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本論文工作的主題是圖像反卷積問題。在很多實際應用，例如生物醫學成像，地震學，天文學，遙感和光學成像中，觀測數據經常會出現令人不愉快的退化現象，這種退化一般由模糊效應(例如光學衍射限條件)和噪聲汙染(比如光子計數噪聲和讀出噪聲)造成的，這兩者都是物理儀器自身的條件限制造成的。作為一個標准的線性反問題，圖像反卷積經常被用作恢複觀測到的模糊的有噪點的圖像。我們旨在基于無偏差風險估計准則研究新的反卷積算法。本論文工作主要分為以下兩大部分。 / 首先，我們考慮在加性高斯白噪聲條件下的圖像非盲反卷積問題，即准確的點擴散函數已知。我們的研究准則是最小化均方誤差的無偏差估計，即SURE. SURE- LET方法最初被應用于圖像降噪問題。本論文工作擴展該方法至討論圖像反卷積問題.我們提出了一個新的SURE-LET算法，用于快速有效地實現圖像複原功能。具體而言，我們將反卷積過程參數化表示為有限個基本函數的線性組合，稱作LET方法。反卷積問題最終簡化為求解該線性組合的最優線性系數。由于SURE的二次項本質和線性參數化表示，求解線性系數可由求解線性方程組而得。實驗結果顯示該論文提出的方法在信噪比，圖像的視覺質量和運算時間等方面均優于其他迄今最優秀的算法。 / 論文的第二部分討論圖像盲複原中的點擴散函數估計問題。我們提出了blur-SURE -一個均方誤差修正版的無偏差估計 - 作為點擴散函數估計的最新准則，即點擴散函數由最小化這個新的目標函數獲得。然後我們利用這個估計的點擴散函數，用第一部分所提出的SURE-LET算法進行圖像的非盲複原。我們以一些典型的點擴散函數形式(高斯函數最為典型)為例詳細闡述該blur-SURE理論框架。實驗結果顯示最小化blur-SURE能夠更准確的估計點擴散函數，從而獲得更加優越的反卷積佳能。相比于圖像非盲複原，盲複原所得的圖片的視覺質量損失可忽略不計。 / 本論文所提出的基于無偏差估計的算法可擴展至其他噪聲模型。由于本論文以SURE基礎的方法在理論上並不僅限于卷積問題，該方法可用于解決數據的其他線性失真問題。 / The subject of this thesis is image deconvolution. In many real applications, e.g. biomedical imaging, seismology, astronomy, remote sensing and optical imaging, undesirable degradations by blurring effect (e.g. optical diffraction-limited condition) and noise corruption (e.g. photon-counting noise and readout noise) are inherent to any physical acquisition device. Image deconvolution, as a standard linear inverse problem, is often applied to recover the images from their blurred and noisy observations. Our interest lies in novel deconvolution algorithms based on unbiased risk estimate. This thesis is organized in two main parts as briefly summarized below. / We first consider non-blind image deconvolution with the corruption of additive white Gaussian noise (AWGN), where the point spread function (PSF) is exactly known. Our driving principle is the minimization of an unbiased estimate of mean squared error (MSE) between observed and clean data, known as "Stein's unbiased risk estimate" (SURE). The SURE-LET approach, which was originally developed for denoising, is extended to the deconvolution problem: a new SURE-LET deconvolution algorithm for fast and efficient implementation is proposed. More specifically, we parametrize the deconvolution process as a linear combination of a small number of known basic processings, which we call the linear expansion of thresholds (LET), and then minimize the SURE over the unknown linear coefficients. Due to the quadratic nature of SURE and the linear parametrization, the optimal linear weights of the combination is finally achieved by solving a linear system of equations. Experiments show that the proposed approach outperforms other state-of-the-art methods in terms of PSNR, SSIM, visual quality, as well as computation time. / The second part of this thesis is concerned with PSF estimation for blind deconvolution. We propose a "blur-SURE" - an unbiased estimate of a filtered version of MSE - as a novel criterion for estimating the PSF, from the observed image only, i.e. the PSF is identified by minimizing this new objective functional, whose validity has been theoretically verified. The blur-SURE framework is exemplified with a number of parametric forms of the PSF, most typically, the Gaussian kernel. Experiments show that the blur-SURE minimization yields highly accurate estimate of PSF parameters. We then perform non-blind deconvolution using the SURE-LET algorithm proposed in Part I, with the estimated PSF. Experiments show that the estimated PSF results in superior deconvolution performance, with a negligible quality loss, compared to the deconvolution with the exact PSF. / One may extend the algorithms based on unbiased risk estimate to other noise model. Since the SURE-based approaches does not restrict themselves to convolution operation, it is possible to extend them to other distortion scenarios. