SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 917-950, June 2024. Abstract. Anomaly detection in the hyperspectral images, which aims to separate interesting sparse anomalies from backgrounds, is a significant topic in remote sensing. In this paper, we propose a generalized nonconvex background representation learning with dictionary constraint (GNBRL) model for hyperspectral anomaly detection. Unlike existing methods that use a specific nonconvex function for a low rank term, GNBRL uses a class of nonconvex functions for both low rank and sparse terms simultaneously, which can better capture the low rank structure of the background and the sparsity of the anomaly. In addition, GNBRL simultaneously learns the dictionary and anomaly tensor in a unified framework by imposing a three-dimensional correlated total variation constraint on the dictionary tensor to enhance the quality of representation. An extrapolated linearized alternating direction method of multipliers (ELADMM) algorithm is then developed to solve the proposed GNBRL model. Finally, a novel coarse to fine two-stage framework is proposed to enhance the GNBRL model by exploiting the nonlocal similarity of the hyperspectral data. Theoretically, we establish an error bound for the GNBRL model and show that this error bound can be superior to those of similar models based on Tucker rank. We prove that the sequence generated by the proposed ELADMM algorithm converges to a Karush–Kuhn–Tucker point of the GNBRL model. This is a challenging task due to the nonconvexity of the objective function. Experiments on hyperspectral image datasets demonstrate that our proposed method outperforms several state-of-the-art methods in terms of detection accuracy.
{"title":"Generalized Nonconvex Hyperspectral Anomaly Detection via Background Representation Learning with Dictionary Constraint","authors":"Quan Yu, Minru Bai","doi":"10.1137/23m157363x","DOIUrl":"https://doi.org/10.1137/23m157363x","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 917-950, June 2024. <br/> Abstract. Anomaly detection in the hyperspectral images, which aims to separate interesting sparse anomalies from backgrounds, is a significant topic in remote sensing. In this paper, we propose a generalized nonconvex background representation learning with dictionary constraint (GNBRL) model for hyperspectral anomaly detection. Unlike existing methods that use a specific nonconvex function for a low rank term, GNBRL uses a class of nonconvex functions for both low rank and sparse terms simultaneously, which can better capture the low rank structure of the background and the sparsity of the anomaly. In addition, GNBRL simultaneously learns the dictionary and anomaly tensor in a unified framework by imposing a three-dimensional correlated total variation constraint on the dictionary tensor to enhance the quality of representation. An extrapolated linearized alternating direction method of multipliers (ELADMM) algorithm is then developed to solve the proposed GNBRL model. Finally, a novel coarse to fine two-stage framework is proposed to enhance the GNBRL model by exploiting the nonlocal similarity of the hyperspectral data. Theoretically, we establish an error bound for the GNBRL model and show that this error bound can be superior to those of similar models based on Tucker rank. We prove that the sequence generated by the proposed ELADMM algorithm converges to a Karush–Kuhn–Tucker point of the GNBRL model. This is a challenging task due to the nonconvexity of the objective function. Experiments on hyperspectral image datasets demonstrate that our proposed method outperforms several state-of-the-art methods in terms of detection accuracy.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140567702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yanran Li, Raymond H. Chan, Lixin Shen, Xiaosheng Zhuang, Risheng Wu, Yijun Huang, Junwei Liu
SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 888-916, June 2024. Abstract. Each coil image in a parallel magnetic resonance imaging (pMRI) system is an imaging slice modulated by the corresponding coil sensitivity. These coil images, structurally similar to each other, are stacked together as 3-dimensional (3D) image data, and their sparsity property can be explored via 3D directional Haar tight framelets. The features of the 3D image data from the 3D framelet systems are utilized to regularize sensitivity encoding (SENSE) pMRI reconstruction. Accordingly, a so-called SENSE3d algorithm is proposed to reconstruct images of high quality from the sampled [math]-space data with a high acceleration rate by decoupling effects of the desired image (slice) and sensitivity maps. Since both the imaging slice and sensitivity maps are unknown, this algorithm repeatedly performs a slice step followed by a sensitivity step by using updated estimations of the desired image and the sensitivity maps. In the slice step, for the given sensitivity maps, the estimation of the desired image is viewed as the solution to a convex optimization problem regularized by the sparsity of its 3D framelet coefficients of coil images. This optimization problem, involving data from the complex field, is solved by a primal-dual three-operator splitting (PD3O) method. In the sensitivity step, the estimation of sensitivity maps is modeled as the solution to a Tikhonov-type optimization problem that favors the smoothness of the sensitivity maps. This corresponding problem is nonconvex and could be solved by a forward-backward splitting method. Experiments on real phantoms and in vivo data show that the proposed SENSE3d algorithm can explore the sparsity property of the imaging slices and efficiently produce reconstructed images of high quality with reduced aliasing artifacts caused by high acceleration rate, additive noise, and the inaccurate estimation of each coil sensitivity. To provide a comprehensive picture of the overall performance of our SENSE3d model, we provide the quantitative index (HaarPSI) and comparisons to some deep learning methods such as VarNet and fastMRI-UNet.
