Electrical impedance tomography (EIT) is a promising technique in medical imaging. With this technique, pathology-related conductivity variation can be visualized. Nevertheless, reconstruction of conductivity distribution is a severely ill-posed inverse problem which poses a great challenge for the EIT technique. Especially in brain EIT, irregular and multi-layered head structure along with low-conductivity skull brings more difficulties for accurate reconstruction. To address such problems, a novel reconstruction method which combines Tikhonov regularization with denoising algorithm is proposed for imaging conductivity distribution in brain EIT. With the proposed method, image reconstruction of intracerebral hemorrhage in different brain lobes of a three-layer head model is conducted. Besides, simultaneous reconstruction of intracerebral hemorrhage and secondary ischemia is performed. Meanwhile, the impact of noise is investigated to evaluate the anti-noise performance. In addition, image reconstructions under head shape deformation are performed. The proposed reconstruction method is also quantitatively estimated by calculating blur radius and structural similarity. Phantom experiments are carried out to further verify the effectiveness of the proposed method. Both qualitative and quantitative results have demonstrated that the proposed combined method is superior to Tikhonov regularization in imaging conductivity distribution. This work would provide an alternative for accurate reconstruction in EIT based medical imaging.
{"title":"Imaging of conductivity distribution based on a combined reconstruction method in brain electrical impedance tomography","authors":"Yanyan Shi, Yajun Lou, Meng Wang, Shuo Zheng, Zhiwei Tian, Feng Fu","doi":"10.3934/ipi.2022060","DOIUrl":"https://doi.org/10.3934/ipi.2022060","url":null,"abstract":"Electrical impedance tomography (EIT) is a promising technique in medical imaging. With this technique, pathology-related conductivity variation can be visualized. Nevertheless, reconstruction of conductivity distribution is a severely ill-posed inverse problem which poses a great challenge for the EIT technique. Especially in brain EIT, irregular and multi-layered head structure along with low-conductivity skull brings more difficulties for accurate reconstruction. To address such problems, a novel reconstruction method which combines Tikhonov regularization with denoising algorithm is proposed for imaging conductivity distribution in brain EIT. With the proposed method, image reconstruction of intracerebral hemorrhage in different brain lobes of a three-layer head model is conducted. Besides, simultaneous reconstruction of intracerebral hemorrhage and secondary ischemia is performed. Meanwhile, the impact of noise is investigated to evaluate the anti-noise performance. In addition, image reconstructions under head shape deformation are performed. The proposed reconstruction method is also quantitatively estimated by calculating blur radius and structural similarity. Phantom experiments are carried out to further verify the effectiveness of the proposed method. Both qualitative and quantitative results have demonstrated that the proposed combined method is superior to Tikhonov regularization in imaging conductivity distribution. This work would provide an alternative for accurate reconstruction in EIT based medical imaging.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134955520","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}
Blurry kernel estimation is a critical yet challenging task for blind deblurring. Most existing works devote to designing end-to-end networks that require a large amount of hard-to-obtain training data. In addition, these methods often ignore the intrinsic effects of blur kernel for blind deblurring. In this work, we present a unified latent image deblur and kernel estimation method based on MAP framework. By revisiting the coarse-to-fine strategy, we introduce a self-supervised multi-scale deblur network(MD-Net), where the multi-scale structure significantly reduce the kernel deviation caused by local area minimization. Specifically, our network commences with random inputs and outputs multi-scale reconstructed images and kernels. By progressively capturing the high-level configuration and low-level details from matching multi-resolution loss functions, the proposed MD-Net enable to capture multi-level image priors. Meanwhile, at each coarse level, we use Feature Extraction(FE) layers to further extract and emphasize features from reconstructed images. Compared with state-of-the-art blind deblurring methods, extensive experiments demonstrate that the proposed approach significantly improves the restoration performance in both quantitative and qualitative evaluations.
