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Title: Set theoretic framework for signal reconstruction
Other Titles: Tu ji ying she li lun ying yong yu xin hao chong jian de yan jiu
Authors: Gan, Xiangchao (甘祥超)
Department: Dept. of Electronic Engineering
Degree: Doctor of Philosophy
Issue Date: 2006
Publisher: City University of Hong Kong
Subjects: Image reconstruction
Set theory
Notes: CityU Call Number: TA1632.G36 2006
Includes bibliographical references (leaves 110-120)
Thesis (Ph.D.)--City University of Hong Kong, 2006
vii, 127 leaves : ill. ; 30 cm.
Type: Thesis
Abstract: In many engineering applications, an optimal solution can be obtained by incorporating all available information in the problem formulation. Conventional estimation techniques, such as the Bayesian analysis and Lagrange multipliers, often involve in computational considerations and mathematical solvability and may lead to compromised estimates which are not consistent with a priori knowledge. Set theoretic framework (STF) is a powerful and reliable theory for incorporating all available information to obtain an estimate. In this framework, each piece of information is transformed into a set in the solution space and the intersection of these sets represents the acceptable solutions. Then, some convergence-guaranteed iterative algorithms can be used to find a point in the intersection of all constructed sets resulting in our estimate which satisfy all a priori information. The key part of STF is the construction of proper sets which capture the a priori knowledge. This thesis is aimed at the development of STF for image reconstruction and DNA microarray gene expression data processing. In Chapter 2 and 3, we propose two different sets for blocking artifacts reduction in a Block Discrete Cosine Transform (BDCT) coded image (JPEG image or a frame in an MPEG stream). Both sets are based on a principle: an image is usually smooth. In the first set, we find that a blocking artifact along edge direction is difficult for human perception while the artifact anti-parallel to the edge direction often fragments the edges and seriously degrades visual quality. By considering the behavior of intensity evolution along and across edges, we propose a new smoothness constraint set. The new constraint set is realized indirectly by limiting the distance between the coded image and the image simulated using edge-adaptive quadrangle meshes. In Chapter 3, based on signal and quantization noise statistics, we propose another smoothness constraint set in the BDCT transform domain via the Wiener filtering concept. Experiments show that STF algorithms using the proposed smoothness constraint sets not only have good convergence but also have better objective and subjective performance. In Chapter 4, we use STF to improve the constrained total least squares (CTLS)-based image restoration algorithm. In image restoration, the region of support of the point spread function (PSF) is often much smaller than the size of the observed degraded image and this property is utilized in many image deconvolution algorithms. For a CTLS-based algorithm, it means that the solution of the CTLS algorithm should retain the block-circulant and sparse structure of the degradation matrix simultaneously. In real image restoration problem, the CTLS method often involves large-scale computation and is often solved using Mesarovic et al.’s algorithm. However, their algorithm cannot preserve the sparse structure of the degradation matrix. We prove that by imposing an extra constraint, the sparse structure can be preserved in their algorithm. Then, we use the STF to find a solution to this extended formulation. Our experimental study indicates that the proposed method performs competitively, and often better, in terms of visual and objective evaluations. In Chapter 5, we use STF for the missing values imputation of gene expression data. Gene expressions measured using microarrays usually suffer from the missing value problem. We propose the STF for missing data imputation and design several convex sets, taking into consideration the biological characteristic of the data: the first set mainly exploit the local correlation structure among genes in microarray data, while the second set captures the global correlation structure among arrays. The third set (a series of sets) exploits the biological phenomenon of synchronization loss in microarray experiments. In cyclic systems, synchronization loss is a very common phenomenon and we construct a series of sets based on this phenomenon for our STF imputation algorithm. Experiments show that our algorithm can achieve a significant error reduction compared to other existing methods.
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