Adaptive discretizations for non-smooth variational vision

Abstract

Variational problems in vision are solved numerically on the pixel lattice because it provides the simplest computational grid to discretize the input images, even though a uniform grid seldom matches the complexity of the solution. To adapt the complexity of the discretization to the solution, it is necessary to adopt finite-element techniques that match the resolution of piecewise polynomial bases to the resolving power of the variational model, but such techniques have been overlooked for nonsmooth variational models. To address this issue, we investigate the pros and cons of finite-element discretizations for nonsmooth variational problems in vision, their multiresolution properties, and the optimization algorithms to solve them. Our 2 and 3D experiments in image segmentation, optical flow, stereo, and depth fusion reveal the conditions where finite-element can outperform finite-difference discretizations by achieving significant computational savings with a minor loss of accuracy.