{"title":"Exploring Structural Sparsity of Coil Images from 3-Dimensional Directional Tight Framelets for SENSE Reconstruction","authors":"Yanran Li, Raymond H. Chan, Lixin Shen, Xiaosheng Zhuang, Risheng Wu, Yijun Huang, Junwei Liu","doi":"10.1137/23m1571150","DOIUrl":"https://doi.org/10.1137/23m1571150","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 888-916, June 2024. <br/> Abstract. Each coil image in a parallel magnetic resonance imaging (pMRI) system is an imaging slice modulated by the corresponding coil sensitivity. These coil images, structurally similar to each other, are stacked together as 3-dimensional (3D) image data, and their sparsity property can be explored via 3D directional Haar tight framelets. The features of the 3D image data from the 3D framelet systems are utilized to regularize sensitivity encoding (SENSE) pMRI reconstruction. Accordingly, a so-called SENSE3d algorithm is proposed to reconstruct images of high quality from the sampled [math]-space data with a high acceleration rate by decoupling effects of the desired image (slice) and sensitivity maps. Since both the imaging slice and sensitivity maps are unknown, this algorithm repeatedly performs a slice step followed by a sensitivity step by using updated estimations of the desired image and the sensitivity maps. In the slice step, for the given sensitivity maps, the estimation of the desired image is viewed as the solution to a convex optimization problem regularized by the sparsity of its 3D framelet coefficients of coil images. This optimization problem, involving data from the complex field, is solved by a primal-dual three-operator splitting (PD3O) method. In the sensitivity step, the estimation of sensitivity maps is modeled as the solution to a Tikhonov-type optimization problem that favors the smoothness of the sensitivity maps. This corresponding problem is nonconvex and could be solved by a forward-backward splitting method. Experiments on real phantoms and in vivo data show that the proposed SENSE3d algorithm can explore the sparsity property of the imaging slices and efficiently produce reconstructed images of high quality with reduced aliasing artifacts caused by high acceleration rate, additive noise, and the inaccurate estimation of each coil sensitivity. To provide a comprehensive picture of the overall performance of our SENSE3d model, we provide the quantitative index (HaarPSI) and comparisons to some deep learning methods such as VarNet and fastMRI-UNet.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140567663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 820-860, June 2024. Abstract.Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach offers the ability to quantify the uncertainty in the solution. In recent years, data-driven techniques for solving inverse problems have also been remarkably successful, due to their superior representation ability. In this work, we incorporate data-based models into a class of Langevin-based sampling algorithms for Bayesian inference in imaging inverse problems. In particular, we introduce NF-ULA (normalizing flow-based unadjusted Langevin algorithm), which involves learning a normalizing flow (NF) as the image prior. We use NF to learn the prior because a tractable closed-form expression for the log prior enables the differentiation of it using autograd libraries. Our algorithm only requires a normalizing flow-based generative network, which can be pretrained independently of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness and nonasymptotic convergence of the resulting NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various image restoration problems such as image deblurring, image inpainting, and limited-angle X-ray computed tomography reconstruction. NF-ULA is found to perform better than competing methods for severely ill-posed inverse problems.
