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Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-12-25 Ziang Long; Penghang Yin; Jack Xin
In this paper, we study the dynamics of gradient descent in learning neural networks for classification problems. Unlike in existing works, we consider the linearly non-separable case where the training data of different classes lie in orthogonal subspaces. We show that when the network has sufficient (but not exceedingly large) number of neurons, (1) the corresponding minimization problem has a desirable
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Stochastic greedy algorithms for multiple measurement vectors Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-11-02 Jing Qin; Shuang Li; Deanna Needell; Anna Ma; Rachel Grotheer; Chenxi Huang; Natalie Durgin
Sparse representation of a single measurement vector (SMV) has been explored in a variety of compressive sensing applications. Recently, SMV models have been extended to solve multiple measurement vectors (MMV) problems, where the underlying signal is assumed to have joint sparse structures. To circumvent the NP-hardness of the $ \ell_0 $ minimization problem, many deterministic MMV algorithms solve
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Fast algorithms for robust principal component analysis with an upper bound on the rank Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-11-02 Ningyu Sha; Lei Shi; Ming Yan
The robust principal component analysis (RPCA) decomposes a data matrix into a low-rank part and a sparse part. There are mainly two types of algorithms for RPCA. The first type of algorithm applies regularization terms on the singular values of a matrix to obtain a low-rank matrix. However, calculating singular values can be very expensive for large matrices. The second type of algorithm replaces
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Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-11-24 Dan Zhu; Rosemary A. Renaut; Hongwei Li; Tianyou Liu
A fast non-convex low-rank matrix decomposition method for potential field data separation is presented. The singular value decomposition of the large size trajectory matrix, which is also a block Hankel matrix, is obtained using a fast randomized singular value decomposition algorithm in which fast block Hankel matrix-vector multiplications are implemented with minimal memory storage. This fast block
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Two-dimensional seismic data reconstruction using patch tensor completion Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-28 Qun Liu; Lihua Fu; Meng Zhang; Wanjuan Zhang
Seismic data are often undersampled owing to physical or financial limitations. However, complete and regularly sampled data are becoming increasingly critical in seismic processing. In this paper, we present an efficient two-dimensional (2D) seismic data reconstruction method that works on texture-based patches. It performs completion on a patch tensor, which folds texture-based patches into a tensor
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Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-28 Chan Liu; Jin Wen; Zhidong Zhang
In this work, an inverse problem in the fractional diffusion equation with random source is considered. The measurements we use are the statistical moments of the realizations of single point observation $ u(x_0,t,\omega). $ We build a representation of the solution $ u $ in the integral sense, then prove some theoretical results like uniqueness and stability. After that, we establish a numerical algorithm
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Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-28 Guanghui Hu; Andrea Mantile; Mourad Sini; Tao Yin
This paper is concerned with the time-harmonic direct and inverse elastic scattering by an extended rigid elastic body surrounded by a finite number of point-like obstacles. We first justify the point-interaction model for the Lamé operator within the singular perturbation approach. For a general family of pointwise-supported singular perturbations, including anisotropic and non-local interactions
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Interactions of semilinear progressing waves in two or more space dimensions Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-28 Antônio Sá Barreto
We analyze the behavior of the singularities of solutions of semilinear wave equations after the interaction of three transversal conormal waves. Our results hold for space dimensions two and higher, and for arbitrary $ {{C}^{\infty }} $ nonlinearity. The case of two space dimensions in which the nonlinearity is a polynomial was studied by the author and Yiran Wang. We also indicate possible applications
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Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-28 Marion Darbas; Jérémy Heleine; Stephanie Lohrengel
This paper concerns the numerical resolution of a data completion problem for the time-harmonic Maxwell equations in the electric field. The aim is to recover the missing data on the inaccessible part of the boundary of a bounded domain from measured data on the accessible part. The non-iterative quasi-reversibility method is studied and different mixed variational formulations are proposed. Well-posedness
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A parallel operator splitting algorithm for solving constrained total-variation retinex Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-10-15 Leyu Hu; Wenxing Zhang; Xingju Cai; Deren Han
An ideal image is desirable to faithfully represent the real-world scene. However, the observed images from imaging system are typically involved in the illumination of light. As the human visual system (HVS) is capable of perceiving identical color with spatially varying illumination, retinex theory is established to probe the rationale of HVS on perceiving color. In the realm of image processing
