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