The problem of reconstruction of digital images from their degraded measurements is regarded as a problem of central importance in various fields of engineering and imaging sciences. In such cases, the degradation is typically caused by the resolution limitations of an imaging device in use and/or by the destructive influence of measurement noise. Specifically, when the noise obeys a Poisson probability law, standard approaches to the problem of image reconstruction are based on using fixed-point algorithms which follow the methodology proposed by Richardson and Lucy in the beginning of the 1970s. The practice of using such methods, however, shows that their convergence properties tend to deteriorate at relatively high noise levels (which typically takes place in so-called low-count settings). This work introduces a novel method for de-noising and/or de-blurring of digital images that have been corrupted by Poisson noise. The proposed method is derived using the framework of MAP estimation, under the assumption that the image of interest can be sparsely represented in the domain of a properly designed linear transform. Consequently, a shrinkage-based iterative procedure is proposed, which guarantees the maximization of an associated maximum-a-posteriori criterion. It is shown in a series of both computer-simulated and real-life experiments that the proposed method outperforms a number of existing alternatives in terms of stability, precision, and computational efficiency.

The problem of reconstruction of digital images from their degraded measurements is regarded as a problem of central importance in various fields of engineering and imaging sciences. In such cases, the degradation is typically caused by the resolution limitations of an imaging device in use and/or by the destructive influence of measurement noise. Specifically, when the noise obeys a Poisson probability law, standard approaches to the problem of image reconstruction are based on using fixed-point algorithms which follow the methodology proposed by Richardson and Lucy in the beginning of the 1970s. The practice of using such methods, however, shows that their convergence properties tend to deteriorate at relatively high noise levels (which typically takes place in so-called low-count settings). This work introduces a novel method for de-noising and/or de-blurring of digital images that have been corrupted by Poisson noise. The proposed method is derived using the framework of MAP estimation, under the assumption that the image of interest can be sparsely represented in the domain of a properly designed linear transform. Consequently, a shrinkage-based iterative procedure is proposed, which guarantees the maximization of an associated maximum-a-posteriori criterion. It is shown in a series of both computer-simulated and real-life experiments that the proposed method outperforms a number of existing alternatives in terms of stability, precision, and computational efficiency.

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第24264号 / 情博第808号 / 新制||情||136(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 山下 信雄, 准教授 福田 秀美, 教授 太田 快人 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Exploiting signal sparsity has recently received considerable attention in a variety of areas including signal and image processing, compressive sensing, machine learning and so on. Many of these applications involve optimization models that are regularized by certain sparsity-promoting metrics. Two most popular regularizers are based on the l1 norm that approximates sparsity of vectorized signals and the total variation (TV) norm that serves as a measure of gradient sparsity of an image. Nevertheless, the l1 and TV terms are merely two representative measures of sparsity. To explore the matter of sparsity further, in this thesis we investigate relaxations of the regularizers to nonconvex terms such as lp and TVp "norms" with 0 <= p < 1. The contributions of the thesis are two-fold. First, several methods to approach globally optimal solutions of related nonconvex problems for improved signal/image reconstruction quality have been proposed. Most algorithms studied in the thesis fall into the category of iterative reweighting schemes for which nonconvex problems are reduced to a series of convex sub-problems. In this regard, the second main contribution of this thesis has to do with complexity improvement of the l1/TV-regularized methodology for which accelerated algorithms are developed. Along with these investigations, new techniques are proposed to address practical implementation issues. These include the development of an lp-related solver that is easily parallelizable, and a matrix-based analysis that facilitates implementation for TV-related optimizations. Computer simulations are presented to demonstrate merits of the proposed models and algorithms as well as their applications for solving general linear inverse problems in the area of signal and image denoising, signal sparse representation, compressive sensing, and compressive imaging. / Graduate