Inverse Problems and Imaging最新文献

{"title":"Stable determination of an anisotropic inclusion in the Schrödinger equation from local Cauchy data","authors":"Sonia Foschiatti, E. Sincich","doi":"10.3934/ipi.2022063","DOIUrl":"https://doi.org/10.3934/ipi.2022063","url":null,"abstract":"We consider the inverse problem of determining an inclusion contained in a body for a Schr\"odinger type equation by means of local Cauchy data. Both the body and the inclusion are made by inhomogeneous and anisotropic materials. Under mild a priori assumptions on the unknown inclusion, we establish a logarithmic stability estimate in terms of the local Cauchy data. In view of possible applications, we also provide a stability estimate in terms of an ad-hoc misfit functional.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42242533","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":"Super-localisation of a point-like emitter in a resonant environment: Correction of the mirage effect","authors":"Lorenzo Baldassari, A. Vanel, Pierre Millien","doi":"10.3934/ipi.2022054","DOIUrl":"https://doi.org/10.3934/ipi.2022054","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we show that it is possible to overcome one of the fundamental limitations of super-resolution microscopy: the necessity to be in an <i>optically homogeneous</i> environment. Using recent modal approximation results from [<xref ref-type=\"bibr\" rid=\"b10\">10</xref>, <xref ref-type=\"bibr\" rid=\"b7\">7</xref>], we show, as a proof of concept, that it is possible to recover the position of a single point-like emitter in a <i>known resonant environment</i> from far-field measurements, with a precision two orders of magnitude below the classical Rayleigh limit. The procedure does not involve solving any partial differential equation, is computationally light (optimisation in <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{R}^d $end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M2\">begin{document}$ d $end{document}</tex-math></inline-formula> of the order of <inline-formula><tex-math id=\"M3\">begin{document}$ 10 $end{document}</tex-math></inline-formula>) and is therefore suited for the recovery of a very large number of single emitters.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47692587","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":"Simultaneous recovery of attenuation and source density in SPECT","authors":"S. Holman, Philip Richardson","doi":"10.3934/ipi.2023005","DOIUrl":"https://doi.org/10.3934/ipi.2023005","url":null,"abstract":"We show that under a certain non-cancellation condition the attenuated Radon transform uniquely determines piecewise constant attenuation $a$ and piecewise $C^2$ source density $f$ with jumps over real analytic boundaries possibly having corners. We also look at numerical examples in which the non-cancellation condition fails and show that unique reconstruction of multi-bang $a$ and $f$ is still appears to be possible although not yet explained by theoretical results.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46207294","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 Regge poles problem on a warped ball","authors":"Jack Borthwick, N. Boussaid, Thierry Daud'e","doi":"10.3934/ipi.2023031","DOIUrl":"https://doi.org/10.3934/ipi.2023031","url":null,"abstract":"In this paper, we study a new type of inverse problem on warped product Riemannian manifolds with connected boundary that we name warped balls. Using the symmetry of the geometry, we first define the set of Regge poles as the poles of the meromorphic continuation of the Dirichlet-to-Neumann map with respect to the complex angular momentum appearing in the separation of variables procedure. These Regge poles can also be viewed as the set of eigenvalues and resonances of a one-dimensional Schr\"odinger equation on the half-line, obtained after separation of variables. Secondly, we find a precise asymptotic localisation of the Regge poles in the complex plane and prove that they uniquely determine the warping function of the warped balls.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44672104","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":"Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data","authors":"J. Railo, Philipp Zimmermann","doi":"10.3934/ipi.2022048","DOIUrl":"https://doi.org/10.3934/ipi.2022048","url":null,"abstract":"<p style='text-indent:20px;'>We construct counterexamples for the partial data inverse problem for the fractional conductivity equation in all dimensions on general bounded open sets. In particular, we show that for any bounded domain <inline-formula><tex-math id=\"M1\">begin{document}$ Omega subset {mathbb R}^n $end{document}</tex-math></inline-formula> and any disjoint open sets <inline-formula><tex-math id=\"M2\">begin{document}$ W_1, W_2 Subset {mathbb R}^n setminus overline{Omega} $end{document}</tex-math></inline-formula> there always exist two positive, bounded, smooth, conductivities <inline-formula><tex-math id=\"M3\">begin{document}$ gamma_1, gamma_2 $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M4\">begin{document}$ gamma_1 neq gamma_2 $end{document}</tex-math></inline-formula>, with equal partial exterior Dirichlet-to-Neumann maps <inline-formula><tex-math id=\"M5\">begin{document}$ Lambda_{gamma_1}f|_{W_2} = Lambda_{gamma_2}f|_{W_2} $end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id=\"M6\">begin{document}$ f in C_c^{infty}(W_1) $end{document}</tex-math></inline-formula>. The proof uses the characterization of equal exterior data from another work of the authors in combination with the maximum principle of fractional Laplacians. The main technical difficulty arises from the requirement that the conductivities should be strictly positive and have a special regularity property <inline-formula><tex-math id=\"M7\">begin{document}$ gamma_i^{1/2}-1 in H^{2s, frac{n}{2s}}( {mathbb R}^n) $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=\"M8\">begin{document}$ i = 1, 2 $end{document}</tex-math></inline-formula>. We also provide counterexamples on domains that are bounded in one direction when <inline-formula><tex-math id=\"M9\">begin{document}$ n geq 4 $end{document}</tex-math></inline-formula> or <inline-formula><tex-math id=\"M10\">begin{document}$ s in (0, n/4] $end{document}</tex-math></inline-formula> when <inline-formula><tex-math id=\"M11\">begin{document}$ n = 2, 3 $end{document}</tex-math></inline-formula> using a modification of the argument on bounded domains.