Kernel non-rigid structure from motion

Non-rigid structure from motion (NRSFM) is a difficult, underconstrained problem in computer vision. The standard 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 applied in standard NRSFM. As a result, we model complex, 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 kernel trick, our model complements the low-rank constraint by capturing non-linear relationships in the shape coefficients of the linear model. The net effect can be seen as using non-linear dimensionality reduction to further compress the (shape) space of possible solutions.