Deterministically Maximizing Feasible Subsystem for Robust Model Fitting with Unit Norm Constraint

Abstract

Many computer vision problems can be accounted for or properly approximated by linearity, and the robust model fitting (parameter estimation) problem in presence of outliers is actually to find the Maximum Feasible Subsystem (MaxFS) of a set of infeasible linear constraints. We propose a deterministic branch and bound method to solve the MaxFS problem with guaranteed global optimality. It can be used in a wide class of computer vision problems, in which the model variables are subject to the unit norm constraint. In contrast to the convex and concave relaxations in existing works, we introduce a piecewise linear relaxation to build very tight under- and over-estimators for square terms by partitioning variable bounds into smaller segments. Based on this novel relaxation technique, our branch and bound method can converge in a few iterations. For homogeneous linear systems, which correspond to some quasi-convex problems based on L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -norm, our method is non-iterative and certainly reaches the globally optimal solution at the root node by partitioning each variable range into two segments with equal length. Throughout this work, we rely on the so-called Big-M method, and successfully avoid potential numerical problems by exploiting proper parametrization and problem structure. Experimental results demonstrate the stability and efficiency of our proposed method.