This thesis applies a fully nonlinear Bayesian inversion methodology to estimate shear-wave velocity (Vs) profiles and uncertainties from surface-wave dispersion data extracted from ambient seismic noise. In the inversion, the Vs profile is parameterized using a Bernstein polynomial basis, which efficiently characterizes general depth-dependent gradients in the soil/sediment column. Bernstein polynomials provide a stable parameterization in that small perturbations to the model parameters (basis-function coefficients) result in only small perturbations to the Vs profile. The inversion solution is defined in terms of the marginal posterior probability for Vs as a function of depth, estimated using Metropolis-Hastings sampling with parallel tempering. This methodology is validated via inversion of synthetic dispersion data as well as previously-considered data inverted using different parameterizations. The approach considered here is better suited than layered modelling approaches in applications where smooth gradients in geophysical parameters are expected, and/or the observed data are diffuse and not sensitive to fine-scale discrete layering (such as surface-wave dispersion). The Bernstein polynomial representation is much more general than other gradient-based models such that the form of the gradients are determined by the data, rather than by subjective parameterization choice. The Bernstein inversion methodology is also applied to dispersion data processed from passive array recordings collected in the coastal community of Kitimat, British Columbia. The region is the proposed site of several large-scale industrial development projects and has great economic and environmental significance for Canada. The inversion results are consistent with findings from other geophysical studies in the region and are used in a site-specific seismic hazard analysis. The level of ground-motion amplification expected to occur during an earthquake due to near-surface Vs structure is probabilistically quantified, and predicted to be significant compared to reference (hard ground) sites. / Graduate

Biomedical imaging at microwave frequencies has shown potential for breast cancer detection and monitoring. The advantages of microwave imaging over current imaging techniques are that it is relatively inexpensive, and uses low-energy, non-ionizing radiation. It also provides a quantitative measurement of the dielectric properties of tissues, which offers the ability to characterize tissue types. Microwave imaging also comes with significant drawbacks. The resolution is poor compared to other imaging modalities, which presents challenges when trying to resolve fine structures. It is also not very sensitive to low contrast objects, and the accuracy of recovered tissue properties can be poor. This thesis shows that the use of prior information in microwave imaging inversion algorithms greatly improves the resulting images by minimizing mathematical difficulties in reconstruction that are due to the ill-posed nature of the inverse problem. The focus of this work is to explore novel methods to obtain and use prior information in the microwave breast imaging problem. We make use of finite element contrast source inversion (FEM-CSI) software formulated in two and three dimensions (2D, 3D). This software has the ability to incorporate prior information as an inhomogeneous numerical background medium. We motivate the usefulness of prior information by developing a simulated annealing technique that segments experimental human forearm images into tissue regions. Tissue types are identified and the resulting map of dielectric properties is used as prior information for the 2D FEM-CSI code. This results in improvements to the reconstructions, demonstrating the ability of prior information to improve breast images. We develop a combined microwave tomography/radar algorithm, and demonstrate that it is able to reconstruct images of superior quality, compared to either technique used alone. The algorithm is applied to data from phantoms containing tumours of decreasing size and can accurately monitor the changes. The combined algorithm is shown to be robust to the choice of immersion medium. This property allows us to design an immersion medium-independent algorithm, in which a numerical background can be used to reduce the contrast. We also develop a novel march-on-background technique that reconstructs high quality images using data collected in multiple immersion media. / October 2016

