Spectral Gradient Algorithms With Applications In Image Compressive Sensing

Compressive Sensing is an emerging methodology in recent years for informationcapturing. The theory illustrates that when the compressible signal is much lower thanthe Nyquist criterion, it can be recovered successfully. Compressive sensing has attractedintensive research activities in many scientific fields, such as in information theory, sig-nal/image processing, medical imaging, pattern recognition, geological exploration, opti-cal/radar imaging, wireless communications etc. Moreover, it recognized as the top10scientific and technological progress in USA by technology review.Compressive sensing uses non-adaptive linear projection to keep the original sig-nal. It also able to re-construct the original signal by solving optimization problems.In this thesis, we propose spectral gradient algorithms for solving1-norm regularizedminimization problems. Under some mild conditions, we show that the proposed algo-rithm converges globally. Numerical experiments are reported, which illustrate that theproposed algorithms is efective and promising for recovering large sparse signals.Firstly, We give the preliminaries of the signal processing, including digital imagesampling and quantization, and the technique to store in a computer. We briefly reviewthe theory of compressive sensing, recent algorithms for solving1-norm regularized min-imization problems. In the mean time, the contributions of this thesis, some importantnotation and symbols which used in the context are also included.Secondly, we present some preliminaries which used in our algorithm, including thedefinition and property of norm for a vector. We also recall the framework of iterative al-gorithm and list some classic methods, such as Newton methods, Quasi-Newton methods,and spectral gradient methods.Thirdly, we propose a spectral gradient algorithms for solving1-norm regularizedminimization problems in compressive sensing. The problem is firstly formulated to a con-vex quadratic program problem, and then to an equivalent non-smooth equation. At eachiteration, a spectral gradient method is applied to the resulting problem without requiringJacobian matrix information. Convergence of the proposed method is followed directlyfrom the results which has already existed. The algorithm is easily performed, where onlymatrix-vector inner product is required at each and every step. Numerical experiments to decode a sparse signal arising in compressive sensing and image de-convolution areperformed. The numerical results illustrate that the proposed method is practical andcompetitive with the well-known solver IST.Fourthly, we further research the spectral gradient algorithm for solving1-norm reg-ularized minimization problems arising from compressive sensing and machine learning.Diferent from the1-norm least squares in the previous chapter, the model in this chapteris more generality. Moreover, the proposed algorithm in this chapter solves the originalmodel directly by making full use of problem’s favorable structures, and thus the equiva-lent transformation is unnecessary in this method. Under some conditions, the proposedalgorithm is shown to be globally convergent. Limited experiments for1-regularizedleast squares problems in compressive sensing verify that our algorithm is efective andpromising.Finally, we conclude this thesis and give some further research topics.