SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 415-440, March 2024. Abstract. In this paper we consider the generalized Radon transform [math] in the plane. Let [math] be a piecewise smooth function, which has a jump across a smooth curve [math]. We obtain a formula, which accurately describes view aliasing artifacts away from [math] when [math] is reconstructed from the data [math] discretized in the view direction. The formula is asymptotic, it is established in the limit as the sampling rate [math]. The proposed approach does not require that [math] be band-limited. Numerical experiments with the classical Radon transform and generalized Radon transform (which integrates over circles) demonstrate the accuracy of the formula.
{"title":"Analysis of View Aliasing for the Generalized Radon Transform in [math]","authors":"Alexander Katsevich","doi":"10.1137/23m1554746","DOIUrl":"https://doi.org/10.1137/23m1554746","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 415-440, March 2024. <br/> Abstract. In this paper we consider the generalized Radon transform [math] in the plane. Let [math] be a piecewise smooth function, which has a jump across a smooth curve [math]. We obtain a formula, which accurately describes view aliasing artifacts away from [math] when [math] is reconstructed from the data [math] discretized in the view direction. The formula is asymptotic, it is established in the limit as the sampling rate [math]. The proposed approach does not require that [math] be band-limited. Numerical experiments with the classical Radon transform and generalized Radon transform (which integrates over circles) demonstrate the accuracy of the formula.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"34 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139949296","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 441-475, March 2024. Abstract. High-dimensional deep features extracted by convolutional neural networks have nonlocal self-similarity. However, incorporating this nonlocal prior of deep features into deep network architectures with an interpretable variational framework is rarely explored. In this paper, we propose a learnable nonlocal self-similarity deep feature network for image denoising. Our method is motivated by the fact that the high-dimensional deep features obey a mixture probability distribution based on the Parzen–Rosenblatt window method. Then a regularizer with learnable nonlocal weights is proposed by considering the dual representation of the log-probability prior of the deep features. Specifically, the nonlocal weights are introduced as dual variables that can be learned by unrolling the associated numerical scheme. This leads to nonlocal modules (NLMs) in newly designed networks. Our method provides a statistical and variational interpretation for the nonlocal self-attention mechanism widely used in various networks. By adopting nonoverlapping window and region decomposition techniques, we can significantly reduce the computational complexity of nonlocal self-similarity, thus enabling parallel computation of the NLM. The solution to the proposed variational problem can be formulated as a learnable nonlocal self-similarity network for image denoising. This work offers a novel approach for constructing network structures that consider self-similarity and nonlocality. The improvements achieved by this method are predictable and partially controllable. Compared with several closely related denoising methods, the experimental results show the effectiveness of the proposed method in image denoising.
{"title":"Learnable Nonlocal Self-Similarity of Deep Features for Image Denoising","authors":"Junying Meng, Faqiang Wang, Jun Liu","doi":"10.1137/22m1536996","DOIUrl":"https://doi.org/10.1137/22m1536996","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 441-475, March 2024. <br/> Abstract. High-dimensional deep features extracted by convolutional neural networks have nonlocal self-similarity. However, incorporating this nonlocal prior of deep features into deep network architectures with an interpretable variational framework is rarely explored. In this paper, we propose a learnable nonlocal self-similarity deep feature network for image denoising. Our method is motivated by the fact that the high-dimensional deep features obey a mixture probability distribution based on the Parzen–Rosenblatt window method. Then a regularizer with learnable nonlocal weights is proposed by considering the dual representation of the log-probability prior of the deep features. Specifically, the nonlocal weights are introduced as dual variables that can be learned by unrolling the associated numerical scheme. This leads to nonlocal modules (NLMs) in newly designed networks. Our method provides a statistical and variational interpretation for the nonlocal self-attention mechanism widely used in various networks. By adopting nonoverlapping window and region decomposition techniques, we can significantly reduce the computational complexity of nonlocal self-similarity, thus enabling parallel computation of the NLM. The solution to the proposed variational problem can be formulated as a learnable nonlocal self-similarity network for image denoising. This work offers a novel approach for constructing network structures that consider self-similarity and nonlocality. The improvements achieved by this method are predictable and partially controllable. Compared with several closely related denoising methods, the experimental results show the effectiveness of the proposed method in image denoising.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"34 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139949362","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 389-414, March 2024. Abstract. Receptive profiles of the primary visual cortex (V1) cortical cells are very heterogeneous and act by differentiating the stimulus image as operators changing from point to point. In this paper we aim to show that the distribution of cells in V1, although not complete to reconstruct the original image, is sufficient to reconstruct the perceived image with subjective constancy. We show that a color constancy image can be reconstructed as the solution of the associated inverse problem, which is a Poisson equation with heterogeneous differential operators. At the neural level the weights of short-range connectivity constitute the fundamental solution of the Poisson problem adapted point by point. A first demonstration of convergence of the result towards homogeneous reconstructions is proposed by means of homogenization techniques.
