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Shape Reconstruction from Multiple Images of the Ocean Surface

Howard Schultz Department of Computer Science University of Massachusetts, Amherst May 20, 1994

Abstract

In many remote sensing applications the detailed shape of a water surface must be determined instantaneously, and under natural lighting conditions. Because of their nondestructive measurement capabilities, stereo image analysis methods seem to be ideally suited for this application. Most of these techniques, however, assume surface re ectance properties which are incompatible with the re ectance properties of water, especially for centimeter and smaller wave lengths. To analyze the ne scale structure of the ocean surface, a new technique (known as Specular Surface Stereo) is proposed that makes use of the unique optical properties of water. In this method, the surface shape is recovered by solving the inverse problem of nding a surface shape (de ned by an elevation and gradient map) that results in a set of synthetic (computer generated) images that closely match a set of observed images. To accomplish this, the

1

algorithm requires an image of the illumination source, three images of the water surface, a few initial elevation estimates, and an image synthesis model that predicts the appearance of an image of the surface. The performance of the Specular Surface Stereo technique was tested by processing simulated data. Based on these numerical results, it is shown that the Specular Surface Stereo technique can potentially recover both surface elevations and gradients over a wide range of wavenumbers. Index Terms: Optical remote sensing, Stereo vision, small scale wave measurements, nondestructive measurements, shape from shading

1 Introduction

Microwave scatter from the ocean is highly sensitive to the small scale structure of the surface, which in turn is modulated by several geophysical processes. Microwave measurements, therefore, are indirectly in uenced by environmental parameters such as atmospheric temperature and pressure, wind speed and direction, sea state and ocean currents. This sensitivity to geophysical processes forms the basis for microwave remote sensing. Interpretation of microwave signatures, therefore, requires an understanding of electromagnetic scattering from rough surfaces, as well as the hydrodynamic processes that in uence the ocean surface. Consequently, many remote sensing research programs have focused on EM scattering theories that predict microwave scatter from the ocean surface, and hydrodynamic theories (air-sea, wave-wave and wave-current) that predict the response of the ocean surface to geophysical forces. Implementation and veri cation of many of these theories requires detailed shape information. For example, Chen (1990 and 1992b) and Nickolaev (1992) , have shown 2

the importance of higher order surface statistics by deriving a relationship between the upwind/downwind asymmetry in the radar cross section of the surface and non-Gaussian terms in the distribution of surface elevations and slopes. Many of these higher order descriptions of the ocean surface require joint elevation{slope maps of the surface over a dense grid of at least 512 512 samples. Because of their nondestructive capabilities, optical remote sensing techniques are ideally suited for this application. Several investigators have developed active optical systems speci cally designed to recover shape information from a smooth water surface. Keller and Gotwols (1983) , and Jahne and Riener (1990) retrieved two-dimensional spectra using a single camera located above the surface and a one-dimensional graduated light source located below the surface. The instantaneous surface slopes in the direction of the light gradient are then inferred from the observed image grey levels and temporal resolution is achieved by analyzing an image sequence. The advantage of these light gradient methods are that the data analysis is simple (and can be performed in real-time) and the techniques can be readily extended to two spatial dimensions. Although these methods have be very successful in the laboratory, interference from ambient light, and the need for a shallow submerged light source limit the number of operational scenarios in the eld. In another approach, Wass and Jahne (1992) made combined slope-height measurements using a Re ective Stereo Slope Gauge (RSSG). This instrument consists of a pair of point light sources and video cameras housed in a single package suspended over the surface. Linear polarization lters are placed in front of the lights and cameras so that only specular re ections from the left light are imaged by the right camera, and only specular re ections from the right light are imaged by the left camera. The apparent displacement of specular re ections between the 3

