is simple, yet ef?cient, and experiments on a range of real and synthetic data show that it produces quanti?ably accurate results. A direct comparison demonstrates that the method generates fundamental matrix estimates of similar quality to those of more complicated alternatives. The fundamental matrix is the input to the reconstruction system presented in Chapter 4. 102

CHAPTER 6. CONCLUSIONS

103

We use Hartley’s method for factorising a fundamental matrix, to construct a pair of camera matrices for the corresponding images. These are used to obtain an estimate for the projective structure of the current view, via back-projection. For the ?rst pair of images in the sequence, ?ve or more ground truth points are required, in order to compute a projective transform matrix with which to upgrade from projective to Euclidean structure. The problem of calculating the projectivity which best ?ts the data is again formulated and solved using the RANSAC scheme. An original feature of the reconstruction system is the way in which remaining image sequence pairs are processed. Each is used to obtain a segment of projective structure, as outlined above, but the upgrade to Euclidean is performed, not with ground truth data, but rather, using previously estimated Euclidean structure. Thus a complete reconstruction for the scene is incrementally obtained, by stitching together the segments of Euclidean structure acquired at each step. Results show that the reconstruction system performs well when recovering the structure of objects which are close to the camera, but copes poorly with distant objects. In the latter case, the image resolution is insuf?cient to be able to detect and match point features accurately, even by hand. As a result of this, very small changes in the image coordinates of a point can lead to large changes in the recovered structure. This is a problem faced by any image feature-based reconstruction system and is not due to the speci?c method used here. At each step of the reconstruction system, new structure is acquired using old structure that was itself estimated. As successive image pairs are processed the error in the recovered structure will accumulate. In a long image sequence there is the possibility that previously viewed parts of the scene will be re-encountered. In this event there will be two structure estimates for the scene segment; the current one and the one obtained the ?rst time it was in view. It is required that a match between these two segments is found and their relative orientation obtained, in order that globally consistent structure can be maintained. Chapter 5 describes an method to tackle this problem, using geometric hashing. The geometric hashing system has been developed to work in tandem with the reconstruction process. Unlike conventional geometric hashing, there is no off-line preprocessing stage. Rather, the models to be recognised are memorised while the system is running, and this is done concurrently with the matching stage. The models in question are patches of recovered Euclidean

CHAPTER 6. CONCLUSIONS

104

structure. At each step in the reconstruction, an additional patch of structure is used to update the hash table. The system is based around 3D Euclidean invariants and a method for obtaining these is described. Also presented is a symmetry-based technique for improving the ef?ciency of the hashing process. The goal of the geometric hashing system is not to produce a unique match between the scene structure and that which is stored in the hash table. Instead, it aims to obtain a manageable number of candidate matches, which can be examined more closely. Unfortunately, this aim has not yet been achieved, as the system either produces large numbers of candidate matches, or not at all. This failure is not due to the geometric hashing implementation, but rather, the data it works with. The problem is caused by the fact that errors in the recovered Euclidean structure accumulate more quickly than expected. For the geometric hashing system to function as intended, the accuracy of the reconstruction must be improved.

6.2 Future Work
The calculation of the projective transform between projective and Euclidean structure is the key to the reconstruction process. The elements of the transform matrix are obtained via the solution of a set of homogeneous linear equations, using the least eigenvector method. This is the same approach taken when computing the fundamental matrix in Chapter 3, and it would be worth investigating the possibility of improving the result of this calculation by developing a normalising transform, based on the analysis in [26]. Another extension to the existing method might be to assign a measure of accuracy to each reconstructed point, and only use the most accurate points in the projectivity calculation. The measure might be based on the distance of the point from the camera, or recursively, on the accuracy of the points used when its structure was recovered. A possible cause of structure recovery problems is the initial reliance on ground truth data. Any measurement error in these points would have an adverse effect on the whole reconstruction. An alternative approach would be to estimate the Euclidean structure directly, using a selfcalibration, as in [25].