{"title":"The Cortical V1 Transform as a Heterogeneous Poisson Problem","authors":"Alessandro Sarti, Mattia Galeotti, Giovanna Citti","doi":"10.1137/23m1555958","DOIUrl":"https://doi.org/10.1137/23m1555958","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 389-414, March 2024. <br/> Abstract. Receptive profiles of the primary visual cortex (V1) cortical cells are very heterogeneous and act by differentiating the stimulus image as operators changing from point to point. In this paper we aim to show that the distribution of cells in V1, although not complete to reconstruct the original image, is sufficient to reconstruct the perceived image with subjective constancy. We show that a color constancy image can be reconstructed as the solution of the associated inverse problem, which is a Poisson equation with heterogeneous differential operators. At the neural level the weights of short-range connectivity constitute the fundamental solution of the Poisson problem adapted point by point. A first demonstration of convergence of the result towards homogeneous reconstructions is proposed by means of homogenization techniques.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"31 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139921622","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}
Antonio Corbo Esposito, Luisa Faella, Gianpaolo Piscitelli, Vincenzo Mottola, Ravi Prakash, Antonello Tamburrino
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 351-388, March 2024. Abstract. This paper refers to an imaging problem in the presence of nonlinear materials. Specifically, the problem we address falls within the framework of Electrical Resistance Tomography and involves two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of development, although breakthroughs are expected in the not-too-distant future. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted [math]-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the [math]-Laplacian in inverse problems with nonlinear materials. Moreover, when [math], this result allows all the imaging methods and algorithms developed for linear materials to be brought into the arena of problems with nonlinear materials. The main result of this work is that for “small” Dirichlet data, (i) one material can be replaced by a perfect electric conductor and (ii) the other material can be replaced by a material giving rise to a weighted [math]-Laplace problem.
{"title":"The [math]-Laplace “Signature” for Quasilinear Inverse Problems","authors":"Antonio Corbo Esposito, Luisa Faella, Gianpaolo Piscitelli, Vincenzo Mottola, Ravi Prakash, Antonello Tamburrino","doi":"10.1137/22m1527192","DOIUrl":"https://doi.org/10.1137/22m1527192","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 351-388, March 2024. <br/> Abstract. This paper refers to an imaging problem in the presence of nonlinear materials. Specifically, the problem we address falls within the framework of Electrical Resistance Tomography and involves two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of development, although breakthroughs are expected in the not-too-distant future. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted [math]-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the [math]-Laplacian in inverse problems with nonlinear materials. Moreover, when [math], this result allows all the imaging methods and algorithms developed for linear materials to be brought into the arena of problems with nonlinear materials. The main result of this work is that for “small” Dirichlet data, (i) one material can be replaced by a perfect electric conductor and (ii) the other material can be replaced by a material giving rise to a weighted [math]-Laplace problem.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"11 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757625","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 334-350, March 2024. Abstract.Data-driven reduced order models (ROMs) have recently emerged as an efficient tool for the solution of inverse scattering problems with applications to seismic and sonar imaging. One requirement of this approach is that it uses the full square multiple-input/multiple-output (MIMO) matrix-valued transfer function as the data for multidimensional problems. The synthetic aperture radar (SAR), however, is limited to the single-input/single-output (SISO) measurements corresponding to the diagonal of the matrix transfer function. Here we present a ROM-based Lippmann–Schwinger approach overcoming this drawback. The ROMs are constructed to match the data for each source-receiver pair separately, and these are used to construct internal solutions for the corresponding source using only the data-driven Gramian. Efficiency of the proposed approach is demonstrated on 2D and 2.5D (3D propagation and 2D reflectors) numerical examples. The new algorithm not only suppresses multiple echoes seen in the Born imaging but also takes advantage of their illumination of some back sides of the reflectors, improving the quality of their mapping.