left and right images (parallax) are proportional to the elevation of the surface facet where the re ection occurred. By computing the parallax for all specular re ections the position and slope of specular points on the surface are found. Because a bright light source is used and the instrument package does not cross the air-sea interface, the instrument is very rugged and can be operated under any lighting conditions, day or night. Although this type of joint height-slope observations adds signi cantly to the understanding of the dynamics of the ocean surface, it is not possible to reconstruct the shape or retrieve high order statistical description of the ocean surface from the data. In another approach Bock and Hara (1992) , Martinsen and Bock(1992) and Lee et al. (1992) employed a scanning laser slope gauge in which a scanning laser beam is located below the surface, and a receiver is located above the surface. Then by recording the instantaneous position of the transmitted and received beams, a two-dimensional slope map of the surface is reconstructed. These instruments have been successfully deployed in the laboratory and eld, and are capable of recovering signi cant statistical information about dynamics of the ocean surface. Although these devices are capable of good temporal resolution, frame rates in excess 30 Hz, they have limited spatial resolution generally less than 100 100 samples. Furthermore, as in the light gradient method, these systems cross the air-sea interface and may disturb the surface. To design a system capable of recovering instantaneous joint elevation-slope maps over a 512 512 grid, while not disturbing the air-surface interface, a variety of passive image analysis techniques were investigated. Only methods that could be adapted to analyze a water surface where considered. In the millimeter to centimeter length scales, a water surface displays specular re ectance properties. That is, at the air-sea boundary, light rays 4

are re ected back into the atmosphere or refracted into the water volume, and multiple re ections and scattering at the surface can be neglected. Consequently, the water surface acts like a partially silvered mirror the re ects the sky brightness into the camera. Most image analysis techniques designed to recover shape information from images (feature matching, photometric stereo, and shape-from-shading) assume Lambertian surface re ectance properties. For a Lambertian type surfaces the apparent brightness of a surface point is strongly dependent on the position of the light source relative to the surface, and weakly dependent on the position of the observer. Feature matching algorithms take advantage of this re ectance property by assuming that small surface features can be identi ed in images taken from di erent positions Horn, 1986] Marr and Poggio, 1976]. Similarly, photometric stereo Horn, 1986] Woodham, 1978] and shape-from-shading algorithms Hartt and Carlotto, 1989] Horn and Brooks, 1989] assume that the pixel brightness depends only on the position of the light source relative to the surface. Several researchers have extended the photometric stereo technique to specular surfaces Blake and Brelsta , 1988] Coleman and Jain, 1982] Ikeuchi, 1981] Tagare and deFigueiredo, 1989]. These methods, however, require a single camera viewing a stationary surface illuminated by a sequence of extended light sources. Shape-from-specularity techniques Healey and Benford, 1987] have successfully recovered shape information from specular surfaces by analyzing the re ections produced by a small number of point light sources Blake and Brelsta , 1988] Blake, 1985] Sanderson et al., 1988] Zisserman et al., 1989]. These methods cannot be extended to natural lighting conditions because it is not possible to isolate individual specular re ection. Optical ow techniques have been employed, recently, to recover the instantaneous shape of a water surface by analyzing the apparent motion of a submerged pattern Murase, 1992]. 5

This technique is limited, however, to gradually undulating surfaces where the maximum slope does not exceed 5 . Because of these restrictions, none of the above methods can be readily adapted to recovering the shape of a moving specular surface (such as water waves) under natural lighting conditions. Thomas et al. (1991) and Hartt et al. (1989) employed a multiple image shape-fromshading technique to recover the shape of a surface with di use re ectance properties. The technique nds a surface elevation map z(x; y) that minimizes a cost function U (z),

" X

U (z(x; y)) =

Z Z

2 ~ Ik (x; y) ? Ik (x; y) +

@z(x; y)2 + 2 @z(x; y)2 + @z(x; y)2 dxdy @x@x @x@y @y@y (1)

!#

~ where z(x; y) is the elevation map, Ik (x; y) and Ik(x; y) are the observed and predicted (or synthetic) pixel intensities in the kth image of the surface and is a constant. The rst term

2 ~ Ik ? Ik is small when the predicted pixel grey levels are close to the observed one, and the

second term,

@z2 @z2 @x@x + 2 @x@y

@z2 + @y@y , is small when the surface is smooth. The second term

acts as a penalty function that limits the roughness of the surface. The purpose of this term is to ensure that the retrieved surface is smooth, resolve grey level ambiguities (cases where more than one elevation results in the same set of pixel intensities), and to ll in regions where either the surface is occluded or the image synthesis model cannot be used. Because of the general form of this algorithm, which includes a mechanism for assimilating images taken from di erent viewing positions, the specular surface stereo algorithm is based on the multiple image shape-from-shading approach. This technique is adapted to analyze specular surface by incorporating a specular surface image synthesis model instead of a di use surface 6

image synthesis model Schultz, 1992] Schultz, 1994]. Furthermore, to reduce the number of hardware dependent operations, units of irradiance are used instead of pixel intensities. To further simplify the algorithm development is set to zero. In addition to removing any smoothness constraints, neglecting the second term in Equation 1 reduces the retrieval process to a point-by-point scheme. Thus, the cost function is minimized by minimizing the di erence term at each node in the retrieval grid.