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Xue, Feng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 119-130). / Abstracts also in Chinese. / Dedication --- p.i / Acknowledgments --- p.iii / Abstract --- p.ix / List of Notations --- p.xi / Contents --- p.xvi / List of Figures --- p.xx / List of Tables --- p.xxii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivations and objectives --- p.1 / Chapter 1.2 --- Mathematical formulation for problem statement --- p.2 / Chapter 1.3 --- Survey of non-blind deconvolution approaches --- p.2 / Chapter 1.3.1 --- Regularization --- p.2 / Chapter 1.3.2 --- Regularized inversion followed by denoising --- p.4 / Chapter 1.3.3 --- Bayesian approach --- p.4 / Chapter 1.3.4 --- Remark --- p.5 / Chapter 1.4 --- Survey of blind deconvolution approaches --- p.5 / Chapter 1.4.1 --- Non-parametric blind deconvolution --- p.5 / Chapter 1.4.2 --- Parametric blind deconvolution --- p.7 / Chapter 1.5 --- Objective assessment of the deconvolution quality --- p.8 / Chapter 1.5.1 --- Peak Signal-to-Noise Ratio (PSNR) --- p.8 / Chapter 1.5.2 --- Structural Similarity Index (SSIM) --- p.8 / Chapter 1.6 --- Thesis contributions --- p.9 / Chapter 1.6.1 --- Theoretical contributions --- p.9 / Chapter 1.6.2 --- Algorithmic contributions --- p.10 / Chapter 1.7 --- Organization --- p.11 / Chapter I --- The SURE-LET Approach to Non-blind Deconvolution --- p.13 / Chapter 2 --- The SURE-LET Framework for Deconvolution --- p.15 / Chapter 2.1 --- Motivations --- p.15 / Chapter 2.2 --- Related work --- p.15 / Chapter 2.3 --- Problem statement --- p.17 / Chapter 2.4 --- Stein's Unbiased Risk Estimate (SURE) for deconvolution --- p.17 / Chapter 2.4.1 --- Original SURE --- p.17 / Chapter 2.4.2 --- Regularized approximation of SURE --- p.18 / Chapter 2.5 --- The SURE-LET approach --- p.19 / Chapter 2.6 --- Summary --- p.20 / Chapter 3 --- Multi-Wiener SURE-LET Approach --- p.23 / Chapter 3.1 --- Problem statement --- p.23 / Chapter 3.2 --- Linear deconvolution: multi-Wiener filtering --- p.23 / Chapter 3.3 --- SURE-LET in orthonormal wavelet representation --- p.24 / Chapter 3.3.1 --- Mathematical formulation --- p.24 / Chapter 3.3.2 --- SURE minimization in orthonormal wavelet domain --- p.26 / Chapter 3.3.3 --- Computational issues --- p.27 / Chapter 3.4 --- SURE-LET approach for redundant wavelet representation --- p.30 / Chapter 3.5 --- Computational aspects --- p.32 / Chapter 3.5.1 --- Periodic boundary extensions --- p.33 / Chapter 3.5.2 --- Symmetric convolution --- p.36 / Chapter 3.5.3 --- Half-point symmetric boundary extensions --- p.36 / Chapter 3.5.4 --- Whole-point symmetric boundary extensions --- p.43 / Chapter 3.6 --- Results and discussions --- p.46 / Chapter 3.6.1 --- Experimental setting --- p.46 / Chapter 3.6.2 --- Influence of the number of Wiener lters --- p.47 / Chapter 3.6.3 --- Influence of the parameters on the deconvolution performance --- p.48 / Chapter 3.6.4 --- Influence of the boundary conditions: periodic vs symmetric --- p.52 / Chapter 3.6.5 --- Comparison with the state-of-the-art --- p.52 / Chapter 3.6.6 --- Analysis of computational complexity --- p.59 / Chapter 3.7 --- Conclusion --- p.60 / Chapter II --- The SURE-based Approach to Blind Deconvolution --- p.63 / Chapter 4 --- The Blur-SURE Framework to PSF Estimation --- p.65 / Chapter 4.1 --- Introduction --- p.65 / Chapter 4.2 --- Problem statement --- p.66 / Chapter 4.3 --- The blur-SURE framework for general linear model --- p.66 / Chapter 4.3.1 --- Blur-MSE: a modified version of MSE --- p.66 / Chapter 4.3.2 --- Blur-MSE minimization --- p.67 / Chapter 4.3.3 --- Blur-SURE: an unbiased estimate of the blur-MSE --- p.67 / Chapter 4.4 --- Application of blur-SURE framework for PSF estimation --- p.68 / Chapter 4.4.1 --- Problem statement in the context of convolution --- p.68 / Chapter 4.4.2 --- Blur-MSE minimization for PSF estimation --- p.69 / Chapter 4.4.3 --- Approximation of exact Wiener filtering --- p.70 / Chapter 4.4.4 --- Blur-SURE minimization for PSF estimation --- p.72 / Chapter 4.5 --- Concluding