{"title":"Self-supervised multi-scale neural network for blind deblurring","authors":"Meina Zhang, Ying Yang, Guoxi Ni, Tingting Wu, Tieyong Zeng","doi":"10.3934/ipi.2023046","DOIUrl":"https://doi.org/10.3934/ipi.2023046","url":null,"abstract":"Blurry kernel estimation is a critical yet challenging task for blind deblurring. Most existing works devote to designing end-to-end networks that require a large amount of hard-to-obtain training data. In addition, these methods often ignore the intrinsic effects of blur kernel for blind deblurring. In this work, we present a unified latent image deblur and kernel estimation method based on MAP framework. By revisiting the coarse-to-fine strategy, we introduce a self-supervised multi-scale deblur network(MD-Net), where the multi-scale structure significantly reduce the kernel deviation caused by local area minimization. Specifically, our network commences with random inputs and outputs multi-scale reconstructed images and kernels. By progressively capturing the high-level configuration and low-level details from matching multi-resolution loss functions, the proposed MD-Net enable to capture multi-level image priors. Meanwhile, at each coarse level, we use Feature Extraction(FE) layers to further extract and emphasize features from reconstructed images. Compared with state-of-the-art blind deblurring methods, extensive experiments demonstrate that the proposed approach significantly improves the restoration performance in both quantitative and qualitative evaluations.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135560496","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}
Optimal transport has gained much attention in image processing fields, such as computer vision, image interpolation and medical image registration. Recently, Bredies et al. (ESAIM: M2AN 54:2351-2382, 2020) and Schmitzer et al. (IEEE T MED IMAGING 39:1626-1635, 2019) established the framework of optimal transport regularization for dynamic inverse problems. In this paper, we incorporate Wasserstein distance, together with total variation, into static inverse problems as a prior regularization. The Wasserstein distance formulated by Benamou-Brenier energy measures the similarity between the given template and the reconstructed image. Also, we analyze the existence of solutions of such variational problems in Radon measure space. Moreover, the first-order primal-dual algorithm is constructed for solving this general imaging problem in a specific grid strategy. Finally, numerical experiments for undersampled MRI reconstruction are presented which show that our proposed model can recover images well with high quality and structure preservation.
最优传输在计算机视觉、图像插值和医学图像配准等图像处理领域受到广泛关注。最近，Bredies等人(ESAIM: M2AN 54:2351- 2382,2020)和Schmitzer等人(IEEE T MED IMAGING 39:1626-1635, 2019)建立了动态逆问题的最优传输正则化框架。本文将Wasserstein距离和总变分作为一种先验正则化方法引入到静态逆问题中。由Benamou-Brenier能量表示的Wasserstein距离度量给定模板与重建图像之间的相似性。同时，我们还分析了这类变分问题在Radon测量空间中解的存在性。在此基础上，构造了一阶原始对偶算法来解决特定网格策略下的一般成像问题。最后，对欠采样的MRI重建进行了数值实验，实验结果表明，该模型能较好地恢复图像，并能保持图像的结构。
{"title":"A Wasserstein distance and total variation regularized model for image reconstruction problems","authors":"Yiming Gao","doi":"10.3934/ipi.2023045","DOIUrl":"https://doi.org/10.3934/ipi.2023045","url":null,"abstract":"Optimal transport has gained much attention in image processing fields, such as computer vision, image interpolation and medical image registration. Recently, Bredies et al. (ESAIM: M2AN 54:2351-2382, 2020) and Schmitzer et al. (IEEE T MED IMAGING 39:1626-1635, 2019) established the framework of optimal transport regularization for dynamic inverse problems. In this paper, we incorporate Wasserstein distance, together with total variation, into static inverse problems as a prior regularization. The Wasserstein distance formulated by Benamou-Brenier energy measures the similarity between the given template and the reconstructed image. Also, we analyze the existence of solutions of such variational problems in Radon measure space. Moreover, the first-order primal-dual algorithm is constructed for solving this general imaging problem in a specific grid strategy. Finally, numerical experiments for undersampled MRI reconstruction are presented which show that our proposed model can recover images well with high quality and structure preservation.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135560504","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}
In many imaging applications where segmented features (e.g. blood vessels) are further used for other numerical simulations (e.g. finite element analysis), the obtained surfaces do not have fine resolutions suitable for the task. Increasing the resolution of such surfaces becomes crucial. This paper proposes a new variational model for solving this problem, based on an Euler-Elastica-based regulariser. Further, we propose and implement two numerical algorithms for solving the model, a projected gradient descent method and the alternating direction method of multipliers. Numerical experiments using real-life examples (including two from outputs of another variational model) have been illustrated for effectiveness. The advantages of the new model are shown through quantitative comparisons by the standard deviation of Gaussian curvatures and mean curvatures from the viewpoint of discrete geometry.
