Generalized SURE for Exponential Families: Applications to Regularization

Stein's unbiased risk estimate (SURE) was proposed by Stein for the independent, identically distributed (iid) Gaussian model in order to derive estimates that dominate least-squares (LS). In recent years, the SURE criterion has been employed in a variety of denoising problems for choosing regularization parameters that minimize an estimate of the mean-squared error (MSE). However, its use has been limited to the iid case which precludes many important applications. In this paper we begin by deriving a SURE counterpart for general, not necessarily iid distributions from the exponential family. This enables extending the SURE design technique to a much broader class of problems. Based on this generalization we suggest a new method for choosing regularization parameters in penalized LS estimators. We then demonstrate its superior performance over the conventional generalized cross validation approach and the discrepancy method in the context of image deblurring and deconvolution. The SURE technique can also be used to design estimates without predefining their structure. However, allowing for too many free parameters impairs the performance of the resulting estimates. To address this inherent tradeoff we propose a regularized SURE objective. Based on this design criterion, we derive a wavelet denoising strategy that is similar in sprit to the standard soft-threshold approach but can lead to improved MSE performance.Comment: to appear in the IEEE Transactions on Signal Processin

Unbiased risk estimate algorithms for image deconvolution.

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.iAcknowledgments --- p.iiiAbstract --- p.ixList of Notations --- p.xiContents --- p.xviList of Figures --- p.xxList of Tables --- p.xxiiChapter 1 --- Introduction --- p.1Chapter 1.1 --- Motivations and objectives --- p.1Chapter 1.2 --- Mathematical formulation for problem statement --- p.2Chapter 1.3 --- Survey of non-blind deconvolution approaches --- p.2Chapter 1.3.1 --- Regularization --- p.2Chapter 1.3.2 --- Regularized inversion followed by denoising --- p.4Chapter 1.3.3 --- Bayesian approach --- p.4Chapter 1.3.4 --- Remark --- p.5Chapter 1.4 --- Survey of blind deconvolution approaches --- p.5Chapter 1.4.1 --- Non-parametric blind deconvolution --- p.5Chapter 1.4.2 --- Parametric blind deconvolution --- p.7Chapter 1.5 --- Objective assessment of the deconvolution quality --- p.8Chapter 1.5.1 --- Peak Signal-to-Noise Ratio (PSNR) --- p.8Chapter 1.5.2 --- Structural Similarity Index (SSIM) --- p.8Chapter 1.6 --- Thesis contributions --- p.9Chapter 1.6.1 --- Theoretical contributions --- p.9Chapter 1.6.2 --- Algorithmic contributions --- p.10Chapter 1.7 --- Organization --- p.11Chapter I --- The SURE-LET Approach to Non-blind Deconvolution --- p.13Chapter 2 --- The SURE-LET Framework for Deconvolution --- p.15Chapter 2.1 --- Motivations --- p.15Chapter 2.2 --- Related work --- p.15Chapter 2.3 --- Problem statement --- p.17Chapter 2.4 --- Stein's Unbiased Risk Estimate (SURE) for deconvolution --- p.17Chapter 2.4.1 --- Original SURE --- p.17Chapter 2.4.2 --- Regularized approximation of SURE --- p.18Chapter 2.5 --- The SURE-LET approach --- p.19Chapter 2.6 --- Summary --- p.20Chapter 3 --- Multi-Wiener SURE-LET Approach --- p.23Chapter 3.1 --- Problem statement --- p.23Chapter 3.2 --- Linear deconvolution: multi-Wiener filtering --- p.23Chapter 3.3 --- SURE-LET in orthonormal wavelet representation --- p.24Chapter 3.3.1 --- Mathematical formulation --- p.24Chapter 3.3.2 --- SURE minimization in orthonormal wavelet domain --- p.26Chapter 3.3.3 --- Computational issues --- p.27Chapter 3.4 --- SURE-LET approach for redundant wavelet representation --- p.30Chapter 3.5 --- Computational aspects --- p.32Chapter 3.5.1 --- Periodic boundary extensions --- p.33Chapter 3.5.2 --- Symmetric convolution --- p.36Chapter 3.5.3 --- Half-point symmetric boundary extensions --- p.36Chapter 3.5.4 --- Whole-point symmetric boundary extensions --- p.43Chapter 3.6 --- Results and discussions --- p.46Chapter 3.6.1 --- Experimental setting --- p.46Chapter 3.6.2 --- Influence of the number of Wiener lters --- p.47Chapter 3.6.3 --- Influence of the parameters on the deconvolution performance --- p.48Chapter 3.6.4 --- Influence of the boundary conditions: periodic vs symmetric --- p.52Chapter 3.6.5 --- Comparison with the state-of-the-art --- p.52Chapter 3.6.6 --- Analysis of computational complexity --- p.59Chapter 3.7 --- Conclusion --- p.60Chapter II --- The SURE-based Approach to Blind Deconvolution --- p.63Chapter 4 --- The Blur-SURE Framework to PSF Estimation --- p.65Chapter 4.1 --- Introduction --- p.65Chapter 4.2 --- Problem statement --- p.66Chapter 4.3 --- The blur-SURE framework for general linear model --- p.66Chapter 4.3.1 --- Blur-MSE: a modified version of MSE --- p.66Chapter 4.3.2 --- Blur-MSE minimization --- p.67Chapter 4.3.3 --- Blur-SURE: an unbiased estimate of the blur-MSE --- p.67Chapter 4.4 --- Application of blur-SURE framework for PSF estimation --- p.68Chapter 4.4.1 --- Problem statement in the context of convolution --- p.68Chapter 4.4.2 --- Blur-MSE minimization for PSF estimation --- p.69Chapter 4.4.3 --- Approximation of exact Wiener filtering --- p.70Chapter 4.4.4 --- Blur-SURE minimization for PSF estimation --- p.72Chapter 4.5 --- Concluding remarks --- p.72Chapter 5 --- The Blur-SURE Approach to Parametric PSF Estimation --- p.75Chapter 5.1 --- Introduction --- p.75Chapter 5.1.1 --- Overview of parametric PSF estimation --- p.75Chapter 5.1.2 --- Gaussian PSF as a typical example --- p.75Chapter 5.1.3 --- Outline of this chapter --- p.76Chapter 5.2 --- Parametric estimation: problem formulation --- p.77Chapter 5.3 --- Examples of PSF parameter estimation --- p.77Chapter 5.3.1 --- Gaussian kernel --- p.77Chapter 5.3.2 --- Non-Gaussian PSF with scaling factor s --- p.78Chapter 5.4 --- Minimization via the approximated function λ = λ (s) --- p.79Chapter 5.5 --- Results and discussions --- p.82Chapter 5.5.1 --- Experimental setting --- p.82Chapter 5.5.2 --- Non-Gaussian functions: estimation of scaling factor s --- p.83Chapter 5.5.3 --- Gaussian function: estimation of standard deviation s --- p.84Chapter 5.5.4 --- Comparison of deconvolution performance with the state-of-the-art --- p.84Chapter 5.5.5 --- Application to real images --- p.87Chapter 5.6 --- Conclusion --- p.90Chapter 6 --- The Blur-SURE Approach to Motion Deblurring --- p.93Chapter 6.1 --- Introduction --- p.93Chapter 6.1.1 --- Background of motion deblurring --- p.93Chapter 6.1.2 --- Related work: parametric estimation of motion blur --- p.93Chapter 6.1.3 --- Outline of this chapter --- p.94Chapter 6.2 --- Parametric estimation of motion blur: problem formulation --- p.94Chapter 6.2.1 --- Parametrized form of linear motion blur --- p.94Chapter 6.2.2 --- The blur-SURE framework to motion blur estimation --- p.94Chapter 6.3 --- An example of the blur-SURE approach to motion blur estimation --- p.95Chapter 6.4 --- Implementation issues --- p.96Chapter 6.4.1 --- Estimation of motion direction --- p.97Chapter 6.4.2 --- Estimation of blur length --- p.97Chapter 6.4.3 --- Short summary --- p.98Chapter 6.5 --- Results and discussions --- p.98Chapter 6.5.1 --- Experimental setting --- p.98Chapter 6.5.2 --- Estimations of blur direction and length --- p.99Chapter 6.5.3 --- Motion deblurring: the synthetic experiments --- p.99Chapter 6.5.4 --- Motion deblurring: the real experiment --- p.101Chapter 6.6 --- Conclusion --- p.103Chapter 7 --- Epilogue --- p.107Chapter 7.1 --- Summary --- p.107Chapter 7.2 --- Perspectives --- p.108Chapter A --- Proof --- p.109Chapter A.1 --- Proof of Theorem 2.1 --- p.109Chapter A.2 --- Proof of Eq.(2.6) in Section 2.4.2 --- p.110Chapter A.3 --- Proof of Eq.(3.5) in Section 3.3.1 --- p.110Chapter A.4 --- Proof of Theorem 3.6 --- p.112Chapter A.5 --- Proof of Theorem 3.12 --- p.112Chapter A.6 --- Derivation of noise variance in 2-D case (Section 3.5.4) --- p.114Chapter A.7 --- Proof of Theorem 4.1 --- p.116Chapter A.8 --- Proof of Theorem 4.2 --- p.11