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Spatial-Frequency domain nonlocal total variation for image denoising Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-10-15 Haijuan Hu; Jacques Froment; Baoyan Wang; Xiequan Fan
Following the pioneering works of Rudin, Osher and Fatemi on total variation (TV) and of Buades, Coll and Morel on non-local means (NL-means), the last decade has seen a large number of denoising methods mixing these two approaches, starting with the nonlocal total variation (NLTV) model. The present article proposes an analysis of the NLTV model for image denoising as well as a number of improvements
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Automatic Extraction of Cell Nuclei Using Dilated Convolutional Network Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-03 Rajendra K C Khatri; Brendan J Caseria; Yifei Lou; Guanghua Xiao; Yan Cao
Pathological examination has been done manually by visual inspection of hematoxylin and eosin (H&E)-stained images. However, this process is labor intensive, prone to large variations, and lacking reproducibility in the diagnosis of a tumor. We aim to develop an automatic workflow to extract different cell nuclei found in cancerous tumors portrayed in digital renderings of the H&E-stained images. For
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LANTERN: Learn analysis transform network for dynamic magnetic resonance imaging Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-04 Shanshan Wang; Yanxia Chen; Taohui Xiao; Lei Zhang; Xin Liu; Hairong Zheng
This paper proposes to learn analysis transform network for dynamic magnetic resonance imaging (LANTERN). Integrating the strength of CS-MRI and deep learning, the proposed framework is highlighted in three components: (ⅰ) The spatial and temporal domains are sparsely constrained by adaptively trained convolutional filters; (ⅱ) We introduce an end-to-end framework to learn the parameters in LANTERN
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Image retinex based on the nonconvex TV-type regularization Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-03 Yuan Wang; Zhi-Feng Pang; Yuping Duan; Ke Chen
Retinex theory is introduced to show how the human visual system perceives the color and the illumination effect such as Retinex illusions, medical image intensity inhomogeneity and color shadow effect etc.. Many researchers have studied this ill-posed problem based on the framework of the variation energy functional for decades. However, to the best of our knowledge, the existing models via the sparsity
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Some Worst-Case Datasets of Deterministic First-Order Methods for Solving Binary Logistic Regression Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-03 Yuyuan Ouyang; Trevor Squires
We present in this paper some worst-case datasets of deterministic first-order methods for solving large-scale binary logistic regression problems. Under the assumption that the number of algorithm iterations is much smaller than the problem dimension, with our worst-case datasets it requires at least $ {{{\mathcal O}}}(1/\sqrt{\varepsilon}) $ first-order oracle inquiries to compute an $ \varepsilon
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A new initialization method based on normed statistical spaces in deep networks Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-03 Hongfei Yang; Xiaofeng Ding; Raymond Chan; Hui Hu; Yaxin Peng; Tieyong Zeng
Training deep neural networks can be difficult. For classical neural networks, the initialization method by Xavier and Yoshua which is later generalized by He, Zhang, Ren and Sun can facilitate stable training. However, with the recent development of new layer types, we find that the above mentioned initialization methods may fail to lead to successful training. Based on these two methods, we will
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Optimal recovery of a radiating source with multiple frequencies along one line Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-03 Tommi Brander; Joonas Ilmavirta; Petteri Piiroinen; Teemu Tyni
We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely determined by these measurements up to averaging over
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Reproducible Kernel Hilbert Space Based Global and Local Image Segmentation Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-03 Liam Burrows; Weihong Guo; Ke Chen; Francesco Torella
Image segmentation is the task of partitioning an image into individual objects, and has many important applications in a wide range of fields. The majority of segmentation methods rely on image intensity gradient to define edges between objects. However, intensity gradient fails to identify edges when the contrast between two objects is low. In this paper we aim to introduce methods to make such weak
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Adversarial Defense via the Data-Dependent Activation, Total Variation Minimization, and Adversarial Training Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-08-03 Bao Wang; Alex Lin; Penghang Yin; Wei Zhu; Andrea L. Bertozzi; Stanley J. Osher
We improve the robustness of Deep Neural Net (DNN) to adversarial attacks by using an interpolating function as the output activation. This data-dependent activation remarkably improves both the generalization and robustness of DNN. In the CIFAR10 benchmark, we raise the robust accuracy of the adversarially trained ResNet20 from $ \sim 46\% $ to $ \sim 69\% $ under the state-of-the-art Iterative Fast
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Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-07-06 Alexey Penenko
