Solving linear Bayesian inverse problems using a fractional total variation-Gaussian (FTG) prior and transport map

Geophysical Inverse Theory and Regularization Problems

The Newsletter is a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting forum by sending to our Bristol office material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome. Book review Introduction to Inverse Problems in Imaging M Bertero and P Boccacci 1998 Bristol: Institute of Physics 362 pp ISBN 0-7503-0435-9 (pbk) £25.00, $49.00 This book shows that several problems in imaging are in fact linear inverse problems. In general inverse problems are ill-posed in the sense that small perturbations in the (measured) data have a significant influence on the output. Consequently, numerical methods developed in a general setting of inverse problems which cure the ill-posedness (such methods are called regularization methods) can be applied to solve problems in imaging. This is an excellent textbook on the principle of linear inverse problems, methods of their numerical solution and practical applications in imaging. It introduces basic ideas and methods for the solution of inverse and ill-posed problems while avoiding the mathematics of functional treatment of operator equations. This goal is achieved by a bottom-up presentation: the authors focus on the two problems of image deconvolution and tomography (bottom) and develop regularization theory specifically for these two problems. Most other books which are concerned with the numerical solution of inverse problems present a top-down approach. That is, a general regularization theory (top) is developed which is then applied to particular problems. The distinguished presentation makes the book very useful to students in applied mathematics, practitioners in engineering science and image processing. This book has the best prerequisites to establish a close link between the mathematical fields of imaging and inverse problems. Mainly the book focuses on two topics: image deconvolution and linear imaging systems. 1. In image deconvolution the goal is to restore a space invariant blurred image. This part of the book contains mathematical tools in image deconvolution, examples of space invariant imaging systems, a comparison of image deconvolution techniques and low-pass filtering techniques, the analysis of the ill-posedness of the deconvolution problem, regularization methods for deconvolution such as constrained least squares regularization, Tikhonov regularization, iterative regularization and statistical methods. Since the authors focus on particular problems they are able to analyse various methods for deconvolution on a very concrete level. As an example I would like to mention that the authors analyse Fourier methods for deconvolution. Thus they provide an optimal starting point to the theory of inverse problems for people working in the area of signal and image processing. 2. The second part of the book considers linear imaging systems that are not of a convolution type such as tomography problems, diffraction problems and inverse scattering problems. Efficient numerical methods for the solution of this class of problems such as singular value decomposition, Tikhonov regularization methods and Fourier-based methods are presented. O Scherzer Universität Linz

Millions of digital images are captured by imaging devices on a daily basis. The way imaging devices operate follows an integral process from which the information of the original scene needs to be estimated. The estimation is done by inverting the integral process of the imaging device with the use of optimization techniques. This linear inverse problem, the inversion of the integral acquisition process, is at the heart of several image processing applications such as denoising, deblurring, inpainting, and super-resolution. We describe in detail the use of linear inverse problems in these applications. We review and compare several state-of-the-art optimization algorithms that invert this integral process. Linear inverse problems are usually very difficult to solve. Therefore, additional prior assumptions need to be introduced to successfully estimate the output signal. Several priors have been suggested in the research literature, with the Total Variation (TV) being one of the most prominent. In this thesis, we review another prior, the l0 pseudo-norm over the gradient domain. This prior allows full control over how many non-zero gradients are retained to approximate prominent structures of the image. We show the superiority of the l0 gradient prior over the TV prior in recovering genuinely piece-wise constant signals. The l0 gradient prior has shown to produce state-of-the-art results in edge-preserving image smoothing. Moreover, this general prior can be applied to several other applications, such as edge extraction, clip-art JPEG artifact removal, non-photorealistic image rendering, detail magnification, and tone mapping. We review and evaluate several state-of-the-art algorithms that solve the optimization problem based on the l0 gradient prior. Subsequently we apply the l0 gradient prior to two applications where we show superior results as compared to the current state-of-the-art. The first application is that of single-image reflection removal. Existing solutions to this problem have shown limited success because of the highly ill-posed nature of the problem. We show that the standard l0 gradient prior with a modified data-fidelity term based on the Laplacian operator is able to sufficiently remove unwanted reflections from images in many realistic scenarios. We conduct extensive experiments and show that our method outperforms the state-of-the-art. In the second application of haze removal from visible-NIR image pairs we propose a novel optimization framework, where the prior term penalizes the number of non-zero gradients of the difference between the output and the NIR image. Due to the longer wavelengths of NIR, an image taken in the NIR spectrum suffers significantly less from haze artifacts. Using this prior term, we are able to transfer details from the haze-free NIR image to the final result. We show that our formulation provides state-of-the-art results compared to haze removal methods that use a single image and also to those that are based on visible-NIR image pairs.