CHAPTER 6. CONCLUSIONS

105

In its present form the geometric hashing system is is based on invariants of a 3D similarity or rigid transform. However, since we are dealing with rigid structures it is possible to obtain more discriminatory information, for example the size of the triangle formed by the three basis points. This shape signature can be used to ensure that only appropriate bases receive votes and thus reduce the number of candidate matches [70]. Also, at present, there is no indication of when the recognition process should take place. Of course, it is possible to attempt recognition after every reconstruction step and hash table update, but this would be wasteful, since it is expected that previously viewed structure will only rarely be re-encountered. The reconstruction process can provide an estimate for the pose of the camera at each step and this odometry information could be used to trigger recognition when approaching structure that has been seen before. In addition, once a match has been found, there is still the question of how best to update the structure, so that it internally consistent. This is of particular importance if the reconstruction is to be of practical use, for example a virtual reality application, and is certain to be the subject of future research.

6.3 Closing Comments
The problem we have tackled is an exceptionally dif?cult one, and the proposed solution brings together elements from a number of areas of machine vision. To date, we have been unable to demonstrate successful resolution of inconsistencies in the reconstructed model, for reasons discussed in Chapter 5. However, the overall approach proposed remains plausible and it is hoped that future work, including some of the suggestions above, will succeed in producing internally consistent models.

Appendix A

Essential Projective Geometry
In this appendix we review some of the most important concepts of projective geometry, which we have used elsewhere in the thesis. For a more thorough introduction, the reader is referred to [11] or the appendix of [47], which provide excellent discussions of the subject, from a machine vision standpoint.

A.1 Homogeneous Coordinates
In projective geometry manipulation of points, lines, planes etc. is carried out using homogeneous coordinates. Projective transformations are linear in homogeneous coordinates, and some problems can be greatly simpli?ed by expressing them in this manner. Consider the case of perspective projection from 3D to 2D, which is important in machine vision as it represents the formation of an image by a camera. Using Cartesian coordinates this transformation is non-linear, but it is linear in homogeneous coordinates [11]. In two dimensions, the Cartesian coordinates of a point are a 2-vector (x; y )T . The same point in projective two-space, vector

(u; v; w)T . The simplest way to obtain the values of these three elements is to set u = x; v = y and w = 1, thus:

P

2

, can be represented in homogeneous coordinates by some 3-

106

APPENDIX A. ESSENTIAL PROJECTIVE GEOMETRY

107

(x; y )T ! (x; y; 1)T

(A.1)

However, an important property of projective geometry is that only the ratios of the elements of the homogeneous coordinates are important. Hence two homogeneous vectors represent the same homogeneous point if one is a multiple of the other. That is:

(x; y; 1)T
where is some non-zero scalar. For example, represent the same homogeneous point.

( x; y; )T

(A.2)

(12; 10; 2)T ; (30; 25; 5)T and (?6; ?5; ?1)T all

Converting from homogeneous back to Cartesian coordinates is equally straightforward. Simply divide through the homogeneous vector by its third element and then remove the third element (which will of course be 1),

( x; y; )T !

x; y;

T

! (x; y; 1)T = (x; y)T

(A.3)

Or, without the explicit scale factor,

above, the three homogeneous vectors all correspond to the Cartesian point (6; 5)T . An additional bene?t of homogeneous coordinates is that they make it possible to represent points located at in?nity on the image plane. There is no such representation in Cartesian coordinates. In homogeneous coordinates a point at in?nity, called an ideal point has its third element equal to zero i.e. it is of the form (u; v; 0)T . Ideal points are treated in exactly the same way as any other points in the image plane. The set of all ideal and non-ideal points in projective 2-space is called the projective plane and is denoted P 2. The equation of a line in two dimensions can be expressed in Cartesian coordinates as

(u; v; w)T ! (u=w; v=w)T .

Thus, in our example

ax + by + c = 0
which can be rewritten in terms of homogeneous coordinates as

(A.4)

APPENDIX A. ESSENTIAL PROJECTIVE GEOMETRY

108

u v a w + b w + c = 0 $ au + bv


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self-calibration and Euclidean reconstruction from image sequences is presented....This was extended in [6] where both focal lengths and principal points ...
...motion and parameter estimation for image sequences with ....pdf
IEEE Image Processing, pages 1509{1519, Nov. 1995. 3] A. Kokaram and S. Godsill. A system for reconstruction of missing data in image sequences ...
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