Research Of Sparse Representation And Its Application

Signal representation is the basic problem in signal processing. Efficient representation of signal contents lies at the foundation of signal processing tasks. Efficiency of a representation refers to the ability to capture significant information of an object of interest in a small description. That is the ability of sparse representation. We can use the sparse representation of signal instead of original signal to reduce costs of signal processing, thus, we can improve the efficiency of signal processing.Sparse representation of signals is presented in twentieth century at the beginning of the nineteen’s, which is a new signal representation method. As part of basic research of signal, it can use for handling various tasks such as denoising, restoration, separation, interpolation and extrapolation, compression, sampling, analysis and synthesis, detection, recognition, and more. This sparse model is intriguing and fascinating because of the beauty of its theoretical foundations, the superior performance it leads to in various applications, its universality and flexibility in serving various data sources, and its unified view, which makes all the above signal processing tasks clear and simple.This paper researched the basic technology of sparse signal representation, m ainly about the decomposition algorithm based on the Lp norm and sparse coding based on the LO norm, and used in classical image denoising problems and handwritten digital recognition respectively. Firstly, the paper introduced the research status of the sparse representation at home and abroad, and then recall the fundamental mathematic theory of sparse representation and the neural network. At last, we emphases research a sparse decomposition algorithm and the application of handwritten digitals recognition, as follows:Firstly, This thesis researched the sparse decomposition algorithm based on Lp norm, which we can extends the existing weighted L1 norm and L2 norm separable surrogate functional (SSF) iterative shrinkage algorithm to approximate the objective function of a weighted Lp norm and L2 norm optimization problem by N one dimensional independent objective functions. However, as these one dimensional independent objective functions are nonconvex, there may be more than one locally optimal solution of the approximated problem. Hence, it is difficult to find its globally optimal solution. To address this difficulty, this thesis further characterizes the regions of the feasible set of the approximated problem where the sign of the convexity of the objective function of the approximated problem within the regions remain unchanged. In this case, the objective function could consist of no more than one stationary point in each region. By finding the optimal solution within each region, the globally optimal solution of the approximated optimization problem is found. Hence, a nearly global optimal solution of the original optimization problem is obtained. Computer numerical simulation results show that our proposed model outperforms the existing weighted L1 norm and L2 norm optimization model and the smooth L0-norm model.Secondly, this thesis also researched handwritten digits recognition based on the sparse coding. As we know, the most common and efficient method of handwritten digits recognition is the sparse autoencoder neural network. However, we find that the features which got from the sparse autoencoder neural network is not as sparse as we want. Here we develop a more efficient method of handwritten digits recognition, that is sparse coding based on LO-norm algorithm. When we fixed the dictionary, we use the OMP to get the sparse representation coefficients of every training sample. When we fixed the sparse representation coefficients, we use the method of direction (MOD) to train the dictionary. Computer numerical simulation results show that our proposed model outperforms the existing sparse autoencoder neural network.