Two Kinds Of Minimization Problem With Mixed Norm

Compressed Sensing is a new method of signal processing which is proposed by Candesin2006, it breakes through the demands of traditional signal processing method on thesampling rate, then we can reconstruct high dimensional signals of high-quality from lowdimensional sampling space and save a lot of cost on sampling.CS can be used in lots of fields,such as Nuclear Magnetic Resonance Imaging, Ground Penetrating Radar, source coding, facerecognition etc. The core of Compressed Sensing including designing matrixes which satisfythe RIP condition, reasonable reconstruction model and exact reconstruction algorithm, ingenerally,the reconstruction model is a minimization problem which contains two or morekinds of norm, we call this kind of minimization problem for minimization problem withmixed norm. Besides be used in compressed sensing, the minimization problem with mixednorm is widely applied in many fields such as Electrical Capacitance Tomography, so theresearch to this type of problems has value on theory and application.In this paper, the minimization problem with mixed norm is researched as the followingtwo aspects:(1) According to the actual situation, a new reconstruction is put forward, and anefficient algorithm, based on transforming the model into the least absolute deviation, isdesigned.Based on the background of compressed sensing and improving to the existingmodels, a new kind of model about minimization problem with mixed norm is proposed inthis paper, namely the minimization problem with1-norm and constraint by least-squaresproblem (in this paper, it is referred to as the first class of minimization problem with mixednorm), and through the study of the general solution to the least-squares problem, it isconverted to a fitting-problem of least absolute deviation; Then, combined with the theory ofleast absolute deviation, an algorithm is gives to the first class of minimization problem withmixed norm; In the last,numerical examples are given to show that the algorithm is effective.(2) A know model is transformed into quadratic programming with boundaryconstraints, and effective algorithms to it are designed.In this paper, the problem whoseobjective function is combinated with the l-norm and2-norm is referred to the second class ofminimization problem with mixed norm, then some algorithms are given by transforming theproblem into a quadratic programming with boundary constraints and combinate the penaltyfunction method and Rosen projection method etc.At last these algorithms are showedeffective through numerical examples.