Kernel non-rigid structure from motion

Non-rigid structure from motion (NRSFM) is a difficult, underconstrained problem in computer vision. The stan-dard approach in NRSFM constrains 3D shape deformation using a linear combination of K basis shapes; the solution is then obtained as the low-rank factorization of an input observation matrix. An important but overlooked problem with this approach is that non-linear deformations are often observed; these deformations lead to a weakened low-rank constraint due to the need to use additional basis shapes to linearly model points that move along curves. Here, we demonstrate how the kernel trick can be ap-plied in standard NRSFM. As a result, we model com-plex, deformable 3D shapes as the outputs of a non-linear mapping whose inputs are points within a low-dimensional shape space. This approach is flexible and can use different kernels to build different non-linear models. Using the ker-nel trick, our model complements the low-rank constraint by capturing non-linear relationships in the shape coeffi-cients of the linear model. The net effect can be seen as using non-linear dimensionality reduction to further com-press the (shape) space of possible solutions. 1