{"title":"Super-resolution surface reconstruction from few low-resolution slices","authors":"Yiyao Zhang, Ke Chen, Shang-Hua Yang","doi":"10.3934/ipi.2023040","DOIUrl":"https://doi.org/10.3934/ipi.2023040","url":null,"abstract":"In many imaging applications where segmented features (e.g. blood vessels) are further used for other numerical simulations (e.g. finite element analysis), the obtained surfaces do not have fine resolutions suitable for the task. Increasing the resolution of such surfaces becomes crucial. This paper proposes a new variational model for solving this problem, based on an Euler-Elastica-based regulariser. Further, we propose and implement two numerical algorithms for solving the model, a projected gradient descent method and the alternating direction method of multipliers. Numerical experiments using real-life examples (including two from outputs of another variational model) have been illustrated for effectiveness. The advantages of the new model are shown through quantitative comparisons by the standard deviation of Gaussian curvatures and mean curvatures from the viewpoint of discrete geometry.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135551334","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}
In this paper, we investigate an inverse source problem for the two-dimensional Helmholtz equation in a two-layered medium. The interface between two media is assumed to be nonlocal and rough, while the compactly supported unknown source is buried in the lower-half medium. For the forward problem, we prove the radiating behaviour of the wave field based on the Angular Spectrum Representation and the asymptotics of Hankel functions. For the inverse problem, using multi-frequency interface measurements, which are limited-aperture, we show an increasing stability estimate which consists of two parts: one part is a Hölder stability estimate, the other part is a logarithmic stability estimate. The latter decreases as the upper bound of the frequency increases. In the derivation of the stability, we require the source function to have an $ H^3 $ regularity to control the high frequency tail of its Fourier transform. To recover the source numerically, we propose a recursive Kaczmarz-Landweber iteration scheme with incomplete data. Numerical examples are presented to justify the theoretical stability estimate and validity of the scheme.
{"title":"Stability for the inverse source problem in a two-layered medium separated by rough interface","authors":"Guanghui Hu, Xiang Xu, Xiaokai Yuan, Yue Zhao","doi":"10.3934/ipi.2023047","DOIUrl":"https://doi.org/10.3934/ipi.2023047","url":null,"abstract":"In this paper, we investigate an inverse source problem for the two-dimensional Helmholtz equation in a two-layered medium. The interface between two media is assumed to be nonlocal and rough, while the compactly supported unknown source is buried in the lower-half medium. For the forward problem, we prove the radiating behaviour of the wave field based on the Angular Spectrum Representation and the asymptotics of Hankel functions. For the inverse problem, using multi-frequency interface measurements, which are limited-aperture, we show an increasing stability estimate which consists of two parts: one part is a Hölder stability estimate, the other part is a logarithmic stability estimate. The latter decreases as the upper bound of the frequency increases. In the derivation of the stability, we require the source function to have an $ H^3 $ regularity to control the high frequency tail of its Fourier transform. To recover the source numerically, we propose a recursive Kaczmarz-Landweber iteration scheme with incomplete data. Numerical examples are presented to justify the theoretical stability estimate and validity of the scheme.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135712576","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}
Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang
{"title":"Corrigendum to \"duality between range and no-response tests and its application for inverse problems\"","authors":"Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang","doi":"10.3934/ipi.2023003","DOIUrl":"https://doi.org/10.3934/ipi.2023003","url":null,"abstract":"","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135534443","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}
This paper is concerned with inverse source problems for the time-harmonic elastic wave equations and Maxwell's equations with a single boundary measurement at a fixed frequency. We show the uniqueness and a Lipschitz-type stability estimate under the assumption that the source function is piecewise constant on a domain which is made of a union of disjoint convex polyhedral subdomains.
