Junren Chen, Zhaoqiang Liu, Meng Ding, Michael K. Ng
SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1909-1977, September 2024. Abstract.This paper studies quantized corrupted sensing where the measurements are contaminated by unknown corruption and then quantized by a dithered uniform quantizer. We establish uniform guarantees for Lasso that ensure the accurate recovery of all signals and corruptions using a single draw of the sub-Gaussian sensing matrix and uniform dither. For signal and corruption with structured priors (e.g., sparsity, low-rankness), our uniform error rate for constrained Lasso typically coincides with the nonuniform one up to logarithmic factors, indicating that the uniformity costs very little. By contrast, our uniform error rate for unconstrained Lasso exhibits worse dependence on the structured parameters due to regularization parameters larger than the ones for nonuniform recovery. These results complement the nonuniform ones recently obtained in Sun, Cui, and Liu [IEEE Trans. Signal Process., 70 (2022), pp. 600–615] and provide more insights for understanding actual applications where the sensing ensemble is typically fixed and the corruption may be adversarial. For signal and corruption living in the ranges of some Lipschitz continuous generative models (referred to as generative priors), we achieve uniform recovery via constrained Lasso with a measurement number proportional to the latent dimensions of the generative models. We present experimental results to corroborate our theories. From the technical side, our treatments to the two kinds of priors are (nearly) unified and share the common key ingredients of a (global) quantized product embedding (QPE) property, which states that the dithered uniform quantization (universally) preserves the inner product. As a by-product, our QPE result refines the one in Xu and Jacques [Inf. Inference, 9 (2020), pp. 543–586] under the sub-Gaussian random matrix, and in this specific instance, we are able to sharpen the uniform error decaying rate (for the projected back-projection estimator with signals in some convex symmetric set) presented therein from [math] to [math].
SIAM 影像科学杂志》，第 17 卷第 3 期，第 1909-1977 页，2024 年 9 月。 摘要：本文研究的是量化损坏传感，即测量被未知损坏污染，然后被抖动均匀量化器量化。我们为 Lasso 建立了统一保证，确保使用亚高斯传感矩阵的单次绘制和统一抖动准确恢复所有信号和损坏。对于具有结构化先验（如稀疏性、低秩性）的信号和损坏，我们的受约束 Lasso 统一错误率通常与非统一错误率重合到对数因子，这表明统一性的成本非常低。相比之下，由于正则化参数大于非均匀恢复的参数，我们的无约束拉索均匀误差率对结构参数的依赖性更差。这些结果补充了 Sun、Cui 和 Liu [IEEE Trans. Signal Process.对于处于某些 Lipschitz 连续生成模型（称为生成先验）范围内的信号和损坏，我们通过约束 Lasso 实现均匀恢复，测量次数与生成模型的潜在维度成正比。我们提出了实验结果来证实我们的理论。从技术层面来看，我们对这两种先验的处理方法（几乎）是统一的，并共享（全局）量化乘积嵌入（QPE）属性的共同关键要素，即抖动均匀量化（普遍）保留了内积。作为副产品，我们的 QPE 结果完善了 Xu 和 Jacques [Inf. Inference, 9 (2020), pp.
{"title":"Uniform Recovery Guarantees for Quantized Corrupted Sensing Using Structured or Generative Priors","authors":"Junren Chen, Zhaoqiang Liu, Meng Ding, Michael K. Ng","doi":"10.1137/23m1578358","DOIUrl":"https://doi.org/10.1137/23m1578358","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1909-1977, September 2024. <br/> Abstract.This paper studies quantized corrupted sensing where the measurements are contaminated by unknown corruption and then quantized by a dithered uniform quantizer. We establish uniform guarantees for Lasso that ensure the accurate recovery of all signals and corruptions using a single draw of the sub-Gaussian sensing matrix and uniform dither. For signal and corruption with structured priors (e.g., sparsity, low-rankness), our uniform error rate for constrained Lasso typically coincides with the nonuniform one up to logarithmic factors, indicating that the uniformity costs very little. By contrast, our uniform error rate for unconstrained Lasso exhibits worse dependence on the structured parameters due to regularization parameters larger than the ones for nonuniform recovery. These results complement the nonuniform ones recently obtained in Sun, Cui, and Liu [IEEE Trans. Signal Process., 70 (2022), pp. 600–615] and provide more insights for understanding actual applications where the sensing ensemble is typically fixed and the corruption may be adversarial. For signal and corruption living in the ranges of some Lipschitz continuous generative models (referred to as generative priors), we achieve uniform recovery via constrained Lasso with a measurement number proportional to the latent dimensions of the generative models. We present experimental results to corroborate our theories. From the technical side, our treatments to the two kinds of priors are (nearly) unified and share the common key ingredients of a (global) quantized product embedding (QPE) property, which states that the dithered uniform quantization (universally) preserves the inner product. As a by-product, our QPE result refines the one in Xu and Jacques [Inf. Inference, 9 (2020), pp. 543–586] under the sub-Gaussian random matrix, and in this specific instance, we are able to sharpen the uniform error decaying rate (for the projected back-projection estimator with signals in some convex symmetric set) presented therein from [math] to [math].","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266244","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 3, Page 1879-1908, September 2024. Abstract.In recent years, patch-based image restoration