Cette thèse porte sur la restauration d'images dégradées à la fois par un flou et par un bruit. Une attention particulière est portée aux images issues de la microscopie confocale et notamment celles de macroscopie. Dans ce contexte, un modèle de bruit Poisson-Gauss apparaît bien adapté car il permet de prendre en compte le faible nombre de photons et le fort bruit enregistrés simultanément par les détecteurs. Cependant, ce type de modèle de bruit a été peu exploité car il pose de nombreuses difficultés tant théoriques que pratiques. Dans ce travail, une approche variationnelle est adoptée pour résoudre le problème de restauration dans le cas où le terme de fidélité exact est considéré. La solution du problème peut aussi être interprétée au sens du Maximum A Posteriori (MAP). L'utilisation d'algorithmes primaux-duaux récemment proposés en optimisation convexe permet d'obtenir de bons résultats comparativement à plusieurs approches existantes qui considèrent des approximations variées du terme de fidélité. En ce qui concerne le terme de régularisation de l'approche MAP, des approximations discrète et continue de la pseudo-norme $ell_0$ sont considérées. Cette mesure, célèbre pour favoriser la parcimonie, est difficile à optimiser car elle est, à la fois, non convexe et non lisse. Dans un premier temps, une méthode basée sur les coupures de graphes est proposée afin de prendre en compte des à priori de type quadratique tronqué. Dans un second temps, un algorithme à mémoire de gradient de type Majoration-Minimisation, dont la convergence est garantie, est considéré afin de prendre en compte des a priori de type norme $ell_2-ell_0$. Cet algorithme permet notamment d'obtenir de bons résultats dans des problèmes de déconvolution. Néanmoins, un inconvénient des approches variationnelles est qu'elles nécessitent la détermination d'hyperparamètres. C'est pourquoi, deux méthodes, reposant sur une approche Espérance-Maximisation (EM) sont proposées, dans ce travail, afin d'estimer les paramètres d'un bruit Poisson-Gauss: (1) à partir d'une série temporelle d'images (dans ce cas, des paramètres de « bleaching » peuvent aussi être estimés) et (2) à partir d'une seule image. De manière générale, cette thèse propose et teste de nombreuses méthodologies adaptées à la prise en compte de bruits et de flous difficiles, ce qui devrait se révéler utile pour des applications variées, au-delà même de la microscopie / This thesis deals with the restoration of images corrupted by blur and noise, with emphasis on confocal microscopy and macroscopy applications. Due to low photon count and high detector noise, the Poisson-Gaussian model is well suited to this context. However, up to now it had not been widely utilized because of theoretical and practical difficulties. In view of this, we formulate the image restoration problem in the presence of Poisson-Gaussian noise in a variational framework, where we express and study the exact data fidelity term. The solution to the problem can also be interpreted as a Maximum A Posteriori (MAP) estimate. Using recent primal-dual convex optimization algorithms, we obtain results that outperform methods relying on a variety of approximations. Turning our attention to the regularization term in the MAP framework, we study both discrete and continuous approximation of the $ell_0$ pseudo-norm. This useful measure, well-known for promoting sparsity, is difficult to optimize due to its non-convexity and its non-smoothness. We propose an efficient graph-cut procedure for optimizing energies with truncated quadratic priors. Moreover, we develop a majorize-minimize memory gradient algorithm to optimize various smooth versions of the $ell_2-ell_0$ norm, with guaranteed convergence properties. In particular, good results are achieved on deconvolution problems. One difficulty with variational formulations is the necessity to tune automatically the model hyperparameters. In this context, we propose to estimate the Poisson-Gaussian noise parameters based on two realistic scenarios: one from time series images, taking into account bleaching effects, and another from a single image. These estimations are grounded on the use of an Expectation-Maximization (EM) approach.Overall, this thesis proposes and evaluates various methodologies for tackling difficult image noise and blur cases, which should be useful in various applicative contexts within and beyond microscopy

Convolution and, as a special case, autoconvolution of functions are important in many branches of mathematics and have found lots of applications, such as in physics, statistics, image processing and others. While it is a relatively easy task to determine the autoconvolution of a function (at least from the numerical point of view), the inverse problem, which consists in reconstructing a function from its autoconvolution is an ill-posed problem. Hence there is no possibility to solve such an inverse autoconvolution problem with a simple algebraic operation. Instead the problem has to be regularized, which means that it is replaced by a well-posed problem, which is close to the original problem in a certain sense. The outline of this thesis is as follows: In the first chapter we give an introduction to the type of inverse problems we consider, including some basic definitions and some important examples of regularization methods for these problems. At the end of the introduction we shortly present some general results about the convergence theory of Tikhonov-regularization. The second chapter is concerned with the autoconvolution of square integrable functions defined on the interval [0, 1]. This will lead us to the classical autoconvolution problems, where the term “classical” means that no kernel function is involved in the autoconvolution operator. For the data situation we distinguish two cases, namely data on [0, 1] and data on [0, 2]. We present some well-known properties of the classical autoconvolution operators. Moreover, we investigate nonlinearity