remarks --- p.72 / Chapter 5 --- The Blur-SURE Approach to Parametric PSF Estimation --- p.75 / Chapter 5.1 --- Introduction --- p.75 / Chapter 5.1.1 --- Overview of parametric PSF estimation --- p.75 / Chapter 5.1.2 --- Gaussian PSF as a typical example --- p.75 / Chapter 5.1.3 --- Outline of this chapter --- p.76 / Chapter 5.2 --- Parametric estimation: problem formulation --- p.77 / Chapter 5.3 --- Examples of PSF parameter estimation --- p.77 / Chapter 5.3.1 --- Gaussian kernel --- p.77 / Chapter 5.3.2 --- Non-Gaussian PSF with scaling factor s --- p.78 / Chapter 5.4 --- Minimization via the approximated function λ = λ (s) --- p.79 / Chapter 5.5 --- Results and discussions --- p.82 / Chapter 5.5.1 --- Experimental setting --- p.82 / Chapter 5.5.2 --- Non-Gaussian functions: estimation of scaling factor s --- p.83 / Chapter 5.5.3 --- Gaussian function: estimation of standard deviation s --- p.84 / Chapter 5.5.4 --- Comparison of deconvolution performance with the state-of-the-art --- p.84 / Chapter 5.5.5 --- Application to real images --- p.87 / Chapter 5.6 --- Conclusion --- p.90 / Chapter 6 --- The Blur-SURE Approach to Motion Deblurring --- p.93 / Chapter 6.1 --- Introduction --- p.93 / Chapter 6.1.1 --- Background of motion deblurring --- p.93 / Chapter 6.1.2 --- Related work: parametric estimation of motion blur --- p.93 / Chapter 6.1.3 --- Outline of this chapter --- p.94 / Chapter 6.2 --- Parametric estimation of motion blur: problem formulation --- p.94 / Chapter 6.2.1 --- Parametrized form of linear motion blur --- p.94 / Chapter 6.2.2 --- The blur-SURE framework to motion blur estimation --- p.94 / Chapter 6.3 --- An example of the blur-SURE approach to motion blur estimation --- p.95 / Chapter 6.4 --- Implementation issues --- p.96 / Chapter 6.4.1 --- Estimation of motion direction --- p.97 / Chapter 6.4.2 --- Estimation of blur length --- p.97 / Chapter 6.4.3 --- Short summary --- p.98 / Chapter 6.5 --- Results and discussions --- p.98 / Chapter 6.5.1 --- Experimental setting --- p.98 / Chapter 6.5.2 --- Estimations of blur direction and length --- p.99 / Chapter 6.5.3 --- Motion deblurring: the synthetic experiments --- p.99 / Chapter 6.5.4 --- Motion deblurring: the real experiment --- p.101 / Chapter 6.6 --- Conclusion --- p.103 / Chapter 7 --- Epilogue --- p.107 / Chapter 7.1 --- Summary --- p.107 / Chapter 7.2 --- Perspectives --- p.108 / Chapter A --- Proof --- p.109 / Chapter A.1 --- Proof of Theorem 2.1 --- p.109 / Chapter A.2 --- Proof of Eq.(2.6) in Section 2.4.2 --- p.110 / Chapter A.3 --- Proof of Eq.(3.5) in Section 3.3.1 --- p.110 / Chapter A.4 --- Proof of Theorem 3.6 --- p.112 / Chapter A.5 --- Proof of Theorem 3.12 --- p.112 / Chapter A.6 --- Derivation of noise variance in 2-D case (Section 3.5.4) --- p.114 / Chapter A.7 --- Proof of Theorem 4.1 --- p.116 / Chapter A.8 --- Proof of Theorem 4.2 --- p.116

A novel technique for blind deconvolution of ultrasound is introduced. Existing deconvolution techniques for ultrasound such as cepstrum-based methods and the work of Adam and Michailovich – based on Discrete Wavelet Transform (DWT) shrinkage of the log-spectrum – exploit the smoothness of the pulse log-spectrum relative to the reflectivity function to estimate the pulse. To reduce the effects of non-stationarity in the ultrasound signal on both the pulse estimation and deconvolution, the log-spectrum is time-localized and represented as the Continuous Wavelet Transform (CWT) log-scalogram in the proposed technique. The pulse CWT coefficients are estimated via DWT shrinkage of the log-scalogram and are then deconvolved by wavelet-domain Wiener filtering. Parameters of the technique are found by heuristic optimization on a training set with various quality metrics: entropy, autocorrelation 6-dB width and fractal dimension. The technique is further enhanced by using different CWT wavelets for estimation and deconvolution, similar to the WienerChop method.