approaches have demonstrated superior performance compared to conventional variational methods. This paper delves into the mathematical foundations underlying patch-based image restoration methods, with a specific focus on establishing restoration guarantees for patch-based image inpainting, leveraging the assumption of self-similarity among patches. To accomplish this, we present a reformulation of the image inpainting problem as structured low-rank matrix completion, accomplished by grouping image patches with potential overlaps. By making certain incoherence assumptions, we establish a restoration guarantee, given that the number of samples exceeds the order of [math], where [math] denotes the size of the image and [math] represents the sum of ranks for each group of image patches. Through our rigorous mathematical analysis, we provide valuable insights into the theoretical foundations of patch-based image restoration methods, shedding light on their efficacy and offering guidelines for practical implementation.
{"title":"Restoration Guarantee of Image Inpainting via Low Rank Patch Matrix Completion","authors":"Jian-Feng Cai, Jae Kyu Choi, Jingyang Li, Guojian Yin","doi":"10.1137/23m1614456","DOIUrl":"https://doi.org/10.1137/23m1614456","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1879-1908, September 2024. <br/> Abstract.In recent years, patch-based image restoration approaches have demonstrated superior performance compared to conventional variational methods. This paper delves into the mathematical foundations underlying patch-based image restoration methods, with a specific focus on establishing restoration guarantees for patch-based image inpainting, leveraging the assumption of self-similarity among patches. To accomplish this, we present a reformulation of the image inpainting problem as structured low-rank matrix completion, accomplished by grouping image patches with potential overlaps. By making certain incoherence assumptions, we establish a restoration guarantee, given that the number of samples exceeds the order of [math], where [math] denotes the size of the image and [math] represents the sum of ranks for each group of image patches. Through our rigorous mathematical analysis, we provide valuable insights into the theoretical foundations of patch-based image restoration methods, shedding light on their efficacy and offering guidelines for practical implementation.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221313","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 3, Page 1844-1878, September 2024. Abstract.We study stability and inclusion of the jump set of minimizers of convex denoising functionals, such as the celebrated “Rudin–Osher–Fatemi” functional, for scalar or vectorial signals. We show that under mild regularity assumptions on the data fidelity term and the regularizer, the jump set of the minimizer is essentially a subset of the original jump set. Moreover, we give an estimate on the magnitude of the jumps in terms of the data. This extends old results, in particular of the first author (with Caselles and Novaga) and of Valkonen, to much more general cases. We also consider the case where the original datum has unbounded variation, and we define a notion of its jump set which, again, must contain the jump set of the solution.
{"title":"Inclusion and Estimates for the Jumps of Minimizers in Variational Denoising","authors":"Antonin Chambolle, Michał Łasica","doi":"10.1137/23m1627948","DOIUrl":"https://doi.org/10.1137/23m1627948","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1844-1878, September 2024. <br/> Abstract.We study stability and inclusion of the jump set of minimizers of convex denoising functionals, such as the celebrated “Rudin–Osher–Fatemi” functional, for scalar or vectorial signals. We show that under mild regularity assumptions on the data fidelity term and the regularizer, the jump set of the minimizer is essentially a subset of the original jump set. Moreover, we give an estimate on the magnitude of the jumps in terms of the data. This extends old results, in particular of the first author (with Caselles and Novaga) and of Valkonen, to much more general cases. We also consider the case where the original datum has unbounded variation, and we define a notion of its jump set which, again, must contain the jump set of the solution.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221314","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}
Jan Hamann, Quoc T. Le Gia, Ian H. Sloan, Robert S. Womersley
SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1820-1843, September 2024. Abstract.The paper analyzes a spectral approach to reconstructing a scalar field on the sphere, given only information about a masked version of the field, together with precise information about the (smooth) mask. The theory is developed for a general mask and later specialized to the case of an axially symmetric mask. Numerical experiments are given for the case of an axial mask motivated by the cosmic microwave background, assuming that the underlying field is a realization of a Gaussian random field with an artificial angular power spectrum of moderate degree ([math]). The recovery is highly satisfactory in the absence of noise and even in the presence of moderate noise.
