Inverse Problems and Imaging最新文献

{"title":"Bayesian neural network priors for edge-preserving inversion","authors":"Chen Li,Matthew Dunlop,Georg Stadler","doi":"10.3934/ipi.2022022","DOIUrl":"https://doi.org/10.3934/ipi.2022022","url":null,"abstract":"<p style='text-indent:20px;'>We consider Bayesian inverse problems wherein the unknown state is assumed to be a function with discontinuous structure a priori. A class of prior distributions based on the output of neural networks with heavy-tailed weights is introduced, motivated by existing results concerning the infinite-width limit of such networks. We show theoretically that samples from such priors have desirable discontinuous-like properties even when the network width is finite, making them appropriate for edge-preserving inversion. Numerically we consider deconvolution problems defined on one- and two-dimensional spatial domains to illustrate the effectiveness of these priors; MAP estimation, dimension-robust MCMC sampling and ensemble-based approximations are utilized to probe the posterior distribution. The accuracy of point estimates is shown to exceed those obtained from non-heavy tailed priors, and uncertainty estimates are shown to provide more useful qualitative information.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"39 7","pages":"0"},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138512436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two single-measurement uniqueness results for inverse scattering problems within polyhedral geometries","authors":"Xinlin Cao,Huaian Diao,Hongyu Liu,Jun Zou","doi":"10.3934/ipi.2022023","DOIUrl":"https://doi.org/10.3934/ipi.2022023","url":null,"abstract":"<p style='text-indent:20px;'>We consider the unique determinations of impenetrable obstacles or diffraction grating profiles in <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{R}^3 $end{document}</tex-math></inline-formula> by a single far-field measurement within polyhedral geometries. We are particularly interested in the case that the scattering objects are of impedance type. We derive two new unique identifiability results by a single measurement for the inverse scattering problem in the aforementioned two challenging setups. The main technical idea is to exploit certain quantitative geometric properties of the Laplacian eigenfunctions which were initiated in our recent works [<xref ref-type=\"bibr\" rid=\"b12\">12</xref>,<xref ref-type=\"bibr\" rid=\"b13\">13</xref>]. In this paper, we derive novel geometric properties that generalize and extend the related results in [<xref ref-type=\"bibr\" rid=\"b13\">13</xref>], which further enable us to establish the new unique identifiability results. It is pointed out that in addition to the shape of the obstacle or the grating profile, we can simultaneously recover the boundary impedance parameters.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"35 1","pages":"0"},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138527250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analytical reconstruction formula with efficient implementation for a modality of Compton scattering tomography with translational geometry","authors":"Cécilia Tarpau, Javier Cebeiro, Geneviève Rollet, Maï K. Nguyen, Laurent Dumas","doi":"10.3934/ipi.2021075","DOIUrl":"https://doi.org/10.3934/ipi.2021075","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we address an alternative formulation for the exact inverse formula of the Radon transform on circle arcs arising in a modality of Compton Scattering Tomography in translational geometry proposed by Webber and Miller (Inverse Problems (36)2, 025007, 2020). The original study proposes a first method of reconstruction, using the theory of Volterra integral equations. The numerical realization of such a type of inverse formula may exhibit some difficulties, mainly due to stability issues. Here, we provide a suitable formulation for exact inversion that can be straightforwardly implemented in the Fourier domain. Simulations are carried out to illustrate the efficiency of the proposed reconstruction algorithm.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"40 4","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138512435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Learning spectral windowing parameters for regularization using unbiased predictive risk and generalized cross validation techniques for multiple data sets","authors":"Michael J. Byrne, R. Renaut","doi":"10.3934/ipi.2023006","DOIUrl":"https://doi.org/10.3934/ipi.2023006","url":null,"abstract":"During the inversion of discrete linear systems noise in data can be amplified and result in meaningless solutions. To combat this effect, characteristics of solutions that are considered desirable are mathematically implemented during inversion, which is a process called regularization. The influence of provided prior information is controlled by non-negative regularization parameter(s). There are a number of methods used to select appropriate regularization parameters, as well as a number of methods used for inversion. New methods of unbiased risk estimation and generalized cross validation are derived for finding spectral windowing regularization parameters. These estimators are extended for finding the regularization parameters when multiple data sets with common system matrices are available. It is demonstrated that spectral windowing regularization parameters can be learned from these new estimators applied for multiple data and with multiple windows. The results demonstrate that these modified methods, which do not require the use of true data for learning regularization parameters, are effective and efficient, and perform comparably to a learning method based on estimating the parameters using true data. The theoretical developments are validated for the case of two dimensional image deblurring. The results verify that the obtained estimates of spectral windowing regularization parameters can be used effectively on validation data sets that are separate from the training data, and do not require known data.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45975717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Pestov identities and X-ray tomography on manifolds of low regularity","authors":"Joonas Ilmavirta, Antti Kykkanen","doi":"10.3934/ipi.2023017","DOIUrl":"https://doi.org/10.3934/ipi.2023017","url":null,"abstract":"We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds $(M,g)$ with $g in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This $C^{1,1}$-regularity is optimal on the H\"older scale. The bulk of the article is devoted to setting up a calculus of differential and curvature operators on the unit sphere bundle atop this non-smooth structure.