SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1587-1618, September 2024. Abstract.The choice of the sensing matrix is crucial in compressed sensing. Random Gaussian sensing matrices satisfy the restricted isometry property, which is crucial for solving the sparse recovery problem using convex optimization techniques. However, tight-frame sensing matrices result in minimum mean-squared-error recovery given oracle knowledge of the support of the sparse vector. If the sensing matrix is not tight, could one achieve the recovery performance assured by a tight frame by suitably designing the recovery strategy? This is the key question addressed in this paper. We consider the analysis-sparse [math]-minimization problem with a generalized [math]-norm-based data-fidelity and show that it effectively corresponds to using a tight-frame sensing matrix. The new formulation offers improved performance bounds when the number of nonzeros is large. One could develop a tight-frame variant of a known sparse recovery algorithm using the proposed formalism. We solve the analysis-sparse recovery problem in an unconstrained setting using proximal methods. Within the tight-frame sensing framework, we rescale the gradients of the data-fidelity loss in the iterative updates to further improve the accuracy of analysis-sparse recovery. Experimental results show that the proposed algorithms offer superior analysis-sparse recovery performance. Proceeding further, we also develop deep-unfolded variants, with a convolutional neural network as the sparsifying operator. On the application front, we consider compressed sensing image recovery. Experimental results on Set11, BSD68, Urban100, and DIV2K datasets show that the proposed techniques outperform the state-of-the-art techniques, with performance measured in terms of peak signal-to-noise ratio and structural similarity index metric.
{"title":"Tight-Frame-Like Analysis-Sparse Recovery Using Nontight Sensing Matrices","authors":"Kartheek Kumar Reddy Nareddy, Abijith Jagannath Kamath, Chandra Sekhar Seelamantula","doi":"10.1137/23m1625846","DOIUrl":"https://doi.org/10.1137/23m1625846","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1587-1618, September 2024. <br/> Abstract.The choice of the sensing matrix is crucial in compressed sensing. Random Gaussian sensing matrices satisfy the restricted isometry property, which is crucial for solving the sparse recovery problem using convex optimization techniques. However, tight-frame sensing matrices result in minimum mean-squared-error recovery given oracle knowledge of the support of the sparse vector. If the sensing matrix is not tight, could one achieve the recovery performance assured by a tight frame by suitably designing the recovery strategy? This is the key question addressed in this paper. We consider the analysis-sparse [math]-minimization problem with a generalized [math]-norm-based data-fidelity and show that it effectively corresponds to using a tight-frame sensing matrix. The new formulation offers improved performance bounds when the number of nonzeros is large. One could develop a tight-frame variant of a known sparse recovery algorithm using the proposed formalism. We solve the analysis-sparse recovery problem in an unconstrained setting using proximal methods. Within the tight-frame sensing framework, we rescale the gradients of the data-fidelity loss in the iterative updates to further improve the accuracy of analysis-sparse recovery. Experimental results show that the proposed algorithms offer superior analysis-sparse recovery performance. Proceeding further, we also develop deep-unfolded variants, with a convolutional neural network as the sparsifying operator. On the application front, we consider compressed sensing image recovery. Experimental results on Set11, BSD68, Urban100, and DIV2K datasets show that the proposed techniques outperform the state-of-the-art techniques, with performance measured in terms of peak signal-to-noise ratio and structural similarity index metric.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722573","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}
Dihan Zheng, Shiqi Tang, R. Wagner, Ronny Ramlau, Chenglong Bao, Raymond H. Chan
{"title":"PhaseNet: A Deep Learning Based Phase Reconstruction Method for Ground-Based Astronomy","authors":"Dihan Zheng, Shiqi Tang, R. Wagner, Ronny Ramlau, Chenglong Bao, Raymond H. Chan","doi":"10.1137/23m1592377","DOIUrl":"https://doi.org/10.1137/23m1592377","url":null,"abstract":"","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141648291","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}
Christina Brandt, T. Kluth, Tobias Knopp, Lena Westen
{"title":"Dynamic Image Reconstruction with Motion Priors in Application to Three Dimensional Magnetic Particle Imaging","authors":"Christina Brandt, T. Kluth, Tobias Knopp, Lena Westen","doi":"10.1137/23m1580401","DOIUrl":"https://doi.org/10.1137/23m1580401","url":null,"abstract":"","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141645591","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 1467-1510, September 2024. Abstract.We consider the inverse problem of fitting atmospheric dispersion parameters based on time-resolved back-scattered differential absorption Lidar (DIAL) measurements. The obvious advantage of light-based remote sensing modalities is their extended spatial range which makes them less sensitive to strictly local perturbations/modelling errors or the distance to the plume source. In contrast to other state-of-the-art DIAL methods, we do not make a single scattering assumption but rather propose a new type modality which includes the collection of multiply scattered photons from wider/multiple fields-of-view and argue that this data, paired with a time dependent radiative transfer model, is beneficial for the reconstruction of certain image features. The resulting inverse problem is solved by means of a semiparametric approach in which the image is reduced to a small number of dispersion related parameters and high-dimensional but computationally convenient nuisance component. This not only allows us to effectively avoid a high-dimensional inverse problem but simultaneously provides a natural regularization mechanism along with parameters which are directly related to the dispersion model. These can be associated with meaningful physical units while spatial concentration profiles can be obtained by means of forward evaluation of the dispersion process.
