Imposing Hard Constraints on Deformable Models through Optimization in Orthogonal Subspaces

An approach is presented for imposing generic hard constraints on deformable models at a low computational cost, while preserving the good convergence properties of snake-like models. We believe this capability to be essential not only for the accurate modeling of individual objects that obey known geometric and semantic constraints but also for the consistent modeling of sets of objects. Many of the approaches to this problem that have appeared in the vision literature rely on adding penalty terms to the objective functions. They rapidly become intractable when the number of constraints increases. Applied mathematicians have developed powerful constrainted optimization algorithms that, in theory, can address this problem. However, these algorithms typically do not take advantage of the specific properties of snakes. We have therefore designed a new algorithm that is closely related to Lagrangian methods but is tailored to accommodate the particular brand of deformable models used in the image understanding community. We demonstrate the validity of our approach first in two dimensions using synthetic images and then in three dimensions using real aerial images to simultaneously model terrain, roads, and ridgelines under consistency constraints.