The structure of sparse representations of images using tight frames.

摘要

We present numerical results on representations of natural images with respect to a particular category of redundant systems: tight wavelet frames and wavelet biframes. The representations are obtained by solving an 11-minimization problem, by using orthogonal projections, as well as by using a thinning method, consisting of alternating between the backward and forward greedy algorithms. We use a number of natural images as our test images; and a total of 23 redundant wavelet frames. We then analyze the ability of the wavelet frames to provide sparse, approximate representations, and also analyze the structure of the representations.; Given an image, an error level and a representation system as above, we find an approximate representation of the image as a linear combination of relatively few elements of the system. The approximate representation has to be accurate up to the given error level. We say that the representation is sparse if it makes use of few elements of the system.; The main conclusions of this thesis are twofold: Firstly, redundant systems can provide sparse representations of natural images, and a large number of redundant frames provide sparser representations than the benchmark non-redundant wavelet system. Furthermore, it is computationally feasible to find such sparse representations. Secondly, the sparse representations that we find have a surprising structure. Mother wavelets with the least vanishing moments absorb most of the energy, and the representations employ fewer elements than expected from the coarser frequency levels.; We also compare the ability of redundant systems to provide sparse representations to their theoretical properties. In this regard, a low approximation order of the frame operator has no negative effects on that ability, and the approximation order of the MRA does not seem to play a role. On the other hand, the frames that have a B-spline as the dual scaling function and have many wavelets with few vanishing moments consistently perform well.