Simultaneous Registration and Modeling of Deformable Shapes
Many natural objects vary the shapes as linear combinations of certain bases. The measurement of such deformable shapes is coupling of rigid similarity transformations between the objects and the measuring systems and non-rigid deformations controlled by the linear bases. Thus registration and modeling of deformable shapes are coupled problems, where registration is to compute the rigid transformations and modeling is to construct the linear bases. The previous methods [3, 2] separate the solution into two steps. The first step registers the measurements regarding the shapes as rigid and the deformations as random noise. The second step constructs the linear model using the registered shapes. Since the deformable shapes do not vary randomly but are constrained by the underlying model, such separate steps result in registration biased by nonrigid deformations and shape models involving improper rigid transformations. We for the first time present this bias problem and formulate that, the coupled registration and modeling problems are essentially a single factorization problem and thus require a simultaneous solution. We then propose the Direct Factorization method that extends a structure from motion method [16]. It yields a linear closedform solution that simultaneously registers the deformable shapes at arbitrary dimensions (2D \to 2D, 3D \to 3D, . . .) and constructs the linear bases. The accuracy and robustness of the proposed approach are demonstrated quantitatively on synthetic data and qualitatively on real shapes.