Sparse Structures in L-Infinity Norm Minimization for Structure and Motion Reconstruction

This paper presents a study on how to numerically solve thefeasibility test problem which is the core of the bisectionalgorithm for minimizing the L ∞ error functions. Weconsider a strategy that minimizes the maximum infeasibility. Theminimization can be performed using several numerical computationmethods, among which the barrier method and the primal-dual methodare examined. In both of the methods, the inequalities aresequentially approximated by log-barrier functions. An initialfeasible solution is found easily by the construction of thefeasibility problem, and Newton-style update computes the optimalsolution iteratively. When we apply the methods to the problem ofestimating the structure and motion, every Newton update requiressolving a very large system of linear equations. We show that thesparse bundle-adjustment technique, previously developed forstructure and motion estimation, can be utilized during the Newtonupdate. In the primal-dual interior-point method, in contrast tothe barrier method, the sparse structure is all destroyed due to anextra constraint introduced for finding an initial solution.However, we show that this problem can be overcome by utilizing thematrix inversion lemma which allows us to exploit the sparsity inthe same manner as in the barrier method. We finally show that thesparsity appears in both of the L ∞ formulations -linear programming and second-order cone programming.