A novel technique for blind deconvolution of ultrasound is introduced. Existing deconvolution techniques for ultrasound such as cepstrum-based methods and the work of Adam and Michailovich – based on Discrete Wavelet Transform (DWT) shrinkage of the log-spectrum – exploit the smoothness of the pulse log-spectrum relative to the reflectivity function to estimate the pulse. To reduce the effects of non-stationarity in the ultrasound signal on both the pulse estimation and deconvolution, the log-spectrum is time-localized and represented as the Continuous Wavelet Transform (CWT) log-scalogram in the proposed technique. The pulse CWT coefficients are estimated via DWT shrinkage of the log-scalogram and are then deconvolved by wavelet-domain Wiener filtering. Parameters of the technique are found by heuristic optimization on a training set with various quality metrics: entropy, autocorrelation 6-dB width and fractal dimension. The technique is further enhanced by using different CWT wavelets for estimation and deconvolution, similar to the WienerChop method.

The purpose of this thesis is to develop a mathematical approach and associated software implementation for deconvolution of two-dimensional Transmission Electron Microscope (TEM) images. The focus is on TEM images of weakly scattering amorphous biological specimens that mainly produce phase contrast. The deconvolution is to remove the distortions introduced by the TEM detector that are modeled by the Modulation Transfer Function (MTF). The report tests deconvolution of the TEM detector MTF by Wiener _ltering and Tikhonov regularization on a range of simulated TEM images with varying degree of noise.The performance of the two deconvolution methods are quanti_ed by means of Figure of Merits (FOMs) and comparison in-between methods is based on statistical analysis of the FOMs.

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published_or_final_version / abstract / toc / Physics / Master / Master of Philosophy

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Long exposure handheld photography is coupled with the problem of blurring, which is difficult to remove without additional information. The goal of this work was to utilize motion sensors contained in modern smartphones to detect exact motion track of the image sensor during the exposure and then to remove the blur from the resulting photograph according to this data. A system was proposed which performs deconvolution using a kernel from the recorded gyroscope data. An implementation on Android platform was proved on a test smartphone device.

This work presents the development, validation and application of a novel deconvolution method based on B-splines for analyzing variable-rate reservoir performance data. Variable-rate deconvolution is a mathematically unstable problem which has been under investigation by many researchers over the last 35 years. While many deconvolution methods have been developed, few of these methods perform well in practice - and the importance of variable-rate deconvolution is increasing due to applications of permanent downhole gauges and large-scale processing/analysis of production data. Under these circumstances, our objective is to create a robust and practical tool which can tolerate reasonable variability and relatively large errors in rate and pressure data without generating instability in the deconvolution process. We propose representing the derivative of unknown unit rate drawdown pressure as a weighted sum of Bsplines (with logarithmically distributed knots). We then apply the convolution theorem in the Laplace domain with the input rate and obtain the sensitivities of the pressure response with respect to individual B-splines after numerical inversion of the Laplace transform. The sensitivity matrix is then used in a regularized least-squares procedure to obtain the unknown coefficients of the B-spline representation of the unit rate response or the well testing pressure derivative function. We have also implemented a physically sound regularization scheme into our deconvolution procedure for handling higher levels of noise and systematic errors. We validate our method with synthetic examples generated with and without errors. The new method can recover the unit rate drawdown pressure response and its derivative to a considerable extent, even when high levels of noise are present in both the rate and pressure observations. We also demonstrate the use of regularization and provide examples of under and over-regularization, and we discuss procedures for ensuring proper regularization. Upon validation, we then demonstrate our deconvolution method using a variety of field cases. Ultimately, the results of our new variable-rate deconvolution technique suggest that this technique has a broad applicability in pressure transient/production data analysis. The goal of this thesis is to demonstrate that the combined approach of B-splines, Laplace domain convolution, least-squares error reduction, and regularization are innovative and robust; therefore, the proposed technique has potential utility in the analysis and interpretation of reservoir performance data.