Signal Sparse Representation Based On Analysis Sparse Model

Sparse signal representations have been the focus of much recent research in signal processing and have become a well-known topic in a wide range of fields.Recently, an alternative model called analysis sparse model has got more and more attention. To explore analysis sparse model’s potential, research focus on analysis dictionary learning(ADL) algorithm. Most existing analysis dictionary learning algorithms assume that the original signals are known or can be estimated if unknown.But it cost too much time in greedy like algorithm to estimate original signals. In order to overcome the shortcoming, we directly employ the observed signals with noisy to learn analysis dictionary, and then used these analysis dictionaries for image denoising. The main work of this paper summarized as follows:1. According to the principle of greedy method, a subset pursuit algorithm has been proposed for analysis dictionary learning(SP-ADL) which directly employs the observed data to compute the approximate analysis sparse representation of the original signals, without having to pre-estimate X. The analysis sparse representation can be exploited to assign the observed data into multiple subsets, which are then used for updating the analysis dictionary. And according to the same principle, an ADL algorithm using K-plane clustering is proposed, which is based on the observation that, the observed data are co-planer in the analysis sparse model. In other words, the columns of the observed data form multi-dimensional subspaces(hyperplanes), and the rows of the analysis dictionary are the normal vectors of the hyper-planes. The normal directions of the K-dimensional concentration hyper-planes can be estimated using the K-plane clustering algorithm, and then the rows of the analysis dictionary which are the normal vectors of the hyper-planes can be obtained.2. According to the principle of majorization method, a recursive least squares algorithm(RLS-ADL) is proposed for ADL directly from the noisy measurements.The algorithm uses the gradient descent method to optimize its objective function, so as to get the dictionary. According to the same principle, a NESTA gradient algorithm is proposed to estimate the analysis dictionary. The algorithm uses maxfunction instead of norm to construct the objective function, and then uses Nesterov gradient method to optimize the objective function to update the dictionary atoms, so as to get the dictionary. And according to the same principle, a majorization minimization algorithm(MM-ADL) is proposed to estimate the analysis dictionary.The algorithm uses Taylor series to construct the objective function and then uses the gradient descent method to optimize it, so as to get the dictionary.