{"title":"Removing the Mask—Reconstructing a Real-Valued Field on the Sphere from a Masked Field by Spherical Fourier Analysis","authors":"Jan Hamann, Quoc T. Le Gia, Ian H. Sloan, Robert S. Womersley","doi":"10.1137/23m1603157","DOIUrl":"https://doi.org/10.1137/23m1603157","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1820-1843, September 2024. <br/> Abstract.The paper analyzes a spectral approach to reconstructing a scalar field on the sphere, given only information about a masked version of the field, together with precise information about the (smooth) mask. The theory is developed for a general mask and later specialized to the case of an axially symmetric mask. Numerical experiments are given for the case of an axial mask motivated by the cosmic microwave background, assuming that the underlying field is a realization of a Gaussian random field with an artificial angular power spectrum of moderate degree ([math]). The recovery is highly satisfactory in the absence of noise and even in the presence of moderate noise.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221315","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}
Elli Karvonen, Matti Lassas, Pekka Pankka, Samuli Siltanen
SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1761-1794, September 2024. Abstract.A novel reconstruction method is introduced for the severely ill-posed inverse problem of limited-angle tomography. It is well known that, depending on the available measurement, angles specify a subset of the wavefront set of the unknown target, while some oriented singularities remain invisible in the data. Topological Interface recovery for Limited-angle Tomography, or TILT, is based on lifting the visible part of the wavefront set under a universal covering map. In the space provided, it is possible to connect the appropriate pieces of the lifted wavefront set correctly using dual-tree complex wavelets, a dedicated metric, and persistent homology. The result is not only a suggested invisible boundary but also a computational representation for all interfaces in the target.
{"title":"TILT: Topological Interface Recovery in Limited-Angle Tomography","authors":"Elli Karvonen, Matti Lassas, Pekka Pankka, Samuli Siltanen","doi":"10.1137/23m1611567","DOIUrl":"https://doi.org/10.1137/23m1611567","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1761-1794, September 2024. <br/> Abstract.A novel reconstruction method is introduced for the severely ill-posed inverse problem of limited-angle tomography. It is well known that, depending on the available measurement, angles specify a subset of the wavefront set of the unknown target, while some oriented singularities remain invisible in the data. Topological Interface recovery for Limited-angle Tomography, or TILT, is based on lifting the visible part of the wavefront set under a universal covering map. In the space provided, it is possible to connect the appropriate pieces of the lifted wavefront set correctly using dual-tree complex wavelets, a dedicated metric, and persistent homology. The result is not only a suggested invisible boundary but also a computational representation for all interfaces in the target.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221317","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 3, Page 1795-1819, September 2024. Abstract.This paper focuses on studying the denoising problem for positive semidefinite fourth-order tensor field estimation from noisy observations. The positive semidefiniteness of the tensor is preserved by mapping the tensor to a 6-by-6 symmetric positive semidefinite matrix where its matrix rank is less than or equal to three. For denoising, we propose to use an anisotropic discrete total variation function over the tensor field as the regularization term. We propose an inexact majorized proximal alternating direction method of multipliers for such a nonconvex and nonsmooth optimization problem. We show that an [math]-stationary solution of the resulting optimization problem can be found in no more than [math] iterations. The effectiveness of the proposed model and algorithm is tested using multifiber diffusion weighted imaging data, and our numerical results demonstrate that our method outperforms existing methods in terms of denoising performance.