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44006328","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":"The enclosure method for the detection of variable order in fractional diffusion equations","authors":"Masaru Ikehata, Yavar Kian","doi":"10.3934/ipi.2022036","DOIUrl":"https://doi.org/10.3934/ipi.2022036","url":null,"abstract":"This paper is concerned with a new type of inverse obstacle problem governed by a variable-order time-fraction diffusion equation in a bounded domain. The unknown obstacle is a region where the space dependent variable-order of fractional time derivative of the governing equation deviates from a known homogeneous background one. The observation data is given by the Neumann data of the solution of the governing equation for a specially designed Dirichlet data. Under a suitable jump condition on the deviation, it is shown that the most recent version of the time domain enclosure method enables one to extract information about the geometry of the obstacle and a qualitative nature of the jump, from the observation data.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45147235","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 spectral target signature for thin surfaces with higher order jump conditions","authors":"F. Cakoni, Heejin Lee, P. Monk, Yangwen Zhang","doi":"10.3934/ipi.2022020","DOIUrl":"https://doi.org/10.3934/ipi.2022020","url":null,"abstract":"<p style='text-indent:20px;'>In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in <inline-formula><tex-math id=\"M1\">begin{document}$ {mathbb R}^m $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M2\">begin{document}$ m = 2, 3 $end{document}</tex-math></inline-formula> from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open <inline-formula><tex-math id=\"M3\">begin{document}$ m-1 $end{document}</tex-math></inline-formula> dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in [<xref ref-type=\"bibr\" rid=\"b20\">20</xref>]. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43992047","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":"Approximation of the elastic Dirichlet-to-Neumann map","authors":"G. Vodev","doi":"10.3934/ipi.2022042","DOIUrl":"https://doi.org/10.3934/ipi.2022042","url":null,"abstract":"<p style='text-indent:20px;'>We study the Dirichlet-to-Neumann map for the stationary linear equation of elasticity in a bounded domain in <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{R}^d $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M2\">begin{document}$ dge 2 $end{document}</tex-math></inline-formula>, with smooth boundary. We show that it can be approximated by a pseudodifferential operator on the boundary with a matrix-valued symbol and we compute the principal symbol modulo conjugation by unitary matrices.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41871863","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":"Uniqueness of the partial travel time representation of a compact Riemannian manifold with strictly convex boundary","authors":"E. Pavlechko, Teemu Saksala","doi":"10.3934/ipi.2022028","DOIUrl":"https://doi.org/10.3934/ipi.2022028","url":null,"abstract":"In this paper a compact Riemannian manifold with strictly convex boundary is reconstructed from its partial travel time data. This data assumes that an open measurement region on the boundary is given, and that for every point in the manifold, the respective distance function to the points on the measurement region is known. This geometric inverse problem has many connections to seismology, in particular to microseismicity. The reconstruction is based on embedding the manifold in a function space. This requires the differentiation of the distance functions. Therefore this paper also studies some global regularity properties of the distance function on a compact Riemannian manifold with strictly convex boundary.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43912046","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 non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification","authors":"L. Afraites, A. Hadri, A. Laghrib, M. Nachaoui","doi":"10.3934/ipi.2022001","DOIUrl":"https://doi.org/10.3934/ipi.2022001","url":null,"abstract":"<p style='text-indent:20px;'>We propose a new variational framework to remove a mixture of Gaussian and impulse noise from images. This framework is based on a non-convex PDE-constrained with a fractional-order operator. The non-convex norm is applied to the impulse component controlled by a weighted parameter <inline-formula><tex-math id=\"M1\">begin{document}$ gamma $end{document}</tex-math></inline-formula>, which depends on the level of the impulse noise and image feature. Furthermore, the fractional operator is used to preserve image texture and edges. In a first part, we study the theoretical properties of the proposed PDE-constrained, and we show some well-posdnees results. In a second part, after having demonstrated how to numerically find a minimizer, a proximal linearized algorithm combined with a Primal-Dual approach is introduced. Moreover, a bi-level optimization framework with a projected gradient algorithm is proposed in order to automatically select the parameter <inline-formula><tex-math id=\"M2\">begin{document}$ gamma $end{document}</tex-math></inline-formula>. Denoising tests confirm that the non-convex term and learned parameter <inline-formula><tex-math id=\"M3\">begin{document}$ gamma $end{document}</tex-math></inline-formula> lead in general to an improved reconstruction when compared to results of convex norm and other competitive denoising methods. Finally, we show extensive denoising experiments on various images and noise intensities and we report conventional numerical results which confirm the validity of the non-convex PDE-constrained, its analysis and also the proposed bi-level optimization with learning data.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"72 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88172859","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}