{"title":"NF-ULA: Normalizing Flow-Based Unadjusted Langevin Algorithm for Imaging Inverse Problems","authors":"Ziruo Cai, Junqi Tang, Subhadip Mukherjee, Jinglai Li, Carola-Bibiane Schönlieb, Xiaoqun Zhang","doi":"10.1137/23m1581807","DOIUrl":"https://doi.org/10.1137/23m1581807","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 820-860, June 2024. <br/> Abstract.Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach offers the ability to quantify the uncertainty in the solution. In recent years, data-driven techniques for solving inverse problems have also been remarkably successful, due to their superior representation ability. In this work, we incorporate data-based models into a class of Langevin-based sampling algorithms for Bayesian inference in imaging inverse problems. In particular, we introduce NF-ULA (normalizing flow-based unadjusted Langevin algorithm), which involves learning a normalizing flow (NF) as the image prior. We use NF to learn the prior because a tractable closed-form expression for the log prior enables the differentiation of it using autograd libraries. Our algorithm only requires a normalizing flow-based generative network, which can be pretrained independently of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness and nonasymptotic convergence of the resulting NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various image restoration problems such as image deblurring, image inpainting, and limited-angle X-ray computed tomography reconstruction. NF-ULA is found to perform better than competing methods for severely ill-posed inverse problems.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140567942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 861-887, June 2024. Abstract.In this paper, we propose a new approach to deformable image registration that captures sliding motions. The large deformation diffeomorphic metric mapping (LDDMM) registration method faces challenges in representing sliding motion since it per construction generates smooth warps. To address this issue, we extend LDDMM by incorporating both zeroth- and first-order momenta with a nondifferentiable kernel. This allows us to represent both discontinuous deformation at switching boundaries and diffeomorphic deformation in homogeneous regions. We provide a mathematical analysis of the proposed deformation model from the viewpoint of discontinuous systems. To evaluate our approach, we conduct experiments on both artificial images and the publicly available DIR-Lab 4DCT dataset. Results show the effectiveness of our approach in capturing plausible sliding motion.
{"title":"Sliding at First-Order: Higher-Order Momentum Distributions for Discontinuous Image Registration","authors":"Lili Bao, Jiahao Lu, Shihui Ying, Stefan Sommer","doi":"10.1137/23m1558665","DOIUrl":"https://doi.org/10.1137/23m1558665","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 861-887, June 2024. <br/> Abstract.In this paper, we propose a new approach to deformable image registration that captures sliding motions. The large deformation diffeomorphic metric mapping (LDDMM) registration method faces challenges in representing sliding motion since it per construction generates smooth warps. To address this issue, we extend LDDMM by incorporating both zeroth- and first-order momenta with a nondifferentiable kernel. This allows us to represent both discontinuous deformation at switching boundaries and diffeomorphic deformation in homogeneous regions. We provide a mathematical analysis of the proposed deformation model from the viewpoint of discontinuous systems. To evaluate our approach, we conduct experiments on both artificial images and the publicly available DIR-Lab 4DCT dataset. Results show the effectiveness of our approach in capturing plausible sliding motion.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140567735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hong Ye Tan, Subhadip Mukherjee, Junqi Tang, Carola-Bibiane Schönlieb
SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 785-819, June 2024. Abstract.Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM, with applications in inverse problems and imaging. Provable PnP methods are a subclass of PnP methods with convergence guarantees, such as fixed point convergence or convergence to critical points of some energy function. Many existing provable PnP methods impose heavy restrictions on the denoiser or fidelity function, such as nonexpansiveness or strict convexity, respectively. In this work, we propose a novel algorithmic approach incorporating quasi-Newton steps into a provable PnP framework based on proximal denoisers, resulting in greatly accelerated convergence while retaining light assumptions on the denoiser. By characterizing the denoiser as the proximal operator of a weakly convex function, we show that the fixed points of the proposed quasi-Newton PnP algorithm are critical points of a weakly convex function. Numerical experiments on image deblurring and super-resolution demonstrate 2–8x faster convergence as compared to other provable PnP methods with similar reconstruction quality.
