查看更多>>摘要：In this paper, we study the L (1)/L (2) minimization on the gradient for imaging applications. Several recent works have demonstrated that L (1)/L (2) is better than the L (1) norm when approximating the L (0) norm to promote sparsity. Consequently, we postulate that applying L (1)/L (2) on the gradient is better than the classic total variation (the L (1) norm on the gradient) to enforce the sparsity of the image gradient. Numerically, we design a specific splitting scheme, under which we can prove subsequential and global convergence for the alternating direction method of multipliers (ADMM) under certain conditions. Experimentally, we demonstrate visible improvements of L (1)/L (2) over L (1) and other nonconvex regularizations for image recovery from low-frequency measurements and two medical applications of magnetic resonance imaging and computed tomography reconstruction. Finally, we reveal some empirical evidence on the superiority of L (1)/L (2) over L (1) when recovering piecewise constant signals from low-frequency measurements to shed light on future works.
查看更多>>摘要：The sketch-and-project, as a general archetypal algorithm for solving linear systems, unifies a variety of randomized iterative methods such as the randomized Kaczmarz and randomized coordinate descent. However, since it aims to find a least-norm solution from a linear system, the randomized sparse Kaczmarz can not be included. This motivates us to propose a more general framework, called sketched Bregman projection (SBP) method, in which we are able to find solutions with certain structures from linear systems. To generalize the concept of adaptive sampling to the SBP method, we show how the progress, measured by Bregman distance, of single step depends directly on a sketched loss function. Theoretically, we provide detailed global convergence results for the SBP method with different adaptive sampling rules. At last, for the (sparse) Kaczmarz methods, a group of numerical simulations are tested, with which we verify that the methods utilizing sampling Kaczmarz-Motzkin rule demands the fewest computational costs to achieve a given error bound comparing to the corresponding methods with other sampling rules.
查看更多>>摘要：Magnetic resonance (MR) images are frequently corrupted by Rician noise during image acquisition and transmission. And it is very challenging to restore MR data because Rician noise is signal-dependent. By exploring the nonlocal self-similarity of natural images and further using the low-rank prior of the matrices formed by nonlocal similar patches for 2D data or cubes for 3D data, we propose in this paper a new nonlocal low-rank regularization (NLRR) method including an optimization model and an efficient iterative algorithm to remove Rician noise. The proposed mathematical model consists of a data fidelity term derived from a maximum a posteriori estimation and a NLRR term using the log-det function. The resulting model in terms of approximated patch/cube matrices is non-convex and non-smooth. To solve this model, we propose an alternating reweighted minimization (ARM) algorithm using the Lipschitz-continuity of the gradient of the fidelity term and the concavity of the logarithmic function in the log-det function. The subproblems of the ARM algorithm have closed-form solutions and its limit points are first-order critical points of the problem. The ARM algorithm is further integrated with a two-stage scheme to enhance the denoising performance of the proposed NLRR method. Experimental results tested on 2D and 3D MR data, including simulated and real data, show that the NLRR method outperforms existing state-of-the-art methods for removing Rician noise.
查看更多>>摘要：We present a microlocal analysis of two novel Radon transforms of interest in Compton scattering tomography, which map compactly supported L-2 functions to their integrals over seven-dimensional sets of apple and lemon surfaces. Specifically, we show that the apple and lemon transforms are elliptic Fourier integral operators, which satisfy the Bolker condition. After an analysis of the full seven-dimensional case, we focus our attention on nD subsets of apple and lemon surfaces with fixed central axis, where n < 7. Such subsets of surface integrals have applications in airport baggage and security screening. When the data dimensionality is restricted, the apple transform is shown to violate the Bolker condition, and there are artifacts which occur on apple-cylinder intersections. The lemon transform is shown to satisfy the Bolker condition, when the support of the function is restricted to the strip {0 < z < 1}.
查看更多>>摘要：We consider increasing stability in the inverse Schrodinger potential problem with power type nonlinearities at a large wavenumber. Two linearization approaches, with respect to small boundary data and small potential function, are proposed and their performance on the inverse Schrodinger potential problem is investigated. It can be observed that higher order linearization for small boundary data can provide an increasing stability for an arbitrary power type nonlinearity term if the wavenumber is chosen large. Meanwhile, linearization with respect to the potential function leads to increasing stability for a quadratic nonlinearity term, which highlights the advantage of nonlinearity in solving the inverse Schrodinger potential problem. Noticing that both linearization approaches can be numerically approximated, we provide several reconstruction algorithms for the quadratic and general power type nonlinearity terms, where one of these algorithms is designed based on boundary measurements of multiple wavenumbers. Several numerical examples shed light on the efficiency of our proposed algorithms.
