Design Of Image Sparse Dictionary And Its Applications

Traditionally, a signal can be represented or approximated by a linear combination of basic signals from a set of orthogonal signals. Although it has its elegant form in mathematics, this representation seems not to be effective or efficient in practical signal processing. The compact representation of a signal is often desired goal in signal processing. The sparsity of signal representation can be obtained by increasing enough basis vectors such that it turns a complete basis to an overcomplete one, which is called sparse redundant dictionary. The sparsity can be measured by l0 norm naturally. The problem of signal sparse representation is in essence a fitting one between the obtained signal and the estimate signal with a linear combination of atoms from sparse dictionary. So, this dissertation starts with these following reasons or motivations:Firstly, solving the sparse problem inevitably involves in sparse dictionary. Secondly, the research on the sparse representation of signal and image is the research on sparse dictionary in some degree. Thirdly, the design and construction theory of sparse dictionary is expected to improve. Finally, a dictionary dealing with image and other high dimensional data has its extraordinary nature.This dissertation dealt with the design and construction of sparse dictionary and its practical applications in solving sparse problem, which presented some creative or innovative works as follows:First, This dissertation studied the history of dictionary and current sparse image dictionary's design and construction systematically and made clear the tendency of development of design and construction of sparse dictionary. It tried to explore the idea of design of current image sparse dictionary by making an investigation of the history of development of sparse dictionary, and summarized some considerable factors about sparse dictionary design:localization, multiresolution, adaptation, geometric invariance and overcompleteness. Then it categorized current dictionaries into three classes:structured dictionary, unions of bases and learned dictionary according to their generating methods. Furthermore, it indicated the deficiency in design and construction of current sparse dictionary and pointed out the direction to work toward. Secondly, this dissertation proposed the design and construction of visual sparse greedy indexed dictionary and orthogonal matching pursuit based on visual greedy indexed dictionary. Considering original matching pursuit employed the full search scheme, which included a number of vector-matrix multiplication and can lead to too much computation. Aimed at the decrease of computation cost, many modified schemes were proposed and had a good effect. However, these existed schemes were confined to structure dictionary, which exploited the wavelet level structure or organized the atom in parametric dictionary into a tree according the parametric index. So, these schemes are infeasible for learned dictionary because learned dictionary is unavailable to these properties. In order to deal with it, this dissertation proposed a sparse greedy indexed dictionary which was attributed to training the atoms in the dictionary by utilizing the greed nature in matching pursuit algorithm and presented two algorithms:indexing algorithms and Orthogonal Matching Pursuit based on indexed dictionary. Moreover, this dissertation made a discussion in theory and in the experiment, which demonstrated that the proposed algorithm arrived at the expected effect.Thirdly, this dissertation proposed the construction of near equiangular tight frame of union of bases involving in image. As we know, tight frame is a special frame which is the concern of academic world, which has better mathematical properties. If it is regarded as a matrix which consists of a set of column vectors, redundant dictionary is in fact a frame. To meet the requirement of uniqueness and equivalence in sparse theory, it is necessary to make some coherent constraints on these column vectors in the dictionary, which results in so-called "incoherent dictionary". Equiangular tight frame is natural incoherent dictionary and seeking for an equiangular tight frame of sparse dictionary is beneficial to solving the sparse problem. Usually, algebraic methods are employed to find the equiangular tight frame which makes use of conference matrix or Gram matrix of dictionary matrix. However, it has been proved that a equiangular tight frame cannot be found under all cases. So, this method don't always lead to success. Another effective method to find the equiangular tight frame is genetic algorithm, but this method is restricted to the performance of hardware because of its exhaustive search. Based on the idea of matrix nearness, this dissertation utilized alternating projection to construct the equiangular tight frame of union of bases and provided an instance. Related experiment indicated that this construction method is very effective. Finally, it demonstrated how to use and choose a proper sparse dictionary in practice. Based on previous design and construction theory of sparse dictionary, this dissertation gave an account of the issue on how to make use of image sparse dictionary, especially the choice of sparse dictionary and demonstrated the importance of choosing proper sparse dictionary combined with image decomposition instance. In the end, it suggested that sparse dictionary with geometric invariance has a promising prospect in image processing.The research findings help to give further impetus to the research on image sparse representation and methods of solving the sparse problem, and expedite the progress of the practical applications of sparse representation in signal and image processing.