Computer vision problems such as contour grouping often lead to quadratic optimization problems. In this talk, we consider two variations of this problem.

The first one is a contour grouping problem first studied by Jianbo Shi and Qihui Zhu. We present a new formulation due to Kennedy, Shi, and Gallier, involving an Hermitian matrix, H(delta), depending on a parameter. To study the variation of the top eigenvalue of this matrix, we show how to compute the derivative of an eigenvalue and of a corresponding eigenvector. It turns out that the eigenvalues of H(delta) are related to the field of values of the matrix P, a concept introduced by Hausdorff and Toeplitz in 1918.

The second problem is to maximize f(x) = x^T A x on the unit sphere under affine constraints. Gander, Golub and von Matt (1989) showed that this amounts to maximizing f(x) = x^ A x + x^T b on the unit sphere, with b nonzero. Unfortunately, the standard Courant-Fisher result does not apply anymore. One can attack the problem using a Lagrangian, and this was done by Gander, Golub and von Matt. Independently, I solved this problem using a method involving the intersection of the unit sphere with an algebraic curve generalizing the hyperbola to n dimensions.