查看更多>>摘要：Ray transforms for vector or tensor fields have been studied for a long time, also in connection with Doppler or polarization tomography. In contrast to the classical Radon transform or x-ray transform their known inversion formulae are rather complicated, and represented, for example, using series expansions or singular value decompositions. Recently, applications of the approximate inverse have been studied. In this paper we present a new class of inversion formulae for compactly supported fields similar to those of the x-ray transform. The derivation is based on two observations. First, we apply Derevtsov's formula relating the ray transforms of vector or tensor fields to a derivative of the x-ray transform of a generating potential, [3]. Second, we apply the calculation of derivatives of a scalar function from its x-ray transform, as it is realized in feature reconstruction, [15, 16]. Generalizations complete the presentation.
查看更多>>摘要：Parameter selection is crucial to regularization-based image restoration methods. Generally speaking, a spatially fixed parameter for the regularization term does not perform well for both edge and smooth areas. A larger parameter for the regularization term reduces noise better in smooth areas but blurs edge regions, while a small parameter sharpens edge but causes residual noise. In this paper, an automated spatially adaptive regularization model, which combines the harmonic and total variation (TV) terms, is proposed for the image reconstruction from noisy and blurred observation. The proposed model detects the edges and then spatially adjusts the parameters of Tikhonov and TV regularization terms for each pixel according to the edge information. Accordingly, the edge information matrix will also be dynamically updated during the iterations. Computationally, the newly-established model is convex, which can be solved by the semi-proximal alternating direction method of multipliers with a linear convergence rate. Numerical simulation results demonstrate that the proposed model effectively preserves the image edges and eliminates the noise and blur at the same time. In comparison to state-of-the-art algorithms, it outperforms other methods in terms of peak signal to noise ratio, structural similarity index and visual quality.
查看更多>>摘要：This paper investigates the inverse problem of estimating a discontinuous parameter in a quasi-variational inequality involving multi-valued terms. We prove that a well-defined parameter-to-solution map admits weakly compact values under some quite general assumptions. The Kakutani-Ky Fan fixed point principle for multi-valued maps is the primary technical tool for this result. Inspired by the total variation regularization for estimating discontinuous parameters, we develop an abstract regularization framework for the inverse problem and provide a new existence result. The theoretical results are applied to identify a parameter in an elliptic mixed boundary value system with the p-Laplace operator, an implicit obstacle, and multi-valued terms involving convex subdifferentials and the generalized subdifferentials in the sense of Clarke.
查看更多>>摘要：In this paper, we generalize inexact Newton regularization methods to solve nonlinear inverse problems from a reflexive Banach space to a Banach space. The image space is not necessarily reflexive so that the method can be used to deal with various types of noise such as the Gaussian noise and the impulsive noise. The method consists of an outer Newton iteration and an inner scheme which provides increments by applying the regularization technique to the local linearized equations. Under some assumptions, in particular, the reflexivity of the image space is not required, we present a novel convergence analysis of the inexact Newton regularization method with inner scheme defined by Landweber iteration. Furthermore, by employing a two-point gradient method as inner regularization scheme to accelerate the convergence, we propose an accelerated version of inexact Newton-Landweber method and present the detailed convergence analysis. The numerical simulations are provided to demonstrate the effectiveness of the proposed methods in handling different kinds of noise and the fast convergence of the accelerated method.
查看更多>>摘要：In electrical impedance tomography (EIT), we aim to solve the conductivity within a target body through electrical measurements made on the surface of the target. This inverse conductivity problem is severely ill-posed, especially in real applications with only partial boundary data available. Thus regularization has to be introduced. Conventionally regularization promoting smooth features is used, however, the Mumford-Shah (M-S) regularizer familiar for image segmentation is more appropriate for targets consisting of several distinct objects or materials. It is, however, numerically challenging. We show theoretically through Gamma-convergence that a modification of the Ambrosio-Tortorelli approximation of the M-S regularizer is applicable to EIT, in particular the complete electrode model of boundary measurements. With numerical and experimental studies, we confirm that this functional works in practice and produces higher quality results than typical regularizations employed in EIT when the conductivity of the target consists of distinct smoothly-varying regions.
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