Quadratic Optimization and Contour Grouping

Abstract

This chapter presents a new and exciting application of quadratic optimization methods to the problem of contour grouping in computer vision. It turns out that this problem leads to finding the local maxima of a Hermitian matrix depending on a parameter. We are thus led to the problem of finding the derivative of an eigenvalue and the derivative of some eigenvector associated with this eigenvalue, in the case of a normal matrix. The problem also leads naturally to the consideration of the field of values of a matrix, a concept studied as early as 1918 by Toeplitz and Hausdorff. We prove that the field of values is convex, a theorem due to Toeplitz and Hausdorff. This fact is helpful in improving the search for local maxima.