2 The Imaging Environment

Before constructing an algorithm, the viewing geometry and image synthesis model must be speci ed. For the purpose of describing the specular surface stereo method, an idealized camera, surface, and illumination models are employed.

2.1 Viewing Geometry

The viewing geometry shown in Figure 1 depicts an ideal camera with a focal length f , viewing a surface described by an elevation map z(x; y) that has continuous derivatives

p(x; y) and q(x; y) with respect to x and y. The camera focal point is located at P , and O

is a point on the surface. The observation vector r is a unit vector that points upward from ^

O to P , the surface normal vector s is perpendicular to the surface at O, and the incidence ^

^ ^ vector t is the re ection of r about s. From Snell's law, t is related to r and s by, ^ ^ ^ ^ ^ ^ s s ^ t = ?r + 2^ (^ r) = ?r + 2^ cos( ) ^ s (2)

where the observation angle (the angle between r and s ) is equal to the incidence angle (the ^ ^ 7

^ angle between s and t). The extension of the observation vector r intersects the image plane ^ ^ of camera k at pixel ij with image space coordinates uijk = (xijk ; yijk ; ?fk ). The object space coordinates of O are related to the image space coordinates uijk by the projective transformation Slama, 1980],

O = Pk +

ijk Mk uijk

(3)

where Pk is the position of the focal point of camera k, Mk is the rotation of the image plane coordinate system of camera k relative to the object space coordinate system, and

ijk

is a

scale factor for pixel ijk. Since the incidence angle is completely speci ed by the coordinates of the pixel ijk that views O, and the orientation (p; q) at O, the incidence angle can be written as

ijk (p; q ).

z' P f (i, j) y' x' Image Plane ^ s θ O θ ^ t Y

Z

^ r

X

Figure 1: Viewing geometry: The focal point of an ideal camera with focal length f is located at P ; the observation point on the surface is located at O; r is the unit vector that points ^ ^ from O to P ; s is the surface normal unit vector; t is the re ection of r about s ^ ^ ^ 8

2.2 Image Synthesis

Once the coordinates of the pixel that views a point on the surface is determined, the irradiance at that pixel must be determined. That is, we need an image synthesis model

Eijk (p; q) that predicts the irradiance at the pixel ijk as a function of the surface orientation

(p; q). The specular surface stereo technique requires only that the image synthesis model predict accurately the irradiance at each pixel, and that the pixel irradiance depends on (p; q). To illustrate the technique, an idealized image synthesis model will be employed that imposes the following conditions on the surface and illumination: 1. The radius of curvature of the surface patch within the pixel eld-of-view is large compared to the dimensions of the patch. Thus, the surface is modeled by a mosaic of small, partially silvered, at mirrors. 2. The illumination originates from a distant, extended source, such that the illumination ^ radiance L(pt; qt) depends only on the orientation (pt; qt) of the incidence vector t. 3. The light incident on the surface is unpolarized. 4. The re ecting material is a homogeneous dielectric. 5. No explicit restrictions are placed on the surface slopes, however, to ensure that most of the incidence vectors come from the sky, the maximum surface slope should not exceed 45 . When light is incident on the surface of a smooth dielectric media some of the light is re ected from the surface and some is refracted through the surface. The Fresnel re ection 9

coe cient for a dielectric media gives the fraction of radiant energy re ected at the surface as a function of the polarization relative to the plane of incidence, the real part of the index of refraction n, and the incidence angle of the incident light are given by, sin2( ? ) Rk( ) = sin2( + 1) ; 1

2 ) R? ( ) = tan2( ? 1) tan ( + 1

Born and Wolf, 1980] Wol and Boult, 1991]. The

re ection coe cients for the parallel Rk( ) and perpendicularly R? ( ) polarized components

(4)

where

1

= sin?1

sin( ) n

!

is the angle of refraction. For unpolarized light, the parallel and perpendicularly polarized components have equal intensities. The re ection coe cient R( ) for unpolarized incident light, therefore, is given by 1 1 R( ) = 2 Rk + 2 R? : (5)