conditions, which are required to show applicability of certain regularization approaches or which lead convergence rates for the Tikhonov regularization. For the inverse autoconvolution problem with data on the interval [0, 1] we show that a convergence rate cannot be shown using the standard convergence rate theory. If the data are given on the interval [0, 2], we can show a convergence rate for Tikhonov regularization if the exact solution satisfies a sparsity assumption. After these theoretical investigations we present various approaches to solve inverse autoconvolution problems. Here we focus on a discretized Lavrentiev regularization approach, for which even a convergence rate can be shown. Finally, we present numerical examples for the regularization methods we presented. In the third chapter we describe a physical measurement technique, the so-called SD-Spider, which leads to an inverse problem of autoconvolution type. The SD-Spider method is an approach to measure ultrashort laser pulses (laser pulses with time duration in the range of femtoseconds). Therefor we first present some very basic concepts of nonlinear optics and after that we describe the method in detail. Then we show how this approach, starting from the wave equation, leads to a kernel-based equation of autoconvolution type. The aim of chapter four is to investigate the equation and the corresponding problem, which we derived in chapter three. As a generalization of the classical autoconvolution we define the kernel-based autoconvolution operator and show that many properties of the classical autoconvolution operator can also be shown in this new situation. Moreover, we will consider inverse problems with kernel-based autoconvolution operator, which reflect the data situation of the physical problem. It turns out that these inverse problems may be locally well-posed, if all possible data are taken into account and they are locally ill-posed if one special part of the data is not available. Finally, we introduce reconstruction approaches for solving these inverse problems numerically and test them on real and artificial data.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. 27 January 2016. / The direct and inverse spectral problems for an inhomogeneous string with an interval of zero density and a concentrated mass at the end point moving with damping are investigated. The partial differential equation is mapped into an ordinary differential equation using separation of variables which in turn is transformed into a Sturm-Liouville differential equation with boundary conditions depending on these parathion variable. The Marchenko approach is employed in the inverse problem to recover the potential, density and other parameters from the knowledge of the two spectra and length of the string.

The main topics of this dissertation are the hardware and the software developments in magnetic induction tomography imaging techniques. In the hardware sections, all the tomography systems developed by the author will be presented and discussed in detail. The developed systems can be divided into two categories, according to the property of the target imaging materials: high conductivity materials and low conductivity materials. Each system has its own suitable application, and each will thus be tested under different circumstances. In terms of the software development, the forward and inverse problems have been studied, including the eddy current problem modeling, sensitivity map formulae derivation and iterative/non-iterative inverse solvers equations. The Biot-Savart Theory was implemented in the ‘two-potential’ method that was used in the eddy current model in order to improve the system’s flexibility. Many different magnetic induction tomography schemes are proposed for the first time in this field of research, their aim being to improve the spatial and temporal resolution of the final reconstructed images. These novel schemes usually involve some modifications of the system hardware and forward/inverse calculations. For example, the rotational scheme can improve the ill-posedness and edge detectability of the system; the volumetric scheme can provide extra spatial resolution in the axial direction; and the temporal scheme can improve the temporal resolution by using the correlation between the consecutive datasets. Volumetric imaging requires an intensive amount of extra computational resources. To overcome the issue of memory constraints when solving large-scale inverse problems, a matrix-free method was proposed, also for the first time in magnetic induction tomography. All the proposed algorithms are verified by the experimental data obtained from suitable tomography systems developed by the author. Although magnetic induction tomography is a new imaging technique, it is believed that the technique is well developed for real-life applications. Several potential applications for magnetic induction tomography are suggested. The initial proof-of-concept study for a challenging low conductivity two-phase flow imaging process is provided. In this thesis, a range of contributions have been made in the field of magnetic induction tomography, which will help the magnetic induction tomography research to be carried on further.