Hierarchical Bayesian sparse image reconstruction with application to MRFM

This paper presents a hierarchical Bayesian model to reconstruct sparse images when the observations are obtained from linear transformations and corrupted by an additive white Gaussian noise. Our hierarchical Bayes model is well suited to such naturally sparse image applications as it seamlessly accounts for properties such as sparsity and positivity of the image via appropriate Bayes priors. We propose a prior that is based on a weighted mixture of a positive exponential distribution and a mass at zero. The prior has hyperparameters that are tuned automatically by marginalization over the hierarchical Bayesian model. To overcome the complexity of the posterior distribution, a Gibbs sampling strategy is proposed. The Gibbs samples can be used to estimate the image to be recovered, e.g. by maximizing the estimated posterior distribution. In our fully Bayesian approach the posteriors of all the parameters are available. Thus our algorithm provides more information than other previously proposed sparse reconstruction methods that only give a point estimate. The performance of our hierarchical Bayesian sparse reconstruction method is illustrated on synthetic and real data collected from a tobacco virus sample using a prototype MRFM instrument.Comment: v2: final version; IEEE Trans. Image Processing, 200

Fast and easy blind deblurring using an inverse filter and PROBE

PROBE (Progressive Removal of Blur Residual) is a recursive framework for blind deblurring. Using the elementary modified inverse filter at its core, PROBE's experimental performance meets or exceeds the state of the art, both visually and quantitatively. Remarkably, PROBE lends itself to analysis that reveals its convergence properties. PROBE is motivated by recent ideas on progressive blind deblurring, but breaks away from previous research by its simplicity, speed, performance and potential for analysis. PROBE is neither a functional minimization approach, nor an open-loop sequential method (blur kernel estimation followed by non-blind deblurring). PROBE is a feedback scheme, deriving its unique strength from the closed-loop architecture rather than from the accuracy of its algorithmic components

Non-parametric PSF estimation from celestial transit solar images using blind deconvolution

Context: Characterization of instrumental effects in astronomical imaging is important in order to extract accurate physical information from the observations. The measured image in a real optical instrument is usually represented by the convolution of an ideal image with a Point Spread Function (PSF). Additionally, the image acquisition process is also contaminated by other sources of noise (read-out, photon-counting). The problem of estimating both the PSF and a denoised image is called blind deconvolution and is ill-posed. Aims: We propose a blind deconvolution scheme that relies on image regularization. Contrarily to most methods presented in the literature, our method does not assume a parametric model of the PSF and can thus be applied to any telescope. Methods: Our scheme uses a wavelet analysis prior model on the image and weak assumptions on the PSF. We use observations from a celestial transit, where the occulting body can be assumed to be a black disk. These constraints allow us to retain meaningful solutions for the filter and the image, eliminating trivial, translated and interchanged solutions. Under an additive Gaussian noise assumption, they also enforce noise canceling and avoid reconstruction artifacts by promoting the whiteness of the residual between the blurred observations and the cleaned data. Results: Our method is applied to synthetic and experimental data. The PSF is estimated for the SECCHI/EUVI instrument using the 2007 Lunar transit, and for SDO/AIA using the 2012 Venus transit. Results show that the proposed non-parametric blind deconvolution method is able to estimate the core of the PSF with a similar quality to parametric methods proposed in the literature. We also show that, if these parametric estimations are incorporated in the acquisition model, the resulting PSF outperforms both the parametric and non-parametric methods.Comment: 31 pages, 47 figure

Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization

The l1/l2 ratio regularization function has shown good performance for retrieving sparse signals in a number of recent works, in the context of blind deconvolution. Indeed, it benefits from a scale invariance property much desirable in the blind context. However, the l1/l2 function raises some difficulties when solving the nonconvex and nonsmooth minimization problems resulting from the use of such a penalty term in current restoration methods. In this paper, we propose a new penalty based on a smooth approximation to the l1/l2 function. In addition, we develop a proximal-based algorithm to solve variational problems involving this function and we derive theoretical convergence results. We demonstrate the effectiveness of our method through a comparison with a recent alternating optimization strategy dealing with the exact l1/l2 term, on an application to seismic data blind deconvolution.Comment: 5 page