SIAM 影像科学期刊》第 17 卷第 1 期第 334-350 页，2024 年 3 月。摘要：数据驱动的降阶模型（ROMs）最近已成为解决反向散射问题的有效工具，并应用于地震和声纳成像。这种方法的一个要求是使用全平方多输入/多输出（MIMO）矩阵值传递函数作为多维问题的数据。然而，合成孔径雷达（SAR）仅限于与矩阵传递函数对角线相对应的单输入/单输出（SISO）测量。在此，我们提出了一种基于 ROM 的李普曼-施温格方法来克服这一缺点。构建 ROM 的目的是分别匹配每对信号源-接收器的数据，然后仅使用数据驱动的格拉米安为相应的信号源构建内部解决方案。在二维和 2.5 维（三维传播和二维反射体）数值示例中演示了所提方法的效率。新算法不仅抑制了 Born 成像中出现的多重回波，还利用了它们对反射体某些背面的照亮，提高了反射体映射的质量。
{"title":"Reduced Order Modeling Inversion of Monostatic Data in a Multi-scattering Environment","authors":"Vladimir Druskin, Shari Moskow, Mikhail Zaslavsky","doi":"10.1137/23m1564365","DOIUrl":"https://doi.org/10.1137/23m1564365","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 334-350, March 2024. <br/>Abstract.Data-driven reduced order models (ROMs) have recently emerged as an efficient tool for the solution of inverse scattering problems with applications to seismic and sonar imaging. One requirement of this approach is that it uses the full square multiple-input/multiple-output (MIMO) matrix-valued transfer function as the data for multidimensional problems. The synthetic aperture radar (SAR), however, is limited to the single-input/single-output (SISO) measurements corresponding to the diagonal of the matrix transfer function. Here we present a ROM-based Lippmann–Schwinger approach overcoming this drawback. The ROMs are constructed to match the data for each source-receiver pair separately, and these are used to construct internal solutions for the corresponding source using only the data-driven Gramian. Efficiency of the proposed approach is demonstrated on 2D and 2.5D (3D propagation and 2D reflectors) numerical examples. The new algorithm not only suppresses multiple echoes seen in the Born imaging but also takes advantage of their illumination of some back sides of the reflectors, improving the quality of their mapping.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"170 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757753","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}
Dominik Narnhofer, Andreas Habring, Martin Holler, Thomas Pock
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 301-333, March 2024. Abstract.In this work, a method for obtaining pixelwise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal prediction in order to obtain coverage guarantees for the error bounds, without making any assumption on the underlying data distribution. It is generally applicable to Bayesian regularization approaches, independent, e.g., of the concrete choice of the prior. Furthermore, the coverage guarantees can also be obtained in case only approximate sampling from the posterior is possible. With this in particular, the proposed framework is able to incorporate any learned prior in a black-box manner. Guaranteed coverage without assumptions on the underlying distributions is only achievable since the magnitude of the error bounds is, in general, unknown in advance. Nevertheless, experiments with multiple regularization approaches presented in the paper confirm that, in practice, the obtained error bounds are rather tight. For realizing the numerical experiments, a novel primal-dual Langevin algorithm for sampling from nonsmooth distributions is also introduced in this work, showing promising results in practice. While a proof of convergence for this primal-dual algorithm is still open, the theoretical guarantees of the proposed method do not require a guaranteed convergence of the sampling algorithm.
