Functions of bounded variation, wavelets, and applications to image processing

A common problem in image processing is to decompose an observed image f into a sum u+v, where u represents the more vital features of the image, i.e. the objects, and v represents the textured areas and any noise that may be present. The benefit of such a decomposition is that the "u" component represents a compressed and noise reduced version of the original image f.; The space BV of functions of bounded variation has been known to work very well as a model space for the objects in an image because indicator functions of sets whose boundary is finite in length belong to BV. This thesis is aimed at investigating the mathematical properties of the space BV while looking at a very well known "u+v" model, called the ROF model, in which it is assumed that u ∈ BV.; More recent work has shown that the optimal pair (u,v) to many decomposition problems can be obtained by expanding a given image f into a wavelet basis and performing simple operations on the wavelet coefficients. This thesis will provide a detailed introduction to the theory of orthonormal wavelets, giving some important examples of their effectiveness, as well as showing comparisons of wavelet bases with classical Fourier series.