{"title":"Provably Convergent Plug-and-Play Quasi-Newton Methods","authors":"Hong Ye Tan, Subhadip Mukherjee, Junqi Tang, Carola-Bibiane Schönlieb","doi":"10.1137/23m157185x","DOIUrl":"https://doi.org/10.1137/23m157185x","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 785-819, June 2024. <br/> Abstract.Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM, with applications in inverse problems and imaging. Provable PnP methods are a subclass of PnP methods with convergence guarantees, such as fixed point convergence or convergence to critical points of some energy function. Many existing provable PnP methods impose heavy restrictions on the denoiser or fidelity function, such as nonexpansiveness or strict convexity, respectively. In this work, we propose a novel algorithmic approach incorporating quasi-Newton steps into a provable PnP framework based on proximal denoisers, resulting in greatly accelerated convergence while retaining light assumptions on the denoiser. By characterizing the denoiser as the proximal operator of a weakly convex function, we show that the fixed points of the proposed quasi-Newton PnP algorithm are critical points of a weakly convex function. Numerical experiments on image deblurring and super-resolution demonstrate 2–8x faster convergence as compared to other provable PnP methods with similar reconstruction quality.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140567734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 756-783, March 2024. Abstract. In this paper, we consider a low-rank tensor recovery problem. Based on the tensor singular value decomposition (t-SVD), we propose the ratio of the tensor nuclear norm and the tensor Frobenius norm (TNF) as a novel nonconvex surrogate of tensor’s tubal rank. The rationale of the proposed model for enforcing a low-rank structure is analyzed as its theoretical properties. Specifically, we introduce a null space property (NSP) type condition, under which a low-rank tensor is a local minimum for the proposed TNF recovery model. Numerically, we consider a low-rank tensor completion problem as a specific application of tensor recovery and employ the alternating direction method of multipliers (ADMM) to secure a model solution with guaranteed subsequential convergence under mild conditions. Extensive experiments demonstrate the superiority of our proposed model over state-of-the-art methods.
{"title":"A Scale-Invariant Relaxation in Low-Rank Tensor Recovery with an Application to Tensor Completion","authors":"Huiwen Zheng, Yifei Lou, Guoliang Tian, Chao Wang","doi":"10.1137/23m1560847","DOIUrl":"https://doi.org/10.1137/23m1560847","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 756-783, March 2024. <br/> Abstract. In this paper, we consider a low-rank tensor recovery problem. Based on the tensor singular value decomposition (t-SVD), we propose the ratio of the tensor nuclear norm and the tensor Frobenius norm (TNF) as a novel nonconvex surrogate of tensor’s tubal rank. The rationale of the proposed model for enforcing a low-rank structure is analyzed as its theoretical properties. Specifically, we introduce a null space property (NSP) type condition, under which a low-rank tensor is a local minimum for the proposed TNF recovery model. Numerically, we consider a low-rank tensor completion problem as a specific application of tensor recovery and employ the alternating direction method of multipliers (ADMM) to secure a model solution with guaranteed subsequential convergence under mild conditions. Extensive experiments demonstrate the superiority of our proposed model over state-of-the-art methods.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 706-755, March 2024. Abstract.This paper proposes a novel method for computing bijective density-equalizing quasiconformal flattening maps for multiply connected open surfaces. In conventional density-equalizing maps, shape deformations are solely driven by prescribed constraints on the density distribution, defined as the population per unit area, while the bijectivity and local geometric distortions of the mappings are uncontrolled. Also, prior methods have primarily focused on simply connected open surfaces but not surfaces with more complicated topologies. Our proposed method overcomes these issues by formulating the density diffusion process as a quasiconformal flow, which allows us to effectively control the local geometric distortion and guarantee the bijectivity of the mapping by solving an energy minimization problem involving the Beltrami coefficient of the mapping. To achieve an optimal parameterization of multiply connected surfaces, we develop an iterative scheme that optimizes both the shape of the target planar circular domain and the density-equalizing quasiconformal map onto it. In addition, landmark constraints can be incorporated into our proposed method for consistent feature alignment. The method can also be naturally applied to simply connected open surfaces. By changing the prescribed population, a large variety of surface flattening maps with different desired properties can be achieved. The method is tested on both synthetic and real examples, demonstrating its efficacy in various applications in computer graphics and medical imaging.