{"title":"Determination of piecewise homogeneous sources for elastic and electromagnetic waves","authors":"Jian Zhai, Yue Zhao","doi":"10.3934/ipi.2022065","DOIUrl":"https://doi.org/10.3934/ipi.2022065","url":null,"abstract":"This paper is concerned with inverse source problems for the time-harmonic elastic wave equations and Maxwell's equations with a single boundary measurement at a fixed frequency. We show the uniqueness and a Lipschitz-type stability estimate under the assumption that the source function is piecewise constant on a domain which is made of a union of disjoint convex polyhedral subdomains.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134955795","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}
The problem of recovery of a potential on a quantum star graph from Weyl's matrix given at a finite number of points is considered. A method for its approximate solution is proposed. It consists in reducing the problem to a two-spectra inverse Sturm-Liouville problem on each edge with its posterior solution. The overall approach is based on Neumann series of Bessel functions (NSBF) representations for solutions of Sturm-Liouville equations, and, in fact, the solution of the inverse problem on the quantum graph reduces to dealing with the NSBF coefficients. The NSBF representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the NSBF representation alone is sufficient for the recovery of the potential. The knowledge of the Weyl matrix at a set of points allows one to calculate a number of the NSBF coefficients at the end point of each edge, which leads to approximation of characteristic functions of two Sturm-Liouville problems and allows one to compute the Dirichlet-Dirichlet and Neumann-Dirichlet spectra on each edge. In turn, for solving this two-spectra inverse Sturm-Liouville problem a system of linear algebraic equations is derived for computing the first NSBF coefficient and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests.
{"title":"Recovery of a potential on a quantum star graph from Weyl's matrix","authors":"S. Avdonin, K. V. Khmelnytskaya, V. Kravchenko","doi":"10.3934/ipi.2023034","DOIUrl":"https://doi.org/10.3934/ipi.2023034","url":null,"abstract":"The problem of recovery of a potential on a quantum star graph from Weyl's matrix given at a finite number of points is considered. A method for its approximate solution is proposed. It consists in reducing the problem to a two-spectra inverse Sturm-Liouville problem on each edge with its posterior solution. The overall approach is based on Neumann series of Bessel functions (NSBF) representations for solutions of Sturm-Liouville equations, and, in fact, the solution of the inverse problem on the quantum graph reduces to dealing with the NSBF coefficients. The NSBF representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the NSBF representation alone is sufficient for the recovery of the potential. The knowledge of the Weyl matrix at a set of points allows one to calculate a number of the NSBF coefficients at the end point of each edge, which leads to approximation of characteristic functions of two Sturm-Liouville problems and allows one to compute the Dirichlet-Dirichlet and Neumann-Dirichlet spectra on each edge. In turn, for solving this two-spectra inverse Sturm-Liouville problem a system of linear algebraic equations is derived for computing the first NSBF coefficient and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42934184","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}
A BSTRACT . We find the necessary and sufficient conditions on the Fourier coefficients of a function g on the torus to be in the range of the X -ray transform of a symmetric tensor of compact support in the plane.
摘要。我们确定了环面上函数g的傅立叶系数在平面上紧支撑对称张量的X射线变换范围内的充要条件。
{"title":"On the range of the $ X $-ray transform of symmetric tensors compactly supported in the plane","authors":"K. Sadiq, A. Tamasan","doi":"10.3934/ipi.2022070","DOIUrl":"https://doi.org/10.3934/ipi.2022070","url":null,"abstract":"A BSTRACT . We find the necessary and sufficient conditions on the Fourier coefficients of a function g on the torus to be in the range of the X -ray transform of a symmetric tensor of compact support in the plane.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47574555","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}
We study the uncoupled space-time fractional operators involving time-dependent coefficients and formulate the corresponding inverse problems. Our goal is to determine the variable coefficients from the exterior partial measurements of the Dirichlet-to-Neumann map. We exploit the integration by parts formula for Riemann-Liouville and Caputo derivatives to derive the Runge approximation property for our space-time fractional operator based on the unique continuation property of the fractional Laplacian. This enables us to extend early unique determination results for space-fractional but time-local operators to the space-time fractional case.
{"title":"On inverse problems for uncoupled space-time fractional operators involving time-dependent coefficients","authors":"Li Li","doi":"10.3934/ipi.2023008","DOIUrl":"https://doi.org/10.3934/ipi.2023008","url":null,"abstract":"We study the uncoupled space-time fractional operators involving time-dependent coefficients and formulate the corresponding inverse problems. Our goal is to determine the variable coefficients from the exterior partial measurements of the Dirichlet-to-Neumann map. We exploit the integration by parts formula for Riemann-Liouville and Caputo derivatives to derive the Runge approximation property for our space-time fractional operator based on the unique continuation property of the fractional Laplacian. This enables us to extend early unique determination results for space-fractional but time-local operators to the space-time fractional case.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44191151","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}
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