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47679469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions","authors":"Zhi-yuan Li, Yikan Liu, Masahiro Yamamoto","doi":"10.3934/ipi.2022027","DOIUrl":"https://doi.org/10.3934/ipi.2022027","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we obtain the sharp uniqueness for an inverse <inline-formula><tex-math id=\"M1\">begin{document}$ x $end{document}</tex-math></inline-formula>-source problem for a one-dimensional time-fractional diffusion equation with a zeroth-order term by the minimum possible lateral Cauchy data. The key ingredient is the unique continuation which holds for weak solutions.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49129213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Time-harmonic diffuse optical tomography: Hölder stability of the derivatives of the optical properties of a medium at the boundary","authors":"Jason Curran, Romina Gaburro, C. Nolan, E. Somersalo","doi":"10.3934/ipi.2022044","DOIUrl":"https://doi.org/10.3934/ipi.2022044","url":null,"abstract":"<p style='text-indent:20px;'>We address the inverse problem in Optical Tomography of stably determining the optical properties of an anisotropic medium <inline-formula><tex-math id=\"M3\">begin{document}$ Omegasubsetmathbb{R}^n $end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id=\"M4\">begin{document}$ ngeq 3 $end{document}</tex-math></inline-formula>, under the so-called <i>diffusion approximation</i>. Assuming that the <i>scattering coefficient</i> <inline-formula><tex-math id=\"M5\">begin{document}$ mu_s $end{document}</tex-math></inline-formula> is known, we prove Hölder stability of the derivatives of any order of the <i>absorption coefficient</i> <inline-formula><tex-math id=\"M6\">begin{document}$ mu_a $end{document}</tex-math></inline-formula> at the boundary <inline-formula><tex-math id=\"M7\">begin{document}$ partialOmega $end{document}</tex-math></inline-formula> in terms of the measurements, in the time-harmonic case, where the anisotropic medium <inline-formula><tex-math id=\"M8\">begin{document}$ Omega $end{document}</tex-math></inline-formula> is interrogated with an input field that is modulated with a fixed harmonic frequency <inline-formula><tex-math id=\"M9\">begin{document}$ omega = frac{k}{c} $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M10\">begin{document}$ c $end{document}</tex-math></inline-formula> is the speed of light and <inline-formula><tex-math id=\"M11\">begin{document}$ k $end{document}</tex-math></inline-formula> is the wave number. The stability estimates are established under suitable conditions that include a range of variability for <inline-formula><tex-math id=\"M12\">begin{document}$ k $end{document}</tex-math></inline-formula> and they rely on the construction of singular solutions of the underlying forward elliptic system, which extend results obtained in J. Differential Equations 84 (2): 252-272 for the single elliptic equation and those obtained in Applicable Analysis DOI:<a href=\"http://dx.doi.org/10.1080/00036811.2020.1758314\" target=\"_blank\">10.1080/00036811.2020.1758314</a>, where a Lipschitz type stability estimate of <inline-formula><tex-math id=\"M13\">begin{document}$ mu_a $end{document}</tex-math></inline-formula> on <inline-formula><tex-math id=\"M14\">begin{document}$ partialOmega $end{document}</tex-math></inline-formula> was established in terms of the measurements.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43885473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A new approach to the inverse discrete transmission eigenvalue problem","authors":"N. Bondarenko, V. Yurko","doi":"10.3934/ipi.2021073","DOIUrl":"https://doi.org/10.3934/ipi.2021073","url":null,"abstract":"A discrete analog is considered for the inverse transmission eigenvalue problem, having applications in acoustics. We provide a well-posed inverse problem statement, develop a constructive procedure for solving this problem, prove uniqueness of solution, global solvability, local solvability, and stability. Our approach is based on the reduction of the discrete transmission eigenvalue problem to a linear system with polynomials of the spectral parameter in the boundary condition.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"118 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77995602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Carleman estimate and an energy method for a first-order symmetric hyperbolic system","authors":"G. Floridia, H. Takase, M. Yamamoto","doi":"10.3934/ipi.2022016","DOIUrl":"https://doi.org/10.3934/ipi.2022016","url":null,"abstract":"<p style='text-indent:20px;'>For a symmetric hyperbolic system of the first order, we prove a Carleman estimate under some positivity condition concerning the coefficient matrices. Next, applying the Carleman estimate, we prove an observability <inline-formula><tex-math id=\"M1\">begin{document}$ L^2 $end{document}</tex-math></inline-formula>-estimate for initial values by boundary data.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47933044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Microlocal analysis of borehole seismic data","authors":"R. Felea, Romina Gaburro, A. Greenleaf, C. Nolan","doi":"10.3934/ipi.2022026","DOIUrl":"https://doi.org/10.3934/ipi.2022026","url":null,"abstract":"Borehole seismic data is obtained by receivers located in a well, with sources located on the surface or another well. Using microlocal analysis, we study possible approximate reconstruction, via linearized, filtered backprojection, of an isotropic sound speed in the subsurface for three types of data sets. The sources may form a dense array on the surface, or be located along a line on the surface (walkaway geometry) or in another borehole (crosswell). We show that for the dense array, reconstruction is feasible, with no artifacts in the absence of caustics in the background ray geometry, and mild artifacts in the presence of fold caustics in a sense that we define. In contrast, the walkaway and crosswell data sets both give rise to strong, nonremovable artifacts.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48704847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}