Since depends on the observation vector and the surface normal, the re ection coe cient can be expresses as a function of the pixel coordinates and surface gradient Rijk (p; q). Likewise, the orientation of the incidence vector can be expressed in terms of (ijk) and ^ (p; q), therefore the radiance directed along t can be written as Lijk (p; q). Provided that the radius of curvature at the surface point is large compared to the surface area within the pixel eld-of-view (see condition 1 above), the radiance re ected into the eld-of-view of the pixel ijk is the product of the radiance Lijk (p; q) and the re ection coe cient Rijk (p; q). The 10

predicted irradiance Eijk (p; q) is the product of the radiance directed along the observation vector and a constant Cijk Horn, 1986] which is a function of the optical properties of the camera, such as the focal length and aperture. The image synthesis model, therefore can be written as,

Eijk (p; q) = Cijk Lijk (p; q)Rijk (p; q):

(6)

The calibration constants Cijk are measured once for each imaging device, the radiance distribution Lijk (p; q) must be measured every time the illumination source changes, and

Rijk (p; q) is a known function of the electrical properties of the water.

3 The Specular Surface Stereo (S3) Algorithm

Starting with at least one known elevation, an iterative process is used to propagate the surface shape. Each iteration is comprised of two steps|surface orientation and surface propagation. For a point on the surface with a known elevation z, the surface orientation algorithm computes the orientation of the surface normal (p; q). Next, the surface propagation algorithm computes the elevations at surrounding points. The process is repeated until all surface elevations and gradients within the retrieval grid are known. In Sections 3.1 and 3.2, the surface orientation and surface propagation procedures are discussed in detail. In Section 3.3, a method is presented for implementing the specular surface stereo technique on a two-dimensional grid. In Section 4, the technique is illustrated with numerical simulations.

11

3.1 Surface Orientation

The surface orientation procedure nds the orientation (p; q) of a point on the surface given ~ the elevation z of the point, N images of the surface Eijk ; k = 1; ; N (which have been converted to units of irradiance), and a radiance map of the illumination L(pt; qt). The rst step is to identify the pixels that view the surface point, by evaluating the inverse of Equation 3 for each camera. Next, a value of (p; q) is found such that the predicted irradiance ~ at these pixels Eijk (p; q); k = 1; ; N match the observed values Eijk ; k = 1; ; N . Although several methods exist for obtaining a match, a weighted least squares objective function of the form

J=

X

N

k=1

~ wijk Eijk ? Eijk (p; q)

2

(7)

is used to illustrate the procedure. The objective function is further simpli ed by setting all weights to unity, wijk = 1 for all ijk. In general, however, wijk may depend on variables such as the signal-to-noise ratio. A simple linear search is employed to minimize J with respect to (p; q). Although several methods exist for performing this minimization, a simple linear search was employed to avoid false solutions that may arise from local minima in (p; q) space, and to facilitate implementing the algorithm on a massively parallel computer. The search was done by dividing the solution space into equally spaced bins, evaluating J at each bin, and selecting the value of (p; q) at the center of the bin where J is a minimum. If a point on the surface is not visible in one or more images, or the image synthesis model does not apply (for example, if the point lies on a rough spot), or the orientation of the incidence vector points outside the domain of L(pt; qt), then the surface orientation 12

algorithm will not predict correctly the surface orientation. Nodes were failures occur are identi ed by comparing the minimum residual of J to a rejection threshold. If the minimum residual is above the rejection threshold, the point is marked as missing and excluded from further analysis.

3.2 Surface Propagation

If at least one surface elevation is known anywhere on the surface, it is possible to recover the surface shape by propagating a solution. Consider a surface point with horizontal coordinates (x0; y0) and a known elevation and gradient (z0; p0; q0). Denoting the horizontal coordinates of a nearby point by x1 = x0 + x and y1 = y0 + y, the elevation z1 is found by evaluating the truncated Taylor series,

z1 = z0 + p0 x + q0 y:

(8)

Initial seed elevations may be obtained in a variety of ways. If, for example, features with Lambertian re ectance properties, such as bubbles or foam, are on the surface, conventional stereo matching techniques can be used to calculate the elevations of these non-specular surface features. If the surface is predominately smooth, electro-optical devices such as the Re ective Stereo Slope Gauge Wass and Jahne, 1992] can be used to obtain seed elevations and slopes. Furthermore, under some circumstances seed orientations may be obtained. On a clear day, for example, the orientation of any pixel where a sun glint is visible is known to within 0:25 (half the angular extent of the sun).

