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Image Synthesis from Projective Displacement: Application to Image-Based Visual Servoing

JAE SEOK PARK and MYUNG JIN CHUNG Department of Electrical Engineering and Computer Science, Kore

a Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon, KOREA. jspark@cheonji.kaist.ac.kr; mjchung@ee.kaist.ac.kr Abstract: - Projective framework provides useful cues of 3-D structure even without metric information or calibration processes. However, it shows a tendency to suffer from the image noises. In this paper, an application of the projective framework to image syntheses of a 3-D object is introduced with a proposition about a noise resistive estimation of the projective displacement. The synthesized images are used to generate image trajectories to handle the limitations of the image-based visual servoing. Results of the simulation demonstrate its effectiveness. Key-Words: - Projective displacement, image-based visual servoing, image synthesis, uncalibrated stereo rig 1 Introduction Use of projective geometry is spreading its range from a fundamental understanding of projective views to a variety of robotic applications. It gives convenient tools to interpret the geometric structures from the uncalibrated images. Since the metric calibration of a vision system is an awesome job, a number of novel approaches not to involve the metric space have been attempted in performing the visionbased applications. The main purpose of this paper is to introduce an approach to handle the known limitations in image-based visual servoing [1] using projective geometry and to consider some computational issues in the approach. In the image-based visual servoing, in spite of many positive aspects, there are some serious limitations. Since the image Jacobian relates only the tangent space of the image plane with the tangent space of the workspace in a speci?c con?guration, convergence is not guaranteed when the initial pose discrepancy is large. Another serious problem under the large initial pose discrepancy is that the feature points may leave the camera’s ?eld of view [4]. Therefore, we have proposed a novel method to overcome these problems in [5]. The problems have been handled by planning a straight path in the image space such that the current con?guration of the manipulator should be always close to the reference con?guration in any time instant. A number of intermediate views of the robot gripper have been synthesized to construct the image trajectories that allow the robot gripper to track a straight path in the 3-D workspace. The contribution of the work was that the method generates the intermediate views of the 3-D structured object only with the image information within the framework of the projective space. However, the proposed method is likely to fail under the in?uence of relatively large image noises or distortions since the framework of the projective space shows a tendency to suffer from the image noise. In the proposed method, a projective displacement of a rigid body has been estimated using the image correspondence. In this paper, an enhanced estimation is proposed to give a better result generating the image space trajectories of the gripper points. It is assumed that uncalibrated stereo cameras are given as in [5]. The cameras are independently ?xed and observe the gripper attached to a manipulator. It is also assumed that the initial and the goal gripper points are within the camera’s ?eld of view. The remainder of this article is structured as follows. Section 2 gives a brief review to the image space path generation method in [5]. Section 3 considers the computational issues and proposes an enhanced estimation of a projective displacement. Image-based visual servoing with the desired image trajectories is presented in section 4 and the results of computer simulations are presented in section 5 to show the feasibility of the proposed approach. Finally, conclusions are given in section 6 with some directions for the further work.

2 Projective Representation of Intermediate

where

(2)

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where are eigenvectors of . In case that all the eigenvectors are not linearly independent, one of two identical eigenvectors can be replaced with an arbitrary linearly independent vector with respect to the remaining eigenvectors. This modi?cation still ensures that the direction of the screw

Although (3) provides far more reliable solution than any other linear equations, it does not guarantee that the undergoing transformation corresponds to the Euclidean displacement of the gripper. In consequence, the resulting transformation may not have its eigenvalues in the form of , which may cause breaking the process of the proposed algorithm. To avoid this situation, additional constraints have to be considered in the cost function (3). More speci?cally, the eigenvalues of should take the desirable form by such constraints. If we assume that the transformation has the desirable form of eigenvalues as

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Suppose that are projective transformations representing the intermediate screw motions enforced on the initial pose of the gripper. If the rotation angles associated with the intermediate screw motions are interpolated as , the eigenvalues of should be in the form of . To ensure that the screw axis of should be as same as that of , the eigenvectors of also have to be as same as those of . Therefore, can be constructed as

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(1)

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In this section, the process of generating the intermediate poses of a rigid body in the projective space [5] is brie?y reviewed. The process is composed of two stages. First, a number of screw motions are constructed and represented in the form of projective transformations. The rotational angle associated with each screw motion is interpolated from the rotational angle associated with the initial projective displacement. Second, appropriate pure translations are added to the constructed screw motions to allow the virtual objects transformed by the original motions to be in a straight path. If the corresponding image points between two cameras and between the initial and the goal gripper points are found, the initial projective displacement as well as the two 3 4 projection matrices conveniently denoted by and can be obtained [7]. Since the projective displacement is conjugated to a Euclidean displacement, has to have the eigen, even though values in the form of the metric representation of the displacement is unknown. Therefore, the rotation angle about the screw axis is found from as

3 Estimation of Projective Displacement In this section, a robust estimation method for the projective displacement is presented. With at least ?ve feature points on the gripper such that no four of them are linearly dependent, that relates two corresponding sets of projective points can be computed by a simple linear equation [7]. However, the linear method usually causes undesirable results because the projective reconstructions from the true images are too sensitive to noise. In the case that more than ?ve points are available, the estimation of can be enhanced by several optimization methods including the one proposed in [9]. The basic idea in [9] is to compare the projections of the transformed and inversely transformed projective points with the true image points. Suppose that and are true image points of the gripper where the capital letters in the subscripts indicate the left or right camera images and the asterisk symbols indicate is then estimated by the go