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xiao_jing_2004_3 - Non-Rigid Shape and Motion Recovery...

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Non-Rigid Shape and Motion Recovery: Degenerate Deformations Jing Xiao Takeo Kanade The Robotics Institute Carnegie Mellon University Pittsburgh, PA 15213 { jxiao, tk } @cs.cmu.edu Abstract This paper studies the problem of 3 D non-rigid shape and motion recovery from a monocular video sequence, under the degenerate deformations. The shape of a deformable object is regarded as a linear combination of certain shape bases. When the bases are non-degenerate, i.e. of full rank 3 , a closed-form solution exists by enforcing linear con- straints on both the camera rotation and the shape bases [18]. In practice, degenerate deformations occur often, i.e. some bases are of rank 1 or 2 . For example, cars moving or pedestrians walking independently on a straight road refer to rank- 1 deformations of the scene. This paper quantita- tively shows that, when the shape is composed of only rank- 3 and rank- 1 bases, i.e. the 3 D points either are static or independently move along straight lines, the linear rotation and basis constraints are sufficient to achieve a unique solu- tion. When the shape bases contain rank- 2 ones, imposing only the linear constraints results in an ambiguous solution space. In such cases, we propose an alternating linear ap- proach that imposes the positive semi-definite constraint to determine the desired solution in the solution space. The performance of the approach is evaluated quantitatively on synthetic data and qualitatively on real videos. 1. Introduction Recovery of 3 D shape and motion from a monocular video sequence is an important task for applications like human computer interaction and robot navigation. The decades of work has led to significant successes on this problem. When the scene is static, reliable systems exist for 3 D reconstruc- tion of the scene structure. In reality, many scenes are dy- namic and non-rigid: expressive faces, cars moving beside buildings, etc. Such scenes often deform with a class of basis structures. For example, the shape of a face can be regarded as a weighted sum of some shape bases, which correspond to various facial expressions [3]. Bregler and his colleagues [5] first introduced the basis representation to the problem of non-rigid structure from motion. Using this representation, in [18], we presented two sets of linear metric constraints, orthonormality constraints on camera rotations ( rotation constraints ) and uniqueness constraints on shape bases ( basis constraints ). We proved that, when the shape deformation is non-degenerate, i.e. all bases are of full rank 3 , enforcing the linear constraints leads to a closed-form solution [18]. In practice, many scenes deform with degenerate bases of rank 1 or 2 . Such bases limit the shape to deform only in a 2 D plane. For instance, if a scene contains pedestrians walking indepen- dently along straight lines, the bases referring to those rank- 1 translations are degenerate. A simple illustration of rank- 3 , 2 , and 1 bases is shown in Figure 1. Under degenerate deformations, enforcing the linear metric constraints is not necessarily sufficient to determine a unique solution.
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