Efficient recovery of low-dimensional structure from high-dimensional data

Many modeling tasks in computer vision, e.g. structure from motion, shape/reflectance from shading, filter synthesis have a low-dimensional intrinsic structure even though the dimension of the input data can be relatively large. We propose a simple but surprisingly effective iterative randomized algorithm that drastically cuts down the time required for recovering the intrinsic structure. The computational cost depends only on the intrinsic dimension of the structure of the task. It is based on the recently proposed Cascade Basis Reduction (CBR) algorithm that was developed in the context of steerable filters. A key feature of our algorithm compared with CBR is that an arbitrary a priori basis for the task is not required. This allows us to extend the applicability of the algorithm to tasks beyond steerable filters such as structure from motion. We prove the convergence for the new algorithm. In practice the new algorithm is much faster than CBR for the same modeling error. We demonstrate this speed-up for the construction of a steerable basis for Gabor filters. We also demonstrate the generality of the new algorithm by applying it to to an example from structure from motion without missing data.