Adaptive Multiscale Discretizations for Vision

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 is seldom wellmatched to the complexity of the solution or the resolution power of the model. To address this issue, we introduce multiresolution discretizations that locally adapt the resolution of a piecewise polynomial solution to the resolving power of the variational model or complexity of its solution. Besides classic triangular finite elements, we investigate quadand octree element grids that efficiently locate pixels and produce multiresolution representations of the solution. We combine our discretization with the optimization algorithms of finite differences to solve the nondifferentiable functionals that characterize vision and develop algorithms easy o parallelize. Our 2 and 3D experiments in image segmentation, optical flow, stereo, and depth fusion validate our method as achieving significant computational savings with a minimal loss of accuracy.