Towards Discrete Solution: A Sparse Preserving Method for Correspondence Problem

Many problems of interest in computer vision can be formulated as a problem of finding consistent correspondences between two feature sets. Feature correspondence (matching) problem with one-to-one mapping constraint is usually formulated as an Integral Quadratic Programming (IQP) problem with permutation (or orthogonal) constraint. Since it is NP-hard, relaxation models are required. One main challenge for optimizing IQP matching problem is how to incorporate the discrete one-to-one mapping (permutation) constraint in its quadratic objective optimization. In this paper, we present a new relaxation model, called Sparse Constraint Preserving Matching (SPM), for IQP matching problem. SPM is motivated by our observation that the discrete permutation constraint can be well encoded via a sparse constraint. Comparing with traditional relaxation models, SPM can incorporate the discrete one-to-one mapping constraint straightly via a sparse constraint and thus provides a tighter relaxation for original IQP matching problem. A simple yet effective update algorithm has been derived to solve the proposed SPM model. Experimental results on several feature matching tasks demonstrate the effectiveness and efficiency of SPM method.