Image analysis using Hilbert space

摘要

Abstract: There has been tremendous progress in the image processing (input: images, output: images) and computer graphics (input: numbers, output: images) area. Unfortunately, progress in image analysis (input: images, output: numbers) has been much slower. In this paper, we introduce the ideas of image analysis using Hilbert space which encodes an image to a small vector. An image can be interpreted as a representation of a vector in a Hilbert space. It is well known that if the eigenvalues of a Hermitian operator is lower-bounded but not upper-bounded, the set of the eigenvectors of the operator is complete and spans a Hilbert space. Sturm-Liouville operators with periodic boundary condition and the first, second, and third classes of boundary conditions are special examples. Any vectors in a Hilbert space can be expanded. If a vector happens to be in a subspace of a Hilbert space where the domain L of the subspace is low (order of 10), the vector can be specified by its norm, an L-vector, and the Hermitian operator which spans the Hilbert space. This establishes a mapping from an image to a set of numbers. This mapping converts an input image to a 4-tuple: P $EQ (norm, T, N, L-vector), where T is a point in an operator parameter space, N is an integer which specify the boundary condition. Unfortunately, the best algorithm for this scheme at this point is a local search which has high time complexity. The search is first conducted for an operator in a parameter space of operators. Then an error function $delta@(t) is computed. The algorithm stops at a local minimum of $delta@(t).!12