Wang Yuliang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 70-75). / Abstracts in English and Chinese. / Chapter 1 --- Introduction to Inverse Problems --- p.1 / Chapter 1.1 --- Typical Examples --- p.1 / Chapter 1.2 --- Major Properties --- p.3 / Chapter 1.3 --- Solution Methods --- p.4 / Chapter 1.4 --- Thesis Outline --- p.4 / Chapter 2 --- Review of the Theory --- p.6 / Chapter 2.1 --- Basic Concepts --- p.6 / Chapter 2.1.1 --- Ill-posedness --- p.6 / Chapter 2.1.2 --- Generalized Inverse --- p.7 / Chapter 2.1.3 --- Compact Operators and SVE --- p.8 / Chapter 2.2 --- Regularization Methods --- p.10 / Chapter 2.2.1 --- An Overview --- p.11 / Chapter 2.2.2 --- Convergence Rates --- p.12 / Chapter 2.2.3 --- Parameter Choice Rules --- p.15 / Chapter 2.2.4 --- Classical Regularization Methods --- p.18 / Chapter 3 --- Ill-posedenss of Typical Inverse Problems --- p.23 / Chapter 3.1 --- Integral Equations --- p.24 / Chapter 3.2 --- Inverse Source Problems --- p.26 / Chapter 3.3 --- Parameter Identification --- p.34 / Chapter 3.4 --- Backward Heat Conduction --- p.37 / Chapter 4 --- Error Estimates for Parameter Identification --- p.39 / Chapter 4.1 --- Overview of Numerical Methods --- p.40 / Chapter 4.2 --- Finite Element Spaces and Standard Estimates --- p.43 / Chapter 4.3 --- Output Least-square Methods --- p.43 / Chapter 4.4 --- Equation Error Methods --- p.50 / Chapter 4.5 --- Hybrid Methods --- p.50 / Chapter 5 --- Numerical Experiments --- p.52 / Chapter 5.1 --- Formulate the Linear Systems --- p.53 / Chapter 5.2 --- Test Problems and Observations --- p.55 / Bibliography --- p.70

心臟表面電位能夠真實反映心肌的活動，因此以重建心臟表面電位為目標的心臟電生理反問題被廣泛研究。心臟電生理反問題是一個不適定問題，因此輸入數據中一個小的噪聲也有可能導致一個高度不穩定的解。因此，通常基於2 範數的規則化方法被用於解決這個病態問題。但是2 範數的懲罰函數會導致一定程度的模糊，使得分辨和定位心臟表面一些不正常或者病變部位不準確。而直接使用1 範數的懲罰函數，會由於其不可微分而增加計算復雜度。 / 我們首先提出一種基於 1 範數的方法來減少計算復雜度和能夠快速收斂。在這個方法中，使用變量分離技術使得1 範數的懲罰函數可微分。然後這個反問題被構造成一個有界約束二次優化問題，從而可以很容易地利用梯度映射法叠代求解。在試驗中，使用合成數據和真實數據來評估提出的方法。實驗表明，提出的方法可以很好地處理測量噪聲和幾何噪聲，而且能夠獲得比以前的1、2 範數方法更準確的實驗結果。 / 盡管提出的 1 範數方法能夠有效克服2 範數存在的問題，但是1 範數方法仍然只是0 範數的近似。因此我們采用了一種平滑0 範數的方法來求解心臟電生理反問題。平滑0 範數使用平滑函數，使得0 範數連續，從而能夠直接求解0 範數的反問題。實驗結果表明，使用平滑0範數方法可以獲得比1、2 範數更好、更準確的心臟表面電位。 / 在以往的心臟反問題研究中，使用的心臟幾何模型都是靜態的，與實際跳動的心臟不符，從而使得反問題方法難以進入臨床。因此我們提出了從動態心臟模型中重建心臟表面電位。動態心臟模型是從一系列核磁共振圖像中重建得到的。體表電位也同步獲得。仿真實驗獲得了很好的心臟表面電位結果。 / 在論文最後，我們提出一個基於心臟電生理反問題的系統，來輔助束支傳導阻滯的治療。在這個系統中，心臟模型和體表模型都從病人的數據中重建獲得，體表電位也得到收集。通過電生理反問題方法，在心臟表面重建電位及其分布。醫生通過觀察重建結果來輔助束支傳導阻滯的診斷和治療。 / The epicardial potentials (EPs) targeted inverse problem of electrocardiography (ECG) has been widely investigated as it is demonstrated that EPs reflect underlying myocardial activity. It is a wellknown ill-posed problem as small noises in input data may yield a highly unstable solution. Traditionally, L2-norm regularization methods have been proposed to solve this ill-posed problem. But L2-norm penalty function inherently leads to considerable smoothing of the solution, which reduces the accuracy of distinguishing abnormalities and locating diseased regions. In this thesis, we propose three new techniques in order to achieve more accurate reconstruction results of EPs and applied these techniques to a clinical application. We first propose a L1-norm regularization method in order to reduce the computational complexity and make rapid convergence possible. Variable splitting is employed to make the