The inverse source problems for nonlinear advection-diffusion-reaction models with image-type measurement data are considered. The use of the sensitivity operators, constructed of the ensemble of adjoint problem solutions, allows transforming the inverse problems stated as the systems of nonlinear PDE to a family of operator equations depending on the given set of functions in the space of measurement
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Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-07-06 Boya Liu
We study the inverse boundary value problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator from the knowledge of the partial Cauchy data set. Our geometric setting is that of a domain whose inaccessible portion of the boundary is contained in a hyperplane, and we are given the Cauchy data set on the complement. The uniqueness and logarithmic stability
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Numerical recovery of magnetic diffusivity in a three dimensional spherical dynamo equation Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-07-06 Djemaa Messaoudi; Osama Said Ahmed; Komivi Souley Agbodjan; Ting Cheng; Daijun Jiang
This paper is concerned with the analysis on a numerical recovery of the magnetic diffusivity in a three dimensional (3D) spherical dynamo equation. We shall transform the ill-posed problem into an output least squares nonlinear minimization by an appropriately selected Tikhonov regularization, whose regularizing effects and mathematical properties are justified. The nonlinear optimization problem
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Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-07-06 Yavar Kian; Alexander Tetlow
We consider the inverse problem of Hölder-stably determining the time- and space-dependent coefficients of the Schrödinger equation on a simple Riemannian manifold with boundary of dimension $ n\geq2 $ from the knowledge of the Dirichlet-to-Neumann map. Assuming the divergence of the magnetic potential is known, we show that the electric and magnetic potentials can be Hölder-stably recovered from these
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Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-07-06 Mourad Bellassoued; Oumaima Ben Fraj
In this work, we study the stable determination of time-dependent coefficients appearing in the Schrödinger equation from partial Dirichlet-to-Neumann map measured on an arbitrary part of the boundary. Specifically, we establish stability estimates up to the natural gauge for the magnetic potential.
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A nonconvex truncated regularization and box-constrained model for CT reconstruction Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-07-06 Weina Wang; Chunlin Wu; Yiming Gao
X-ray computed tomography has been a useful technology in cancer detection and radiation therapy. However, high radiation dose during CT scans may increase the underlying risk of healthy organs. Usually, sparse-view X-ray projection is an effective method to reduce radiation. In this paper, we propose a constrained nonconvex truncated regularization model for this low-dose CT reconstruction. It preserves
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Nonlocal regularized CNN for image segmentation Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-07-06 Fan Jia; Xue-Cheng Tai; Jun Liu
Non-local dependency is a very important prior for many image segmentation tasks. Generally, convolutional operations are building blocks that process one local neighborhood at a time which means the convolutional neural networks(CNNs) usually do not explicitly make use of the non-local prior on image segmentation tasks. Though the pooling and dilated convolution techniques can enlarge the receptive
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Convexification for a 1D hyperbolic coefficient inverse problem with single measurement data Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-07-06 Alexey Smirnov; Michael Klibanov; Loc Nguyen
A version of the convexification numerical method for a Coefficient Inverse Problem for a 1D hyperbolic PDE is presented. The data for this problem are generated by a single measurement event. This method converges globally. The most important element of the construction is the presence of the Carleman Weight Function in a weighted Tikhonov-like functional. This functional is strictly convex on a certain
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On the simultaneous recovery of the conductivity and the nonlinear reaction term in a parabolic equation Inverse Probl. Imaging (IF 1.373) Pub Date : 2020-07-06 Barbara Kaltenbacher; William Rundell
This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity $ a(x) $ and the nonlinear reaction term $ f(u) $ in a reaction-diffusion equation from overposed data. These measurements can consist of: the value of two different solution measurements taken at a later time $ T $; time-trace profiles from two solutions; or both final time and time-trace measurements
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EMPIRICAL AVERAGE-CASE RELATION BETWEEN UNDERSAMPLING AND SPARSITY IN X-RAY CT. Inverse Probl. Imaging (IF 1.373) Pub Date : 2016-03-29 Jakob S Jørgensen,Emil Y Sidky,Per Christian Hansen,Xiaochuan Pan
In X-ray computed tomography (CT) it is generally acknowledged that reconstruction methods exploiting image sparsity allow reconstruction from a significantly reduced number of projections. The use of such reconstruction methods is inspired by recent progress in compressed sensing (CS). However, the CS framework provides neither guarantees of accurate CT reconstruction, nor any relation between sparsity
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