{"title":"An Inexact Majorized Proximal Alternating Direction Method of Multipliers for Diffusion Tensors","authors":"Hong Zhu, Michael K. Ng","doi":"10.1137/23m1607015","DOIUrl":"https://doi.org/10.1137/23m1607015","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1795-1819, September 2024. <br/> Abstract.This paper focuses on studying the denoising problem for positive semidefinite fourth-order tensor field estimation from noisy observations. The positive semidefiniteness of the tensor is preserved by mapping the tensor to a 6-by-6 symmetric positive semidefinite matrix where its matrix rank is less than or equal to three. For denoising, we propose to use an anisotropic discrete total variation function over the tensor field as the regularization term. We propose an inexact majorized proximal alternating direction method of multipliers for such a nonconvex and nonsmooth optimization problem. We show that an [math]-stationary solution of the resulting optimization problem can be found in no more than [math] iterations. The effectiveness of the proposed model and algorithm is tested using multifiber diffusion weighted imaging data, and our numerical results demonstrate that our method outperforms existing methods in terms of denoising performance.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221316","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}
Matthias J. Ehrhardt, Lorenz Kuger, Carola-Bibiane Schönlieb
SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1729-1760, September 2024. Abstract. In order to solve tasks like uncertainty quantification or hypothesis tests in Bayesian imaging inverse problems, we often have to draw samples from the arising posterior distribution. For the usually log-concave but high-dimensional posteriors, Markov chain Monte Carlo methods based on time discretizations of Langevin diffusion are a popular tool. If the potential defining the distribution is nonsmooth, these discretizations are usually of an implicit form leading to Langevin sampling algorithms that require the evaluation of proximal operators. For some of the potentials relevant in imaging problems this is only possible approximately using an iterative scheme. We investigate the behavior of a proximal Langevin algorithm under the presence of errors in the evaluation of proximal mappings. We generalize existing nonasymptotic and asymptotic convergence results of the exact algorithm to our inexact setting and quantify the bias between the target and the algorithm’s stationary distribution due to the errors. We show that the additional bias stays bounded for bounded errors and converges to zero for decaying errors in a strongly convex setting. We apply the inexact algorithm to sample numerically from the posterior of typical imaging inverse problems in which we can only approximate the proximal operator by an iterative scheme and validate our theoretical convergence results.
{"title":"Proximal Langevin Sampling with Inexact Proximal Mapping","authors":"Matthias J. Ehrhardt, Lorenz Kuger, Carola-Bibiane Schönlieb","doi":"10.1137/23m1593565","DOIUrl":"https://doi.org/10.1137/23m1593565","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1729-1760, September 2024. <br/> Abstract. In order to solve tasks like uncertainty quantification or hypothesis tests in Bayesian imaging inverse problems, we often have to draw samples from the arising posterior distribution. For the usually log-concave but high-dimensional posteriors, Markov chain Monte Carlo methods based on time discretizations of Langevin diffusion are a popular tool. If the potential defining the distribution is nonsmooth, these discretizations are usually of an implicit form leading to Langevin sampling algorithms that require the evaluation of proximal operators. For some of the potentials relevant in imaging problems this is only possible approximately using an iterative scheme. We investigate the behavior of a proximal Langevin algorithm under the presence of errors in the evaluation of proximal mappings. We generalize existing nonasymptotic and asymptotic convergence results of the exact algorithm to our inexact setting and quantify the bias between the target and the algorithm’s stationary distribution due to the errors. We show that the additional bias stays bounded for bounded errors and converges to zero for decaying errors in a strongly convex setting. We apply the inexact algorithm to sample numerically from the posterior of typical imaging inverse problems in which we can only approximate the proximal operator by an iterative scheme and validate our theoretical convergence results.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141864204","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 3, Page 1690-1728, September 2024. Abstract.Intensity inhomogeneity is a challenging task in image registration. Few past works have addressed the case of intensity inhomogeneity due to texture noise. To address this difficulty, we propose a novel three-stage approach for 2D/3D diffeomorphic multimodality image registration. The proposed approach contains three stages: (1) [math] decomposition which decomposes the image pairs into texture, noise, and smooth component; (2) Blake–Zisserman homogenization which transforms the geometric features from different modalities into approximately the same modality in terms of the first-order and second-order edge information; (3) image registration which combines the homogenized geometric features and mutual information. Based on the proposed approach, the greedy matching for multimodality image registration is discussed and a coarse-to-fine algorithm is also proposed. Furthermore, several numerical tests are performed to validate the efficiency of the proposed approach.