13

3.3 Implementation

On a two-dimensional grid, the specular surface stereo technique is implemented as a dilation lter. That is, the surface grows around any surface point where the elevation is known. At each iteration, a 3 3 window is passed over the retrieval grid, and the action taken at the window center depends on the state of the nodes that lie within the window. De ning a complete node as any node where (z; p; q) are known, the possible actions taken at iteration

m and node ij are:

1. If (z; p; q)m?1 are known, then set (z; p; q)m = (z; p; q)m?1. ij ij ij

m m m 2. If zij ?1 is known and (p; q)m?1 are missing, then set zij = zij ?1 and use the surface ij

orientation algorithm to compute (p; q)m. ij

m 3. If zij ?1 is missing and (p; q)m?1 are known, and there are no complete surrounding ij

nodes, then set (z; p; q)m = (z; p; q)m?1. ij ij

m 4. If zij ?1 is missing and (p; q)m?1 are known, and there are at least nc complete surij

rounding nodes, then for each complete node use the surface propagation algorithm

m to estimate the elevation at the window center, set zij equal to the average elevation

estimate, and set (p; q)m = (p; q)m?1. ij ij 5. If (z; p; q)m?1 are missing, and there are no complete surrounding nodes, then set ij (z; p; q)m = (z; p; q)m?1. ij ij 6. If (z; p; q)m?1 are missing, and there are at least nc complete surrounding nodes, then ij for each complete node use the surface propagation algorithm to estimate the elevation 14

m at the window center, set zij equal to the average elevation estimate, and use the surface

orientation algorithm to compute (p; q)m. ij Initially nc is set to two, thus, at least two complete nodes are required to estimate the elevation at the window center. By requiring a minimum of two complete nodes instead of one, numerical errors grow more slowly as the surfaces propagates away from the seed point. If no new elevation are computed after an iteration, then nc is reduced to one, and a new iteration is started. If, again, no new elevations are computed with nc = 1, then the procedure has converged. Otherwise, if at least one new elevation was computed, nc is reset to two, and a new iteration is started. The procedure converges when no new elevations are computed, or the maximum number of iterations is exceeded. Since only two out of the eight surrounding nodes must be complete to estimate the elevation at the window center, solutions can grow around regions of bad or missing data. If, for example, a small region on the surface is occluded in some of the images, no elevations or gradients will be recovered in the occluded region. Along the edges of the occluded region, however, where at least two complete nodes are found, the algorithm will successfully compute (z; p; q).

4 Numerical Simulations

The S3 technique was tested by processing numerical simulations. For each simulation, the sky radiance map, surface elevation map, and camera positions were speci ed, and a set of three synthetic images of the surface generated. Then from the synthetic images, the sky radiance map and a few seed elevations, the surface elevation and gradient maps were 15

retrieved using the S3 technique and compared to the original ones. The same sky radiance map and camera placement were used for all simulations. The focal length of each lens was 170mm, the focal points were located on the vertices of a 3 3 3 meter equilateral triangle elevated 10m over the mean surface, and the cameras were oriented so that the optic axes passed through the center of the retrieval grid. The surface elevation maps were de ned on a 512 512 grid in object space that covered a horizontal area of 1m 1m. The radiance map

L(p; q) was de ned on a 512 512 grid in gradient space where ?4 p 4 and ?4 q 4.

~ The computer generated images Eijk , k=1,2,3 were de ned on 512 512 pixel grid in the image coordinate system of each camera and covered an area of 17.4mm simulated elevation map and three synthesized images are shown in Figure 3. 17.4mm. The camera placement is shown in Figure 2, and the sky radiance map along with a sample

16

z

3m

10 m

y

x 1m

Figure 2: Camera placement for the numerical simulations The simulated elevation maps were produced by generating a two-dimensional sequence of Fourier coe cients in wavenumber space (kx; ky ) with a known amplitude spectrum and random phases, taking an inverse Fourier transform, and storing the results in a two-dimensional array z(x; y). To study the relationship between spectral shape and retrieval accuracy three spectral forms were used. Form 1 had no dependency on the wind direction,

jH j = 0:0007Uk?4 ; jH j = 0;

17

for k 1:0cm?1 for

k > 1:0cm?1

(9)

Sky

4 2 0 -2 -4 -4 -2 0 p 2

8.7

Surface

0.5

y [m]

0.0

q

4

-0.5 -0.5

Image 1

0.0 x [m]

