Inverse Problems最新文献

Microlocal analysis of non-linear operators arising in Compton CT. 康普顿CT非线性算子的微局部分析。

IF 2.1 2区数学

Inverse Problems Pub Date : 2026-02-27 Epub Date: 2026-02-10 DOI: 10.1088/1361-6420/ae3acc

James W Webber, Sean Holman

{"title":"Microlocal analysis of non-linear operators arising in Compton CT.","authors":"James W Webber, Sean Holman","doi":"10.1088/1361-6420/ae3acc","DOIUrl":"10.1088/1361-6420/ae3acc","url":null,"abstract":"We present a novel microlocal analysis of a non-linear ray transform, <math> <mrow><mrow><mi>R</mi></mrow> </mrow> </math> , arising in Compton scattering tomography (CST). Due to attenuation effects in CST, the integral weights depend on the reconstruction target, f, which has singularities. Thus, standard linear Fourier integral operator (FIO) theory does not apply as the weights are non-smooth. The V-line (or broken ray) transform, <math> <mrow><mrow><mi>V</mi></mrow> </mrow> </math> , can be used to model the attenuation of incoming and outgoing rays. Through novel analysis of <math> <mrow><mrow><mi>V</mi></mrow> </mrow> </math> , we characterize the location and strength of the singularities of the ray transform weights. In conjunction, we provide new results which quantify the strength of the singularities of distributional products based on the Sobolev order of the individual components. By combining this new theory, our analysis of <math> <mrow><mrow><mi>V</mi></mrow> </mrow> </math> , and classical linear FIO theory, we determine the Sobolev order of the singularities of <math> <mrow><mrow><mi>R</mi></mrow> <mi>f</mi></mrow> </math> . The strongest (lowest Sobolev order) singularities of <math> <mrow><mrow><mi>R</mi></mrow> <mi>f</mi></mrow> </math> are shown to correspond to the wavefront set elements of the classical Radon transform applied to f, and we use this idea and known results on the Radon transform to prove injectivity results for <math> <mrow><mrow><mi>R</mi></mrow> </mrow> </math> . In addition, we present novel reconstruction methods based on our theory, and we validate our results using simulated image reconstructions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"42 2","pages":"025007"},"PeriodicalIF":2.1,"publicationDate":"2026-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12887694/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146167489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A PINN-driven game-theoretic framework in limited data photoacoustic tomography. 有限数据光声层析成像中pin驱动的博弈论框架。

IF 2.1 2区数学

Inverse Problems Pub Date : 2025-11-28 Epub Date: 2025-11-14 DOI: 10.1088/1361-6420/ae1bcd

Souvik Roy, Suvra Pal

引用次数: 0

An accelerated preconditioned proximal gradient algorithm with a generalized Nesterov momentum for PET image reconstruction. 基于广义Nesterov动量的PET图像重构加速预条件近端梯度算法。

IF 2.1 2区数学

Inverse Problems Pub Date : 2025-04-01 Epub Date: 2025-03-14 DOI: 10.1088/1361-6420/adbd6a