{"title":"Posterior-Variance–Based Error Quantification for Inverse Problems in Imaging","authors":"Dominik Narnhofer, Andreas Habring, Martin Holler, Thomas Pock","doi":"10.1137/23m1546129","DOIUrl":"https://doi.org/10.1137/23m1546129","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 301-333, March 2024. <br/> Abstract.In this work, a method for obtaining pixelwise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal prediction in order to obtain coverage guarantees for the error bounds, without making any assumption on the underlying data distribution. It is generally applicable to Bayesian regularization approaches, independent, e.g., of the concrete choice of the prior. Furthermore, the coverage guarantees can also be obtained in case only approximate sampling from the posterior is possible. With this in particular, the proposed framework is able to incorporate any learned prior in a black-box manner. Guaranteed coverage without assumptions on the underlying distributions is only achievable since the magnitude of the error bounds is, in general, unknown in advance. Nevertheless, experiments with multiple regularization approaches presented in the paper confirm that, in practice, the obtained error bounds are rather tight. For realizing the numerical experiments, a novel primal-dual Langevin algorithm for sampling from nonsmooth distributions is also introduced in this work, showing promising results in practice. While a proof of convergence for this primal-dual algorithm is still open, the theoretical guarantees of the proposed method do not require a guaranteed convergence of the sampling algorithm.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"170 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757707","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 273-300, March 2024. Abstract. Intensity-based image registration is critical for neuroimaging tasks, such as 3D reconstruction, times-series alignment, and common coordinate mapping. The gradient-based optimization methods commonly used to solve this problem require a careful selection of step-length. This limitation imposes substantial time and computational costs. Here we propose a gradient-independent rigid-motion registration algorithm based on the majorization-minimization (MM) principle. Each iteration of our intensity-based MM algorithm reduces to a simple point-set rigid registration problem with a closed form solution that avoids the step-length issue altogether. The details of the algorithm are presented, and an error bound for its more practical truncated form is derived. The performance of the MM algorithm is shown to be more effective than gradient descent on simulated images and Nissl stained coronal slices of mouse brain. We also compare and contrast the similarities and differences between the MM algorithm and another gradient-free registration algorithm called the block-matching method. Finally, extensions of this algorithm to more complex problems are discussed.
SIAM 影像科学杂志》第 17 卷第 1 期第 273-300 页，2024 年 3 月。 摘要基于强度的图像配准对于三维重建、时间序列配准和共坐标映射等神经成像任务至关重要。通常用于解决这一问题的基于梯度的优化方法需要仔细选择步长。这一限制带来了大量的时间和计算成本。在此，我们提出了一种与梯度无关的刚性运动配准算法，该算法基于大化最小化（MM）原理。我们基于强度的 MM 算法的每次迭代都会简化为一个简单的点集刚性配准问题，其闭合形式解完全避免了步长问题。本文介绍了该算法的细节，并推导出更实用的截断形式的误差边界。在模拟图像和 Nissl 染色的小鼠大脑冠状切片上，MM 算法的性能比梯度下降算法更有效。我们还比较了 MM 算法和另一种无梯度配准算法（即块匹配法）之间的异同。最后，我们还讨论了该算法在更复杂问题上的扩展。
{"title":"A Majorization-Minimization Algorithm for Neuroimage Registration","authors":"Gaiting Zhou, Daniel Tward, Kenneth Lange","doi":"10.1137/22m1516907","DOIUrl":"https://doi.org/10.1137/22m1516907","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 273-300, March 2024. <br/> Abstract. Intensity-based image registration is critical for neuroimaging tasks, such as 3D reconstruction, times-series alignment, and common coordinate mapping. The gradient-based optimization methods commonly used to solve this problem require a careful selection of step-length. This limitation imposes substantial time and computational costs. Here we propose a gradient-independent rigid-motion registration algorithm based on the majorization-minimization (MM) principle. Each iteration of our intensity-based MM algorithm reduces to a simple point-set rigid registration problem with a closed form solution that avoids the step-length issue altogether. The details of the algorithm are presented, and an error bound for its more practical truncated form is derived. The performance of the MM algorithm is shown to be more effective than gradient descent on simulated images and Nissl stained coronal slices of mouse brain. We also compare and contrast the similarities and differences between the MM algorithm and another gradient-free registration algorithm called the block-matching method. Finally, extensions of this algorithm to more complex problems are discussed.