The Research On Signals’ Recovery Algorithms Based On The Cosparse Analysis Model

In recent years, with the uninterrupted development of the information technology, signal model is becoming increasingly important in the field of signal processing. So far, for solving the problem of the signal’s sparse recovery, there are two main signal models:sparse synthesis model and cosparse analysis model. Many domestic and foreign researchers have researched these models relatively from different perspectives, especially, for the signal recovery problem based on sparse synthesis signal model, a series of algorithms and theory have been proposed. Recently, the cosparse analysis model has also been very impressive, the signal recovery problem based on the cosparse analysis model has become an emerging research topic. This dissertation will study the cosparse analysis model, and give some improvements and innovations based on existing relevant algorithms and theory.Firstly, the basic theory of the signal’s sparse representation are described. Then, we introduce two main signal models of the signal’s sparse recovery problem, and the relative algorithms for each model are elaborated respectively. Besides, we propose two optimiza-tion models and algorithms for the signal recovery problem based on the cosparse analysis model:one is that using the cosparsity inducing function to approximate I0 norm; the other is to regard the maximum entropy function as an approximative alternative for l1 norm. The specific work is summarized as follows:In the first work, we give a new alternative way to replace the l0 norm based on the nonconvex cosparsity inducing function, which is closer to l0 norm than l1 norm and l2 norm. Firstly, we construct the new objective function and then give a constrained opti-mal model based on this function. Moreover, we give the first order approximation of the cosparsity inducing function to make the nonconvex problem into the easy convex case. Then the objective function is converted into the unconstrained optimization problem by Lagrangian multiplier method. Finally, we propose Cosparsity Inducing Function (CIF) al-gorithm, which belongs to a two-layer optimization algorithm:we firstly obtain a temporary optimal variable via the subgradient algorithm; secondly, a new cosupport is given by the temporary optimal variable, and the desired signal is estimated on the new cosupport. The theory analysis and simulations on the recovering of the unknown signal indicate the better performance of CIF Algorithm.In the second work, considering the signal recovery l1 norm problem based on the cosparse analysis model:convex but non-differentiable at zero point. Therefore, we give the maximum entropy function of l1 norm, which is continuous and differentiable.First of all, we establish the differentiable constrained optimization model based on the maximum entropy function. Next, the optimization model is transformed into unconstrained situation through Lagrangian multiplier method. A series of the gradient optimization algorithm can be applied for this model. The experiments show that the results of the maximum entropy function are as well as l1 norm. Thus, using the maximum entropy function to approximate l1 minimization problem is a better choice.Finally, we summarize the main work of this dissertation, and give our future research-es in the last chapter.