The paper is devoted to the study of the solvability of inverse coefficient problems for degenerate parabolic equations of the second order. We study both linear inverse problems – the problems of determining an unknown right-hand side (external influence), and nonlinear problems of determining an unknown coefficient of the equation itself. The peculiarity of the studied work is that its unknown coefficients are functions of a time variable only. The work aims to prove the existence and uniqueness of regular solutions to the studied problems (having all the generalized in the sense of S.L. Sobolev derivatives entering the equation).

A MATLAB implementation of the minimum relative entropy method for linear inverse problems

We study the convergence rates of a non-stationary asymptotical regularization method for linear statistical inverse problems from indirect noisy measurements. The convergence rates are usually derived under some range-type source conditions, which are difficult to be verified rigorously in general, especially for the models governed by some partial differential equations. We shall propose a new strategy associated with conditional stability estimates to obtain the error bounds in terms of mean-squared error. To the best of our knowledge, it is the first time to apply conditional stability estimates for convergence rates in statistical inverse problems. As an application, an inverse source problem in a wave equation is considered and its conditional stability estimate is verified rigorously, and therefore the convergence rates are derived immediately. Several numerical experiments for the inverse source problem are also provided to demonstrate the accuracy and efficiency of the non-stationary asymptotical regularization method.

Inverse problems in geophysics require the introduction of complex a priori information and are solved using computationally expensive Monte Carlo techniques (where large portions of the model space are explored). The geostatistical method allows for fast integration of complex a priori information in the form of covariance functions and training images. We combine geostatistical methods and inverse problem theory to generate realizations of the posterior probability density function of any Gaussian linear inverse problem, honoring a priori information in the form of a covariance function describing the spatial connectivity of the model space parameters. This is achieved using sequential Gaussian simulation, a well-known, noniterative geostatisticalmethod for generating samples of a Gaussian random field with a given covariance function. This work is a contribution to both linear inverse problem theory and geostatistics. Our main result is an efficient method to generate realizations, actual solutions rather than the conventional least-squares-based approach, to any Gaussian linear inverse problem using a noniterative method. The sequential approach to solving linear and weakly nonlinear problems is computationally efficient compared with traditional least-squares-based inversion. The sequential approach also allows one to solve the inverse problem in only a small part of the model space while conditioned to all available data. From a geostatistical point of view, the method can be used to condition realizations of Gaussian random fields to the possibly noisy linear average observations of the model space.

Medical image reconstruction is typically an ill-posed inverse problem. In order to address such ill-posed problems, the prior distribution of the sought after object property is usually incorporated by means of some sparsity-promoting regularization. Recently, prior distributions for images estimated using generative adversarial networks (GANs) have shown great promise in regularizing some of these image reconstruction problems. In this work, we apply an image-adaptive GAN-based reconstruction method (IAGAN) to reconstruct high fidelity images from incomplete medical imaging data. It is observed that the IAGAN method can potentially recover fine structures in the object that are relevant for medical diagnosis but may be oversmoothed in reconstructions with traditional sparsity-promoting regularization.

The unique solvability of linear inverse coefficient problems for the evolutionary equation in a Banach space with the Caputo Fabrizio derivative is studied. An operator at the unknown function in the equation is assumed to be bounded, the equation is endowed with the Cauchy condition. For the inverse problem with a constant unknown coefficient and with an integral overdefinition condition in the sense of Riemann Stieltjes, which includes the condition of final overdefinition as a special case, a well-posedness criterion is obtained. Sufficient conditions for unique solvability and an estimate of the well-posedness for the solution are obtained for a linear inverse problem with a time-dependent unknown coefficient. The abstract results obtained are used in the study of inverse problems with an unknown coefficient depending only on spatial variables or only on time, for equations with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.

Score-based diffusion models (SDMs) have emerged as a powerful tool for sampling from the posterior distribution in Bayesian inverse problems. However, existing methods often require multiple evaluations of the forward mapping to generate a single sample, resulting in significant computational costs for large-scale inverse problems. To address this, we propose an unconditional representation of the conditional score function (UCoS) tailored to linear inverse problems, which avoids forward model evaluations during sampling by shifting com putational effort to an offline training phase. In this phase, a task-dependent score function is learned based on the linear forward operator. Crucially, we show that the conditional score can be derived exactly from a trained (unconditional) score using affine transformations, eliminating the need for conditional score approximations. Our approach is formulated in infinite-dimensional function spaces, making it inherently discretization-invariant. We support this formulation with a rigorous convergence analysis that justifies UCoS beyond any specific discretization. Finally we validate UCoS through high-dimensional computed tomography (CT) and image deblurring experiments, demonstrating both scalability and accuracy.

Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree of ambiguity that may exist about its solution. Traditional measures of ill-posedness, such as the condition number of a matrix, provide characterizations that are global in nature. While such characterizations can be powerful, they can also fail to provide full insight into situations where certain entries of the solution vector are more or less ambiguous than others. In this work, we derive novel theoretical lower- and upper-bounds that apply to individual entries of the solution vector, and are valid for all potential solution vectors that are nearly data-consistent. These bounds are agnostic to the noise statistics and the specific method used to solve the inverse problem, and are also shown to be tight. In addition, our results also lead us to introduce an entrywise version of the traditional condition number, which provides a substantially more nuanced characterization of scenarios where certain elements of the solution vector are less sensitive to perturbations than others. Our results are illustrated in an application to magnetic resonance imaging reconstruction, and we include discussions of practical computation methods for large-scale inverse problems, connections between our new theory and the traditional Cramér-Rao bound under statistical modeling assumptions, and potential extensions to cases involving constraints beyond just data-consistency.

Глобальная деформационная модель по предлагаемой методике строиться в два этапа. На первом этапе, в предположении волновой природы сейсмичности, строится модель первичных син- гулярных источников деформационного процесса. Для этого применяется алгоритм прямой локации семейства источников деформационных волн по глобальному сейсмическому каталогу или по каталогу комплексных геодинамических индикаторов. На втором этапе, на основе решения линейной обратной задачи, семейство сингулярных источников наделяется амплитудными характеристиками. Для этого используются мировые каталоги скоростей GPS/GNSS, на основе которых ставится и решается линей- ная обратная задача. Таким образом, было построено семейство первичных сингулярных источников возможных деформационных фронтов для Земли. Основная масса этих источников сосредоточена вну- три ядра (R<3000 км, 93.7 %). Применение обратной четырехмерной засечки для найденного множе- ства источников первичных сингулярных деформаций позволяет восстановить с хорошей точностью исходный каталог землетрясений в земной коре и верхней мантии. Даются формулы для расчета всех элементов тензора деформаций и трех главных инвариант деформационного тензора. Описан способ снижения размерности обратной линейной задачи. Изучается структура полученного решения путем расчета различных элементов динамического деформационного тензора. The global deformation model according to the proposed method is built in two stages. At the fi rst stage, under the assumption of the wave nature of seismicity, a model of primary singular sources of the deformation process is built. To do this, the algorithm of direct location of the family of sources of deformation waves on the global seismic catalog or on the catalog of complex geodynamic indicators is used. At the second stage, based on the solution of the linear inverse problem, the family of singular sources is endowed with amplitude characteristics. For this, world directories of GPS/GNSS speeds are used, on the basis of which a linear inverse problem is set and solved. Thus, a family of primary singular sources of possible deformation fronts for the Earth was built. The bulk of these sources are concentrated inside the core (R < 3000 km, 93.7%). The use of inverse four-dimensional serifs for the found set of sources of primary singular deformations makes it possible to restore with good accuracy the original catalog of earthquakes in the earth’s crust and upper mantle. Formulas are given for calculation of all elements of strain tensor and three main invariants of strain tensor. Described is a method of reducing the dimension of an inverse linear problem. The structure of the resulting solution is studied by calculating various elements of the dynamic strain tensor.

Article Stable numerical solution to linear inverse heat conduction problems by the conjugate gradient method was published on January 1, 1995 in the journal Journal of Inverse and Ill-posed Problems (volume 3, issue 6).

Recent studies in linear inverse problems have recognized the sparse representation of unknown signal in a certain basis as an useful and effective prior information to solve those problems. In many multiscale bases (e.g. wavelets), signals of interest (e.g. piecewise-smooth signals) not only have few significant coefficients, but also <i>those significant coefficients are well-organized in trees</i>. We propose to exploit the <i>tree-structured sparse representation</i> as additional prior information for linear inverse problems with limited numbers of measurements. We present numerical results showing that exploiting the sparse <i>tree</i> representations lead to better reconstruction while requiring less time compared to methods that only assume sparse representations.

Fourier phase retrieval problem is equivalent to the recovery of a two-dimensional image from its autocorrelation measurements. This problem is generally nonlinear and nonconvex. Good initialization and prior information about the support or sparsity of the target image are often critical for a robust recovery. In this paper, we show that the presence of a known reference image can help us solve the nonlinear phase retrieval problem as a sequence of small linear inverse problems. Instead of recovering the entire image at once, our sequential method recovers a small number of rows or columns by solving a linear deconvolution problem at every step. Existing methods for the reference-based (holographic) phase retrieval either assume that the reference and target images are sufficiently separated so that the recovery problem is linear or recover the image via nonlinear optimization. In contrast, our proposed method does not require the separation condition. We performed an extensive set of simulations to demonstrate that our proposed method can successfully recover images from autocorrelation data under different settings of reference placement and noise.