A Total Fractional-Order Variation Model for Image Restoration with Non-homogeneous Boundary Conditions and its Numerical Solution

To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total $\alpha$-order variation model, which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-order variations for image restoration; however first no analysis is done yet and second all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while non-zero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total $\alpha$-order variational model rigorously. It then develops four algorithms for solving the variational problem, one based on the variational Split-Bregman idea and three based on direct solution of the discretise-optimization problem. Numerical experiments show that, in terms of restoration quality and solution efficiency, the proposed model can produce highly competitive results, for smooth images, to two established high order models: the mean curvature and the total generalized variation.Comment: 26 page

Computational Multispectral Endoscopy

Minimal Access Surgery (MAS) is increasingly regarded as the de-facto approach in interventional medicine for conducting many procedures this is due to the reduced patient trauma and consequently reduced recovery times, complications and costs. However, there are many challenges in MAS that come as a result of viewing the surgical site through an endoscope and interacting with tissue remotely via tools, such as lack of haptic feedback; limited field of view; and variation in imaging hardware. As such, it is important best utilise the imaging data available to provide a clinician with rich data corresponding to the surgical site. Measuring tissue haemoglobin concentrations can give vital information, such as perfusion assessment after transplantation; visualisation of the health of blood supply to organ; and to detect ischaemia. In the area of transplant and bypass procedures measurements of the tissue tissue perfusion/total haemoglobin (THb) and oxygen saturation (SO2) are used as indicators of organ viability, these measurements are often acquired at multiple discrete points across the tissue using with a specialist probe. To acquire measurements across the whole surface of an organ one can use a specialist camera to perform multispectral imaging (MSI), which optically acquires sequential spectrally band limited images of the same scene. This data can be processed to provide maps of the THb and SO2 variation across the tissue surface which could be useful for intra operative evaluation. When capturing MSI data, a trade off often has to be made between spectral sensitivity and capture speed. The work in thesis first explores post processing blurry MSI data from long exposure imaging devices. It is of interest to be able to use these MSI data because the large number of spectral bands that can be captured, the long capture times, however, limit the potential real time uses for clinicians. Recognising the importance to clinicians of real-time data, the main body of this thesis develops methods around estimating oxy- and deoxy-haemoglobin concentrations in tissue using only monocular and stereo RGB imaging data

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

A Comprehensive Survey on Test-Time Adaptation under Distribution Shifts

Machine learning methods strive to acquire a robust model during training that can generalize well to test samples, even under distribution shifts. However, these methods often suffer from a performance drop due to unknown test distributions. Test-time adaptation (TTA), an emerging paradigm, has the potential to adapt a pre-trained model to unlabeled data during testing, before making predictions. Recent progress in this paradigm highlights the significant benefits of utilizing unlabeled data for training self-adapted models prior to inference. In this survey, we divide TTA into several distinct categories, namely, test-time (source-free) domain adaptation, test-time batch adaptation, online test-time adaptation, and test-time prior adaptation. For each category, we provide a comprehensive taxonomy of advanced algorithms, followed by a discussion of different learning scenarios. Furthermore, we analyze relevant applications of TTA and discuss open challenges and promising areas for future research. A comprehensive list of TTA methods can be found at \url{https://github.com/tim-learn/awesome-test-time-adaptation}.Comment: Discussions, comments, and questions are all welcomed in \url{https://github.com/tim-learn/awesome-test-time-adaptation

Blind image deconvolution: nonstationary Bayesian approaches to restoring blurred photos

High quality digital images have become pervasive in modern scientific and everyday life — in areas from photography to astronomy, CCTV, microscopy, and medical imaging. However there are always limits to the quality of these images due to uncertainty and imprecision in the measurement systems. Modern signal processing methods offer the promise of overcoming some of these problems by postprocessing these blurred and noisy images. In this thesis, novel methods using nonstationary statistical models are developed for the removal of blurs from out of focus and other types of degraded photographic images. The work tackles the fundamental problem blind image deconvolution (BID); its goal is to restore a sharp image from a blurred observation when the blur itself is completely unknown. This is a “doubly illposed” problem — extreme lack of information must be countered by strong prior constraints about sensible types of solution. In this work, the hierarchical Bayesian methodology is used as a robust and versatile framework to impart the required prior knowledge. The thesis is arranged in two parts. In the first part, the BID problem is reviewed, along with techniques and models for its solution. Observation models are developed, with an emphasis on photographic restoration, concluding with a discussion of how these are reduced to the common linear spatially-invariant (LSI) convolutional model. Classical methods for the solution of illposed problems are summarised to provide a foundation for the main theoretical ideas that will be used under the Bayesian framework. This is followed by an indepth review and discussion of the various prior image and blur models appearing in the literature, and then their applications to solving the problem with both Bayesian and nonBayesian techniques. The second part covers novel restoration methods, making use of the theory presented in Part I. Firstly, two new nonstationary image models are presented. The first models local variance in the image, and the second extends this with locally adaptive noncausal autoregressive (AR) texture estimation and local mean components. These models allow for recovery of image details including edges and texture, whilst preserving smooth regions. Most existing methods do not model the boundary conditions correctly for deblurring of natural photographs, and a Chapter is devoted to exploring Bayesian solutions to this topic. Due to the complexity of the models used and the problem itself, there are many challenges which must be overcome for tractable inference. Using the new models, three different inference strategies are investigated: firstly using the Bayesian maximum marginalised a posteriori (MMAP) method with deterministic optimisation; proceeding with the stochastic methods of variational Bayesian (VB) distribution approximation, and simulation of the posterior distribution using the Gibbs sampler. Of these, we find the Gibbs sampler to be the most effective way to deal with a variety of different types of unknown blurs. Along the way, details are given of the numerical strategies developed to give accurate results and to accelerate performance. Finally, the thesis demonstrates state of the art results in blind restoration of synthetic and real degraded images, such as recovering details in out of focus photographs