Yizun Lin, Yongxin He, C Ross Schmidtlein, Deren Han

{"title":"An accelerated preconditioned proximal gradient algorithm with a generalized Nesterov momentum for PET image reconstruction.","authors":"Yizun Lin, Yongxin He, C Ross Schmidtlein, Deren Han","doi":"10.1088/1361-6420/adbd6a","DOIUrl":"10.1088/1361-6420/adbd6a","url":null,"abstract":"This paper presents an accelerated preconditioned proximal gradient algorithm (APPGA) for effectively solving a class of positron emission tomography (PET) image reconstruction models with differentiable regularizers. We establish the convergence of APPGA with the generalized Nesterov (GN) momentum scheme, demonstrating its ability to converge to a minimizer of the objective function with rates of <math><mi>o</mi> <mfenced><mrow><mn>1</mn> <mo>/</mo> <msup><mrow><mi>k</mi></mrow> <mrow><mn>2</mn> <mi>ω</mi></mrow> </msup> </mrow> </mfenced> </math> and <math><mi>o</mi> <mfenced><mrow><mn>1</mn> <mo>/</mo> <msup><mrow><mi>k</mi></mrow> <mrow><mi>ω</mi></mrow> </msup> </mrow> </mfenced> </math> in terms of the function value and the distance between consecutive iterates, respectively, where <math><mi>ω</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo></math> is the power parameter of the GN momentum. To achieve an efficient algorithm with high-order convergence rate for the higher-order isotropic total variation (ITV) regularized PET image reconstruction model, we replace the ITV term by its smoothed version and subsequently apply APPGA to solve the smoothed model. Numerical results presented in this work indicate that as <math><mi>ω</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo></math> increase, APPGA converges at a progressively faster rate. Furthermore, APPGA exhibits superior performance compared to the preconditioned proximal gradient algorithm and the preconditioned Krasnoselskii-Mann algorithm. The extension of the GN momentum technique for solving a more complex optimization model with multiple nondifferentiable terms is also discussed.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"41 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12456403/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145139132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Optimizing quantitative photoacoustic imaging systems: the Bayesian Cramér-Rao bound approach. 优化定量光声成像系统：贝叶斯克拉梅-拉奥约束方法。

IF 2 2区数学

Inverse Problems Pub Date : 2024-12-01 Epub Date: 2024-11-20 DOI: 10.1088/1361-6420/ad910a

Evan Scope Crafts, Mark A Anastasio, Umberto Villa

{"title":"Optimizing quantitative photoacoustic imaging systems: the Bayesian Cramér-Rao bound approach.","authors":"Evan Scope Crafts, Mark A Anastasio, Umberto Villa","doi":"10.1088/1361-6420/ad910a","DOIUrl":"10.1088/1361-6420/ad910a","url":null,"abstract":"Quantitative photoacoustic computed tomography (qPACT) is an emerging medical imaging modality that carries the promise of high-contrast, fine-resolution imaging of clinically relevant quantities like hemoglobin concentration and blood-oxygen saturation. However, qPACT image reconstruction is governed by a multiphysics, partial differential equation (PDE) based inverse problem that is highly non-linear and severely ill-posed. Compounding the difficulty of the problem is the lack of established design standards for qPACT imaging systems, as there is currently a proliferation of qPACT system designs for various applications and it is unknown which ones are optimal or how to best modify the systems under various design constraints. This work introduces a novel computational approach for the optimal experimental design of qPACT imaging systems based on the Bayesian Cramér-Rao bound (CRB). Our approach incorporates several techniques to address challenges associated with forming the bound in the infinite-dimensional function space setting of qPACT, including priors with trace-class covariance operators and the use of the variational adjoint method to compute derivatives of the log-likelihood function needed in the bound computation. The resulting Bayesian CRB based design metric is computationally efficient and independent of the choice of estimator used to solve the inverse problem. The efficacy of the bound in guiding experimental design was demonstrated in a numerical study of qPACT design schemes under a stylized two-dimensional imaging geometry. To the best of our knowledge, this is the first work to propose Bayesian CRB based design for systems governed by PDEs.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"40 12","pages":"125012"},"PeriodicalIF":2.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11577155/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142689375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Linearized boundary control method for density reconstruction in acoustic wave equations. 用于声波方程密度重建的线性化边界控制法

IF 2 2区数学

Inverse Problems Pub Date : 2024-12-01 Epub Date: 2024-12-13 DOI: 10.1088/1361-6420/ad98bc

Lauri Oksanen, Tianyu Yang, Yang Yang

引用次数: 0

A microlocal and visual comparison of 2D Kirchhoff migration formulas in seismic imaging * 地震成像中二维基尔霍夫迁移公式的微观和视觉比较 *

IF 2.1 2区数学

Inverse Problems Pub Date : 2024-09-18 DOI: 10.1088/1361-6420/ad797b

Kevin Ganster, Eric Todd Quinto and Andreas Rieder