{"title":"Bijective Density-Equalizing Quasiconformal Map for Multiply Connected Open Surfaces","authors":"Zhiyuan Lyu, Gary P. T. Choi, Lok Ming Lui","doi":"10.1137/23m1594376","DOIUrl":"https://doi.org/10.1137/23m1594376","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 706-755, March 2024. <br/> Abstract.This paper proposes a novel method for computing bijective density-equalizing quasiconformal flattening maps for multiply connected open surfaces. In conventional density-equalizing maps, shape deformations are solely driven by prescribed constraints on the density distribution, defined as the population per unit area, while the bijectivity and local geometric distortions of the mappings are uncontrolled. Also, prior methods have primarily focused on simply connected open surfaces but not surfaces with more complicated topologies. Our proposed method overcomes these issues by formulating the density diffusion process as a quasiconformal flow, which allows us to effectively control the local geometric distortion and guarantee the bijectivity of the mapping by solving an energy minimization problem involving the Beltrami coefficient of the mapping. To achieve an optimal parameterization of multiply connected surfaces, we develop an iterative scheme that optimizes both the shape of the target planar circular domain and the density-equalizing quasiconformal map onto it. In addition, landmark constraints can be incorporated into our proposed method for consistent feature alignment. The method can also be naturally applied to simply connected open surfaces. By changing the prescribed population, a large variety of surface flattening maps with different desired properties can be achieved. The method is tested on both synthetic and real examples, demonstrating its efficacy in various applications in computer graphics and medical imaging.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140314602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 672-705, March 2024. Abstract. We develop a boundary integral equation–based numerical method to solve for the electrostatic potential in two dimensions, inside a medium with piecewise constant conductivity, where the boundary condition is given by the complete electrode model (CEM). The CEM is seen as the most accurate model of the physical setting where electrodes are placed on the surface of an electrically conductive body, currents are injected through the electrodes, and the resulting voltages are measured again on these same electrodes. The integral equation formulation is based on expressing the electrostatic potential as the solution to a finite number of Laplace equations which are coupled through boundary matching conditions. This allows us to re-express the solution in terms of single-layer potentials; the problem is thus recast as a system of integral equations on a finite number of smooth curves. We discuss an adaptive method for the solution of the resulting system of mildly singular integral equations. This forward solver is both fast and accurate. We then present a numerical inverse solver for electrical impedance tomography which uses our forward solver at its core. To demonstrate the applicability of our results we test our numerical methods on an open electrical impedance tomography data set provided by the Finnish Inverse Problems Society.
{"title":"A Boundary Integral Equation Method for the Complete Electrode Model in Electrical Impedance Tomography with Tests on Experimental Data","authors":"Teemu Tyni, Adam R. Stinchcombe, Spyros Alexakis","doi":"10.1137/23m1585696","DOIUrl":"https://doi.org/10.1137/23m1585696","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 672-705, March 2024. <br/> Abstract. We develop a boundary integral equation–based numerical method to solve for the electrostatic potential in two dimensions, inside a medium with piecewise constant conductivity, where the boundary condition is given by the complete electrode model (CEM). The CEM is seen as the most accurate model of the physical setting where electrodes are placed on the surface of an electrically conductive body, currents are injected through the electrodes, and the resulting voltages are measured again on these same electrodes. The integral equation formulation is based on expressing the electrostatic potential as the solution to a finite number of Laplace equations which are coupled through boundary matching conditions. This allows us to re-express the solution in terms of single-layer potentials; the problem is thus recast as a system of integral equations on a finite number of smooth curves. We discuss an adaptive method for the solution of the resulting system of mildly singular integral equations. This forward solver is both fast and accurate. We then present a numerical inverse solver for electrical impedance tomography which uses our forward solver at its core. To demonstrate the applicability of our results we test our numerical methods on an open electrical impedance tomography data set provided by the Finnish Inverse Problems Society.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Julien Flamant, Konstantin Usevich, Marianne Clausel, David Brie
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 632-671, March 2024. Abstract. This work introduces polarimetric Fourier phase retrieval (PPR), a physically inspired model to leverage polarization of light information in Fourier phase retrieval problems. We provide a complete characterization of its uniqueness properties by unraveling equivalencies with two related problems, namely, bivariate phase retrieval and a polynomial autocorrelation factorization problem. In particular, we show that the problem admits a unique solution, which can be formulated as a greatest common divisor (GCD) of measurement polynomials. As a result, we propose algebraic solutions for PPR based on approximate GCD computations using the null-space properties of Sylvester matrices. Alternatively, existing iterative algorithms for phase retrieval, semidefinite positive relaxation and Wirtinger flow, are carefully adapted to solve the PPR problem. Finally, a set of numerical experiments permits a detailed assessment of the numerical behavior and relative performances of each proposed reconstruction strategy. They further demonstrate the fruitful combination of algebraic and iterative approaches toward a scalable, computationally efficient, and robust to noise reconstruction strategy for PPR.
