Optimization Of Dictionary And Sensing Matrix For Compressed Sensing Systems

Compressed sensing(CS)is an emerging theory that offers a framework for simultaneous sensing and compression of finite-dimensional vectors.It is an interdiscipline of mathematics,electronic engineering,computer science,and physics.CS is named from the assumption of combining the sensing and compression procedures.This assumption is realizable as most natural signals are sparse or can be sparsely represented.Though the measurements are highly incomplete,sparsity makes it possible to solve the underdetermined problem for the original high-dimensional signals by using proper recovery algorithms.In practice,there already exist many of such algorithms that can be implemented for reconstructing the signals with high probability.CS is mainly composed of three components:(1).Sparse representations of the high-dimensional signals;(2).Linear dimensionality reduction method;(3).Efficient recovery algorithm.In this thesis,we focus on the investigations of the first two components.The first one is closely related with the dictionary learning problem which concerns training a dictionary with some training samples to sparsely represent a kind of signals.The dimensionality reduction method,referring to the design of sensing(projection)matrix,determines whether the measurements contain the crucial information to reconstruct the original signals and also affects the precision of the recovery algorithm.The main contributions are five folds:1.Constructing incoherent frames and equiangular tight frames.CS is a rapidly growing theory and the research field of CS draws from a variety of other areas.One of such important areas is frame theory.Our work on this involves:· Construct incoherent frames using alternating optimization algorithm.The solutions of incoherent frames are derived analytically within a commonly used convex structural constraint space.Compared with traditional algorithms,the proposed method converges faster and the atoms of the resulted frames show lower coherence.· Define a new spectral constraint space to ensure the frame tightness.Based on the alternating projection,an algorithm for constructing equiangular tight frames is developed and the closed-form solution set of a matrix nearness problem that arises from the construction process is derived.The generated frames enjoy reduced coherence while maintain the tightness property as well.2.Research on incoherent dictionary learning.We investigate the dictionary learning problem with the coherence taken into consideration.The coherence is constrained by making the Gram matrix of the desired dictionary approximate to an identity matrix(the simplest Gram matrix of equiangular tight frame)of proper dimension,and the solution is derived analytically.Noting that there exist two degrees of freedom in the expression,an alternating optimization-based algorithm is proposed to minimize the sparse representation error.Such model and algorithm lead to a dictionary that represents signals sparsely,and improves the precision of the recovery algorithm.3.Designing robust sensing matrix.In order to promote the signals reconstruction accuracy,two robust sensing matrix design methods are proposed:· A weighting model that combines the Gram matrix of an incoherent frame and the Gram matrix of the fixed dictionary as the target Gram is proposed to design the sensing matrix.An iterative algorithm is developed to handle such a design problem.When updating the sensing matrix,the analytical solution is derived.A sensing matrix designed this way is not only suitable for exactly sparse scenarios,but also robust against the sparse representation error.· Directly take the sparse representation error into consideration.Furthermore,different from the first work,the equiangular tight frame is employed as the target frame to ensure the incoherence and tightness properties.Such a new model is solved by an iterative algorithm.The designed sensing matrix directly optimizes the sparse representation error and makes the corresponding equivalent dictionary possess good frame properties.4.Optimizing the dictionary and sensing matrix jointly.Design the sensing matrix based on the dictionary and also embed the sensing matrix in the dictionary learning process to promote their interaction.An alternating minimization-based algorithm is proposed to solve the joint optimization model which consists of two main steps:(1).For fixed dictionary,update the robust sensing matrix with the method discussed before;(2).Derive the analytical solution of the dictionary with sensing matrix embedded.The performance of CS systems designed along this direction can be greatly enhanced.5.The application of CS theory.Two application scenarios are considered:· Research the compressive detection problem based on sensing matrix design.Firstly,we prove that the optimum sensing matrix related to the NeymanPearson detector is determined only by the orthonormal matrix formed by its right side singular vectors,and derive the relational expression between this orthonormal matrix and the detection probability.In order to improve the detection probability,the orthonormal matrix is designed such that the Gram matrix of the corresponding equivalent dictionary is as close to the Gram matrix of an equiangular tight frame as possible.The resulted sensing matrix boosts the detection probability effectively.· Investigate the sparse parameter vector estimation problem in a distributed compressed system.In such a system,the estimation of the parameter vector is conducted in compressed domain and the adaptive recursive least-squares algorithm can be applied to calculate local estimators.Besides,designing incoherent sensing matrix off-line for the system can not only reduce complexity,but also improve the estimation accuracy.The experiments for a wireless sensor network application show that the proposed scheme achieves satisfactory results in terms of convergence rate and estimation performance.