Efficient Multiresolution Counterparts to Variational Methods for Surface Reconstruction

Variational methods have been employed with considerable success in computer vision, particularly for surface reconstruction problems. Formulations of this type require the solution of computationally complex Euler–Lagrange partial differential equations (PDEs) to obtain the desired reconstructions. Further, the calculation of reconstruction error covariances for such approaches are usually neglected. In this paper we describe a computationally efficient multiscale approach to surface reconstruction which differs fundamentally from other multiresolution methods that are used to solve the Euler–Lagrange PDEs. Instead, we interpret the variational problem as a statistical estimation problem in order to define a nearby, but slightlydifferent, multiscale estimation problem that admits efficient solutions for both surface reconstructionandthe calculation of error statistics. In particular, the membrane and thin-plate variational models for surfaces are interpreted as 1/f2prior statistical models for the surface and its gradients, respectively. Such 1/f2behavior is then achieved using a recently introduced class of multiresolution models that admits algorithms with constant per-pixel computational complexity.