{"title":"Polarimetric Fourier Phase Retrieval","authors":"Julien Flamant, Konstantin Usevich, Marianne Clausel, David Brie","doi":"10.1137/23m1570971","DOIUrl":"https://doi.org/10.1137/23m1570971","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 632-671, March 2024. <br/> Abstract. This work introduces polarimetric Fourier phase retrieval (PPR), a physically inspired model to leverage polarization of light information in Fourier phase retrieval problems. We provide a complete characterization of its uniqueness properties by unraveling equivalencies with two related problems, namely, bivariate phase retrieval and a polynomial autocorrelation factorization problem. In particular, we show that the problem admits a unique solution, which can be formulated as a greatest common divisor (GCD) of measurement polynomials. As a result, we propose algebraic solutions for PPR based on approximate GCD computations using the null-space properties of Sylvester matrices. Alternatively, existing iterative algorithms for phase retrieval, semidefinite positive relaxation and Wirtinger flow, are carefully adapted to solve the PPR problem. Finally, a set of numerical experiments permits a detailed assessment of the numerical behavior and relative performances of each proposed reconstruction strategy. They further demonstrate the fruitful combination of algebraic and iterative approaches toward a scalable, computationally efficient, and robust to noise reconstruction strategy for PPR.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140108004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 540-594, March 2024. Abstract. For problems in image processing and many other fields, a large class of effective neural networks has encoder-decoder-based architectures. Although these networks have shown impressive performance, mathematical explanations of their architectures are still underdeveloped. In this paper, we study the encoder-decoder-based network architecture from the algorithmic perspective and provide a mathematical explanation. We use the two-phase Potts model for image segmentation as an example for our explanations. We associate the segmentation problem with a control problem in the continuous setting. Then, the continuous control model is time discretized by an operator-splitting scheme, the PottsMGNet, and space discretized by the multigrid method. We show that the resulting discrete PottsMGNet is equivalent to an encoder-decoder-based network. With minor modifications, it is shown that a number of the popular encoder-decoder-based neural networks are just instances of the proposed PottsMGNet. By incorporating the soft-threshold-dynamics into the PottsMGNet as a regularizer, the PottsMGNet has shown to be robust with the network parameters such as network width and depth and has achieved remarkable performance on datasets with very large noise. In nearly all our experiments, the new network always performs better than or as well as on accuracy and dice score compared to existing networks for image segmentation.
{"title":"PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks","authors":"Xue-Cheng Tai, Hao Liu, Raymond Chan","doi":"10.1137/23m1586355","DOIUrl":"https://doi.org/10.1137/23m1586355","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 540-594, March 2024. <br/> Abstract. For problems in image processing and many other fields, a large class of effective neural networks has encoder-decoder-based architectures. Although these networks have shown impressive performance, mathematical explanations of their architectures are still underdeveloped. In this paper, we study the encoder-decoder-based network architecture from the algorithmic perspective and provide a mathematical explanation. We use the two-phase Potts model for image segmentation as an example for our explanations. We associate the segmentation problem with a control problem in the continuous setting. Then, the continuous control model is time discretized by an operator-splitting scheme, the PottsMGNet, and space discretized by the multigrid method. We show that the resulting discrete PottsMGNet is equivalent to an encoder-decoder-based network. With minor modifications, it is shown that a number of the popular encoder-decoder-based neural networks are just instances of the proposed PottsMGNet. By incorporating the soft-threshold-dynamics into the PottsMGNet as a regularizer, the PottsMGNet has shown to be robust with the network parameters such as network width and depth and has achieved remarkable performance on datasets with very large noise. In nearly all our experiments, the new network always performs better than or as well as on accuracy and dice score compared to existing networks for image segmentation.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140075565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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