L1- norm penalty function differentiable based on the observation that both positive and negative potentials exist on the epicardial surface. Then, the inverse problem of ECG is further formulated as a boundconstrained quadratic problem, which can be efficiently solved by gradient projection in an iterative manner. Extensive experiments conducted on both synthetic data and real data demonstrate that the proposed method can handle both measurement noise and geometry noise and obtain more accurate results than previous L2- and L1- norm regularization methods, especially when the noises are large. / Although L1 norm regularization achieves better reconstructed results compared with L2 norm regularization, L1 norm is still an approximation of L0 norm which is more accurate than L1 norm. We further presented a smoothed L0 norm technique in order to directly solve the L0 norm constrained problem. Our method employs a smoothing function to make the L0 norm continuous. Extensive experiments showed that the proposed method reconstructs more accurate epicardial potentials compared with L1 norm and L2 norm. / In current research of ECG inverse problem, epicardial potentials are reconstructed from a static heart model which blocks the techniques to clinic applications. A novel strategy is presented to recovii er epicardial potentials using a dynamic heart model built from MRI image sequences and ECG data. We used MRI images to estimate the current density and visualize it on the surface of the heart model. The ECG data also be used to achieve the time synchronization when the propagation of the current density. Experiments are conducted on a set of real time MRI images, also with the real ECG data, and we get favorable results. / Finally, a non-invasive system is presented for enhancing the diagnosis of Bundle Branch Block (BBB). In this system, epicardial potential is estimated and visualized in the 3D heart model to improve the diagnosis of BBB. Using patient CT and BSPM data, the system is able to reconstruct details of the complete electrical activity of BBB on the 3D heart model. Through the analysis of the epicardial potential mapping in this system, patients with BBB are easily and accurately distinguished instead of from empirically checking ECG. Therefore the diagnosis of BBB is improved using this system. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Wang, Liansheng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 103-124). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Inverse Problem of ECG --- p.6 / Chapter 2.1 --- Background --- p.6 / Chapter 2.2 --- Problem Formulations --- p.8 / Chapter 2.2.1 --- Potential Reconstruction Problem --- p.8 / Chapter 2.2.2 --- Coefficient Reconstruction Problem --- p.11 / Chapter 2.3 --- Solving Methods --- p.11 / Chapter 2.3.1 --- Regularization Methods --- p.11 / Chapter 2.3.2 --- Non-quadratic Regularization --- p.12 / Chapter 2.3.3 --- Activation Wavefronts Solution --- p.14 / Chapter 3 --- L1-Norm to EPs Reconstruction --- p.16 / Chapter 3.1 --- Related Work --- p.16 / Chapter 3.2 --- Method --- p.21 / Chapter 3.3 --- Experimental Results and Validation --- p.24 / Chapter 3.3.1 --- Error Evaluation --- p.26 / Chapter 3.3.2 --- Synthetic Data Cases --- p.26 / Chapter 3.3.3 --- Real Data Cases --- p.32 / Chapter 3.4 --- Discussion --- p.44 / Chapter 