0.5

y [mm]

0.0

-8.7 -8.7

0.0 x [mm] 8.7

8.7

Image 2

8.7

Image 3

y [mm]

0.0

y [mm]

0.0

-8.7 -8.7

0.0 x [mm]

8.7

-8.7 -8.7

0.0 x [mm]

8.7

Figure 3: The sky radiance map (used for all simulations), a sample surface elevation map, and a set of three synthesized images. For the sky radiance map, light pixels represent higher radiance values and dark pixels represent lower radiance values. For the surface elevation map, light pixels represent higher elevations and dark pixels represent lower elevations form 2 had a moderate dependency on the wind direction

jH j = 0:0007Uk?4 0:5 + 0:5 cos2( ? 0)] ; jH j = 0;

and form 3 had a strong dependency on the wind direction

for k 1:0cm?1 for

k > 1:0cm?1

(10)

18

jH j = 0:0007Uk?4 cos2( ? 0); jH j = 0;

where

q

for k 1:0cm?1 for

k > 1:0cm?1

h

(11)

ky i kx

0

2 2 is the wind direction, k = kx + ky is the wavenumber, and = tan?1

is the

wave direction. In all cases, the wind speed U was set to 10 m/sec and the wind direction

0

was set to 0 . The shapes of the elevation and slope spectra for each form are shown in

Figure 4, and sample elevation maps generated using spectral forms 1, 2 and 3 are shown in Figure 5.

Form 1

128 64 128 64 0 -64 0 64 128 -128 -128 -64 128 64 0 -64 0 64 128 -128 -128 -64 128 64 0 -64 0 64 128 -128 -128 -64 0 64 128 0 64 128 0 64 128

Form 2

128 64 0 -64

Form 3

z

0 -64 -128 -128 -64 128 64

-128 -128 -64 128 64 0 -64 -128 -128 -64 128 64 0 -64 -128 -128 -64

0

64 128

p

0 -64 -128 -128 -64 128 64

0

64 128

q

0 -64 -128 -128 -64

0

64 128

Figure 4: Contour maps of the elevation and slope wavenumber spectra for forms 1, 2 and 3 To achieve statistically reliability twenty realizations of each spectral form were produced and analyzed. For each simulated surface, three synthetic images were generated and 16 19

Form 1

Form 2

Form 3

Figure 5: Surface realizations of spectral forms 1, 2 and 3. Higher elevations are depicted by lighter shades and lower elevations are depicted by darker shades. equally spaced seed elevations were selected. The surface elevation and gradient maps were then retrieved using the S3 technique. The residual threshold, used by the surface orientation algorithm, was set so that approximately 85 percent of the retrieved points had a residual below the threshold. In other words, after convergence approximately 15 percent of the nodes in the retrieval grid were classi ed as missing. Afterwards missing points in the elevation and gradient maps were lled in using a median replacement lter, which replaced missing data with the median of the surrounding non-missing data. To illustrate this procedure Figure 6 shows an example of an input (simulated) and retrieved elevation and gradient maps, as well as the di erence between the input and retrieved maps. The accuracy of the retrieval was tested by computing the coherency spectrum between the known (simulated) and recovered elevation and slope maps. The coherency spectrum ?z (kx ; ky ) for the elevation maps is de ned by

N ?1 X

?z (kx; ky ) = v N ?1 u

u X t

m=0

~ Zm Zm

N ?1 X m=0

m=0

~ ~ Zm Zm

ZmZm

(12)

20

Input

Retrieved

Error

z

p

q

Figure 6: Sample input and retrieved (z; p; q) maps as well as the di erence between the input and retrieved maps. For the error maps low errors are shown in lighter shades and higher errors are shown in dark shades. where N is the number of simulations, zm(x; y) is the m'th simulated elevation map, zm(x; y) ~ ~ is the m'th retrieved elevation map, w(x; y) is a spectral window, and Zm and Zm are the discrete Fourier transforms of zm(x; y) w(x; y) and zm(x; y) w(x; y) respectively. Simi~ lar expression exist for the slope coherency spectra ?p (kx; ky ) and ?q (kx ; ky ). A coherency spectrum may be interpreted as the correlation coe cient for each wavenumber, between the input spectrum (derived from the simulated surface) and the output spectrum (derived from the retrieved surface). A signi cance test may be applied to the coe cients ?z (kx; ky ) 21

to determine which spectral components are correlated. In the above examples, ?z (kx; ky ) was computed using 20 degrees of freedom, which has a minimum signi cance value of 0.28 Jenkins and Watts, 1968] Julian, 1975]. Thus, when ?z (kx ; ky ) 0:28 the input and output spectral amplitudes at (kx ; ky ) are correlated.