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"23 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757631","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 248-272, March 2024. Abstract. The Mumford–Shah model is a classical segmentation model, but its objective function is nonconvex. The smoothing and thresholding (SaT) approach is a convex variant of the Mumford–Shah model, which seeks a smoothed approximation solution to the Mumford–Shah model. The SaT approach separates the segmentation into two stages: first, a convex energy function is minimized to obtain a smoothed image; then, a thresholding technique is applied to segment the smoothed image. The energy function consists of three weighted terms and the weights are called the regularization parameters. Selecting appropriate regularization parameters is crucial to achieving effective segmentation results. Traditionally, the regularization parameters are chosen by trial-and-error, which is a very time-consuming procedure and is not practical in real applications. In this paper, we apply a Bayesian inference approach to infer the regularization parameters and estimate the smoothed image. We analyze the convex variant Mumford–Shah variational model from a statistical perspective and then construct a hierarchical Bayesian model. A mean field variational family is used to approximate the posterior distribution. The variational density of the smoothed image is assumed to have a Gaussian density, and the hyperparameters are assumed to have Gamma variational densities. All the parameters in the Gaussian density and Gamma densities are iteratively updated. Experimental results show that the proposed approach is capable of generating high-quality segmentation results. Although the proposed approach contains an inference step to estimate the regularization parameters, it requires less CPU running time to obtain the smoothed image than previous methods.
{"title":"Image Segmentation Using Bayesian Inference for Convex Variant Mumford–Shah Variational Model","authors":"Xu Xiao, Youwei Wen, Raymond Chan, Tieyong Zeng","doi":"10.1137/23m1545379","DOIUrl":"https://doi.org/10.1137/23m1545379","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 248-272, March 2024. <br/> Abstract. The Mumford–Shah model is a classical segmentation model, but its objective function is nonconvex. The smoothing and thresholding (SaT) approach is a convex variant of the Mumford–Shah model, which seeks a smoothed approximation solution to the Mumford–Shah model. The SaT approach separates the segmentation into two stages: first, a convex energy function is minimized to obtain a smoothed image; then, a thresholding technique is applied to segment the smoothed image. The energy function consists of three weighted terms and the weights are called the regularization parameters. Selecting appropriate regularization parameters is crucial to achieving effective segmentation results. Traditionally, the regularization parameters are chosen by trial-and-error, which is a very time-consuming procedure and is not practical in real applications. In this paper, we apply a Bayesian inference approach to infer the regularization parameters and estimate the smoothed image. We analyze the convex variant Mumford–Shah variational model from a statistical perspective and then construct a hierarchical Bayesian model. A mean field variational family is used to approximate the posterior distribution. The variational density of the smoothed image is assumed to have a Gaussian density, and the hyperparameters are assumed to have Gamma variational densities. All the parameters in the Gaussian density and Gamma densities are iteratively updated. Experimental results show that the proposed approach is capable of generating high-quality segmentation results. Although the proposed approach contains an inference step to estimate the regularization parameters, it requires less CPU running time to obtain the smoothed image than previous methods.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"217 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139585765","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 225-247, March 2024. Abstract. We study the tensor robust principal component analysis (TRPCA) problem, a tensorial extension of matrix robust principal component analysis, which aims to split the given tensor into an underlying low-rank component and a sparse outlier component. This work proposes a fast algorithm, called robust tensor CUR decompositions (RTCUR), for large-scale nonconvex TRPCA problems under the Tucker rank setting. RTCUR is developed within a framework of alternating projections that projects between the set of low-rank tensors and the set of sparse tensors. We utilize the recently developed tensor CUR decomposition to substantially reduce the computational complexity in each projection. In addition, we develop four variants of RTCUR for different application settings. We demonstrate the effectiveness and computational advantages of RTCUR against state-of-the-art methods on both synthetic and real-world datasets.