3.5 --- Summary --- p.48 / Chapter 4 --- L0-Norm to EPs Reconstruction --- p.49 / Chapter 4.1 --- Related Work --- p.49 / Chapter 4.2 --- Smoothed L0-norm Method --- p.54 / Chapter 4.3 --- Experimental Results and Protocols --- p.57 / Chapter 4.3.1 --- Data --- p.57 / Chapter 4.3.2 --- Evaluation Protocol --- p.60 / Chapter 4.3.3 --- Experiments and Results --- p.60 / Chapter 4.4 --- Discussion --- p.68 / Chapter 4.5 --- Summary --- p.69 / Chapter 5 --- EPs Reconstruction in A Dynamic Model --- p.71 / Chapter 5.1 --- Related Work --- p.71 / Chapter 5.2 --- Forward Model --- p.73 / Chapter 5.3 --- Parameters Estimation for Inverse Problem of ECG --- p.75 / Chapter 5.4 --- Experiments and Results --- p.77 / Chapter 5.5 --- Summary --- p.80 / Chapter 6 --- Diagnosis of BBB: an Application --- p.82 / Chapter 6.1 --- Related Work --- p.82 / Chapter 6.2 --- Method --- p.84 / Chapter 6.2.1 --- Data --- p.85 / Chapter 6.2.2 --- Signal Preprocessing of BSPM --- p.87 / Chapter 6.2.3 --- Epicardial Potential Estimation and Imaging --- p.88 / Chapter 6.3 --- Experiments and Results --- p.89 / Chapter 6.3.1 --- Population Under Study --- p.89 / Chapter 6.3.2 --- Results --- p.89 / Chapter 6.4 --- Summary --- p.92 / Chapter 7 --- Conclusion --- p.94 / Chapter 7.1 --- Summary of Contributions --- p.94 / Chapter 7.2 --- Future Works --- p.96 / Chapter A --- Barzilai and Borwein Approach --- p.97 / Chapter B --- List of Publications --- p.99 / Bibliography --- p.103

The main goal of this thesis is to investigate the frequentist asymptotic properties of nonparametric Bayesian procedures in inverse problems and the Gaussian white noise model. In the first part, we study the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. This rate provides a quantitative measure of the quality of statistical estimation of the procedure. A theorem is proved in a general Hilbert space setting under approximation-theoretic assumptions on the prior. The result is applied to non-conjugate priors, notably sieve and wavelet series priors, as well as in the conjugate setting. In the mildly ill-posed setting, minimax optimal rates are obtained, with sieve priors being rate adaptive over Sobolev classes. In the severely ill-posed setting, oversmoothing the prior yields minimax rates. Previously established results in the conjugate setting are obtained using this method. Examples of applications include deconvolution, recovering the initial condition in the heat equation and the Radon transform. In the second part of this thesis, we investigate Bernstein--von Mises type results for adaptive nonparametric Bayesian procedures in both the Gaussian white noise model and the mildly ill-posed inverse setting. The Bernstein--von Mises theorem details the asymptotic behaviour of the posterior distribution and provides a frequentist justification for the Bayesian approach to uncertainty quantification. We establish weak Bernstein--von Mises theorems in both a Hilbert space and multiscale setting, which have applications in $L^2$ and $L^\infty$ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets using a Bayesian approach. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the different geometries involved.

by Yu Zhenli. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (p. 193-202). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.