{"title":"Three-Stage Approach for 2D/3D Diffeomorphic Multimodality Image Registration with Textural Control","authors":"Ke Chen, Huan Han","doi":"10.1137/23m1583971","DOIUrl":"https://doi.org/10.1137/23m1583971","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1690-1728, September 2024. <br/> Abstract.Intensity inhomogeneity is a challenging task in image registration. Few past works have addressed the case of intensity inhomogeneity due to texture noise. To address this difficulty, we propose a novel three-stage approach for 2D/3D diffeomorphic multimodality image registration. The proposed approach contains three stages: (1) [math] decomposition which decomposes the image pairs into texture, noise, and smooth component; (2) Blake–Zisserman homogenization which transforms the geometric features from different modalities into approximately the same modality in terms of the first-order and second-order edge information; (3) image registration which combines the homogenized geometric features and mutual information. Based on the proposed approach, the greedy matching for multimodality image registration is discussed and a coarse-to-fine algorithm is also proposed. Furthermore, several numerical tests are performed to validate the efficiency of the proposed approach.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141773716","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 3, Page 1650-1689, September 2024. Abstract.A classical approach to designing binary image operators is mathematical morphology (MM). We propose the Discrete Morphological Neural Networks (DMNN) for binary image analysis to represent W-operators and estimate them via machine learning. A DMNN architecture, which is represented by a morphological computational graph, is designed as in the classical heuristic design of morphological operators, in which the designer should combine a set of MM operators and Boolean operations based on prior information and theoretical knowledge. Then, once the architecture is fixed, instead of adjusting its parameters (i.e., structuring elements or maximal intervals) by hand, we propose a lattice descent algorithm (LDA) to train these parameters based on a sample of input and output images under the usual machine learning approach. We also propose a stochastic version of the LDA that is more efficient, is scalable, and can obtain small error in practical problems. The class represented by a DMNN can be quite general or specialized according to expected properties of the target operator, i.e., prior information, and the semantic expressed by algebraic properties of classes of operators is a differential relative to other methods. The main contribution of this paper is the merger of the two main paradigms for designing morphological operators: classical heuristic design and automatic design via machine learning. As a proof-of-concept, we apply the DMNN to recognize the boundary of digits with noise, and we discuss many topics for future research.
{"title":"Discrete Morphological Neural Networks","authors":"Diego Marcondes, Junior Barrera","doi":"10.1137/23m1598477","DOIUrl":"https://doi.org/10.1137/23m1598477","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1650-1689, September 2024. <br/> Abstract.A classical approach to designing binary image operators is mathematical morphology (MM). We propose the Discrete Morphological Neural Networks (DMNN) for binary image analysis to represent W-operators and estimate them via machine learning. A DMNN architecture, which is represented by a morphological computational graph, is designed as in the classical heuristic design of morphological operators, in which the designer should combine a set of MM operators and Boolean operations based on prior information and theoretical knowledge. Then, once the architecture is fixed, instead of adjusting its parameters (i.e., structuring elements or maximal intervals) by hand, we propose a lattice descent algorithm (LDA) to train these parameters based on a sample of input and output images under the usual machine learning approach. We also propose a stochastic version of the LDA that is more efficient, is scalable, and can obtain small error in practical problems. The class represented by a DMNN can be quite general or specialized according to expected properties of the target operator, i.e., prior information, and the semantic expressed by algebraic properties of classes of operators is a differential relative to other methods. The main contribution of this paper is the merger of the two main paradigms for designing morphological operators: classical heuristic design and automatic design via machine learning. As a proof-of-concept, we apply the DMNN to recognize the boundary of digits with noise, and we discuss many topics for future research.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141773778","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}
Giovanni S. Alberti, Romain Petit, Matteo Santacesaria
SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1619-1649, September 2024. Abstract.We consider the inverse scattering problem for time-harmonic acoustic waves in a medium with pointwise inhomogeneities. In the Foldy–Lax model, the estimation of the scatterers’ locations and intensities from far field measurements can be recast as the recovery of a discrete measure from nonlinear observations. We propose a “linearize and locally optimize” approach to perform this reconstruction. We first solve a convex program in the space of measures (known as the Beurling LASSO), which involves a linearization of the forward operator (the far field pattern in the Born approximation). Then, we locally minimize a second functional involving the nonlinear forward map, using the output of the first step as initialization. We provide guarantees that the output of the first step is close to the sought-after measure when the scatterers have small intensities and are sufficiently separated. We also provide numerical evidence that the second step still allows for accurate recovery in settings that are more involved.
{"title":"Localization of Point Scatterers via Sparse Optimization on Measures","authors":"Giovanni S. Alberti, Romain Petit, Matteo Santacesaria","doi":"10.1137/24m1636265","DOIUrl":"https://doi.org/10.1137/24m1636265","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1619-1649, September 2024. <br/> Abstract.We consider the inverse scattering problem for time-harmonic acoustic waves in a medium with pointwise inhomogeneities. In the Foldy–Lax model, the estimation of the scatterers’ locations and intensities from far field measurements can be recast as the recovery of a discrete measure from nonlinear observations. We propose a “linearize and locally optimize” approach to perform this reconstruction. We first solve a convex program in the space of measures (known as the Beurling LASSO), which involves a linearization of the forward operator (the far field pattern in the Born approximation). Then, we locally minimize a second functional involving the nonlinear forward map, using the output of the first step as initialization. We provide guarantees that the output of the first step is close to the sought-after measure when the scatterers have small intensities and are sufficiently separated. We also provide numerical evidence that the second step still allows for accurate recovery in settings that are more involved.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141773715","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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