Form 1

128 64 128 64 0 -64 -128 -128 -64 128 64 0 -64 -128 -128 -64 128 64 0 -64 -128 -128 -64

Form 2

128 64 0 -64

Form 3

z

0 -64 -128 -128 -64 128 64

0

64 128

0

64 128

-128 -128 -64 128 64 0 -64 -128 -128 -64 128 64 0 -64 -128 -128 -64

0

64 128

p

0 -64 -128 -128 -64 128 64

0

64 128

0

64 128

0

64 128

q

0 -64 -128 -128 -64

0

64 128

0

64 128

0

64 128

Coherency Squared

0.0 0.2 0.4 0.6 0.8 1.0

Figure 7: Coherency spectra between the simulated and retrieved elevation and slope maps for spectral forms 1, 2 and 3. The coherency spectra for spectral forms 1, 2 and 3 are shown in Figure 7. When the variance, or spectral amplitude, associated with a particular wavenumber is small, the variability of the spectral amplitude become large, which results in a poor correlation (low coherency) between the input and retrieved elevation and slope maps. Thus, when the 22

spectral amplitudes are small it becomes harder to reject the null hypothesis|to determine if two spectral amplitudes are correlated. Inspection of Figures 4 and 7 clearly shows this relationship between spectral shape and coherency. It also is apparent from Figure 7 that the S3 technique recovers slopes better than elevations. This is not unexpected because the appearance of an image of a water surface is primarily a function of the surface slopes.

5 Discussion and Conclusions

The specular surface stereo technique is speci cally designed to retrieve the shape of a water surface under natural lighting conditions. The technique consists of two parts|a surface orientation procedure, that computes the gradient at a point on the surface by minimizing the di erence between observed and synthesized irradiance values, and a surface propagation procedure. The surface orientation procedure incorporates an image synthesis model, that predicts the irradiance at each image pixel as a function of the source radiance, re ectance properties of the surface, and the optical properties of the imaging device. There are four general requirements for using the specular surface stereo method, (1) the existence of an image synthesis model where the brightness of the surface depends on the orientation of the surface facets, (2) a radiance map of the illumination source, (3) a su cient number of images of surface so that the surface orientation procedure usually has a unique solution, and (4) a few seed elevation estimates. To test the feasibility the specular surface stereo technique, an idealized surface, radiance map and camera were speci ed, and three images of the surface were synthesized. Then from the radiance map, synthesized images and sixteen seed elevation estimates, the surface 23

elevations and gradients were retrieved. Under these controlled circumstances, the results demonstrated the feasibility of the method. When applied to real world situations, additional factors must be included in the image synthesis and illumination models. In a eld environment, for example, sun glitter, sky polarization, rough surface features, nite radius of curvature, and the absorption and multiple scattering of light within the water volume must be included in the models.

6 Acknowledgements

This research was sponsored by a grant (N00014-89-J-3229) from the O ce of Naval Research.

References

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Fig. 1. Viewing geometry. The focal point of an idea camera with focal length f is located at P ; the observation point on the surface is located at O; r is the unit vector that ^ ^ points from O to P ; s is the surface normal unit vector; t is the re ection of r about s. ^ ^ ^ Fig. 2. Camera placement for the numerical simulations. Fig. 3. The sky radiance map (used for all simulations), a sample surface elevation map, and a set of three synthesized images. For the sky radiance map, light pixels represent higher radiance values and dark pixels represent lower radiance values. For the surface elevation map, light pixels represent higher elevations and dark pixels represent lower elevations. Fig. 4. Contour maps of the elevation and slope wavenumber spectra for forms 1, 2 and 3. Fig. 5. Surface realizations of spectral forms 1, 2 and 3. Higher elevations are depicted by lighter shades and lower elevations are depicted by darker shades. Fig. 6. Sample input and retrieved (z; p; q) maps as well as the di erence between the input and retrieved maps. For the error maps low errors are shown in lighter shades and higher errors are shown in dark shades. Fig. 7. Coherency spectra between the simulated and retrieved elevation and slope maps for spectral forms 1, 2 and 3.

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