{"title":"Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruptions","authors":"HanQin Cai, Zehan Chao, Longxiu Huang, Deanna Needell","doi":"10.1137/23m1574282","DOIUrl":"https://doi.org/10.1137/23m1574282","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 225-247, March 2024. <br/> Abstract. We study the tensor robust principal component analysis (TRPCA) problem, a tensorial extension of matrix robust principal component analysis, which aims to split the given tensor into an underlying low-rank component and a sparse outlier component. This work proposes a fast algorithm, called robust tensor CUR decompositions (RTCUR), for large-scale nonconvex TRPCA problems under the Tucker rank setting. RTCUR is developed within a framework of alternating projections that projects between the set of low-rank tensors and the set of sparse tensors. We utilize the recently developed tensor CUR decomposition to substantially reduce the computational complexity in each projection. In addition, we develop four variants of RTCUR for different application settings. We demonstrate the effectiveness and computational advantages of RTCUR against state-of-the-art methods on both synthetic and real-world datasets.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554888","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 188-224, March 2024. Abstract. This paper is concerned with the problem of inverse scattering of time-harmonic acoustic plane waves by a two-layered medium with a locally rough interface in two dimensions. A direct imaging method is proposed to reconstruct the locally rough interface from the phaseless total-field data measured on the upper half of the circle with a large radius at a fixed frequency or from the phased far-field data measured on the upper half of the unit circle at a fixed frequency. The presence of the locally rough interface poses challenges in the theoretical analysis of the imaging methods. To address these challenges, a technically involved asymptotic analysis is provided for the relevant oscillatory integrals involved in the imaging methods, based mainly on the techniques and results in our recent work [L. Li, J. Yang, B. Zhang, and H. Zhang, arXiv:2208.00456, 2022] on the uniform far-field asymptotics of the scattered field for acoustic scattering in a two-layered medium. Finally, extensive numerical experiments are conducted to demonstrate the feasibility and robustness of our imaging algorithms.
SIAM 影像科学杂志》第 17 卷第 1 期第 188-224 页，2024 年 3 月。 摘要本文研究时谐平面波在二维具有局部粗糙界面的两层介质中的反向散射问题。本文提出了一种直接成像方法，可根据在固定频率下在大半径圆的上半部分测量的无相全场数据或在固定频率下在单位圆的上半部分测量的有相远场数据重建局部粗糙界面。局部粗糙界面的存在给成像方法的理论分析带来了挑战。为了应对这些挑战，主要基于我们最近关于两层介质中声散射的均匀远场渐近分析的工作[L. Li, J. Yang, B. Zhang, and H. Zhang, arXiv:2208.00456, 2022]中的技术和结果，对成像方法中涉及的相关振荡积分进行了渐近分析。最后，我们进行了大量的数值实验，以证明我们的成像算法的可行性和鲁棒性。
{"title":"Direct Imaging Methods for Reconstructing a Locally Rough Interface from Phaseless Total-Field Data or Phased Far-Field Data","authors":"Long Li, Jiansheng Yang, Bo Zhang, Haiwen Zhang","doi":"10.1137/23m1571393","DOIUrl":"https://doi.org/10.1137/23m1571393","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 188-224, March 2024. <br/> Abstract. This paper is concerned with the problem of inverse scattering of time-harmonic acoustic plane waves by a two-layered medium with a locally rough interface in two dimensions. A direct imaging method is proposed to reconstruct the locally rough interface from the phaseless total-field data measured on the upper half of the circle with a large radius at a fixed frequency or from the phased far-field data measured on the upper half of the unit circle at a fixed frequency. The presence of the locally rough interface poses challenges in the theoretical analysis of the imaging methods. To address these challenges, a technically involved asymptotic analysis is provided for the relevant oscillatory integrals involved in the imaging methods, based mainly on the techniques and results in our recent work [L. Li, J. Yang, B. Zhang, and H. Zhang, arXiv:2208.00456, 2022] on the uniform far-field asymptotics of the scattered field for acoustic scattering in a two-layered medium. Finally, extensive numerical experiments are conducted to demonstrate the feasibility and robustness of our imaging algorithms.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"255 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139555072","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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