{"title":"Imaging of Atmospheric Dispersion Processes with Differential Absorption Lidar","authors":"Robert Lung, Nick Polydorides","doi":"10.1137/23m1598404","DOIUrl":"https://doi.org/10.1137/23m1598404","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1467-1510, September 2024. <br/> Abstract.We consider the inverse problem of fitting atmospheric dispersion parameters based on time-resolved back-scattered differential absorption Lidar (DIAL) measurements. The obvious advantage of light-based remote sensing modalities is their extended spatial range which makes them less sensitive to strictly local perturbations/modelling errors or the distance to the plume source. In contrast to other state-of-the-art DIAL methods, we do not make a single scattering assumption but rather propose a new type modality which includes the collection of multiply scattered photons from wider/multiple fields-of-view and argue that this data, paired with a time dependent radiative transfer model, is beneficial for the reconstruction of certain image features. The resulting inverse problem is solved by means of a semiparametric approach in which the image is reduced to a small number of dispersion related parameters and high-dimensional but computationally convenient nuisance component. This not only allows us to effectively avoid a high-dimensional inverse problem but simultaneously provides a natural regularization mechanism along with parameters which are directly related to the dispersion model. These can be associated with meaningful physical units while spatial concentration profiles can be obtained by means of forward evaluation of the dispersion process.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141609697","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}
Michael Wilson, Tom Needham, Chiwoo Park, Suparteek Kundu, Anuj Srivastava
SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1433-1466, September 2024. Abstract.This paper uses sample data to study the problem of comparing populations on finite-dimensional parallelizable Riemannian manifolds and more general trivial vector bundles. Utilizing triviality, our framework represents populations as mixtures of Gaussians on vector bundles and estimates the population parameters using a mode-based clustering algorithm. We derive a Wasserstein-type metric between Gaussian mixtures, adapted to the manifold geometry, in order to compare estimated distributions. Our contributions include an identifiability result for Gaussian mixtures on manifold domains and a convenient characterization of optimal couplings of Gaussian mixtures under the derived metric. We demonstrate these tools on some example domains, including the preshape space of planar closed curves, with applications to the shape space of triangles and populations of nanoparticles. In the nanoparticle application, we consider a sequence of populations of particle shapes arising from a manufacturing process and utilize the Wasserstein-type distance to perform change-point detection.
{"title":"A Wasserstein-Type Distance for Gaussian Mixtures on Vector Bundles with Applications to Shape Analysis","authors":"Michael Wilson, Tom Needham, Chiwoo Park, Suparteek Kundu, Anuj Srivastava","doi":"10.1137/23m1620363","DOIUrl":"https://doi.org/10.1137/23m1620363","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1433-1466, September 2024. <br/> Abstract.This paper uses sample data to study the problem of comparing populations on finite-dimensional parallelizable Riemannian manifolds and more general trivial vector bundles. Utilizing triviality, our framework represents populations as mixtures of Gaussians on vector bundles and estimates the population parameters using a mode-based clustering algorithm. We derive a Wasserstein-type metric between Gaussian mixtures, adapted to the manifold geometry, in order to compare estimated distributions. Our contributions include an identifiability result for Gaussian mixtures on manifold domains and a convenient characterization of optimal couplings of Gaussian mixtures under the derived metric. We demonstrate these tools on some example domains, including the preshape space of planar closed curves, with applications to the shape space of triangles and populations of nanoparticles. In the nanoparticle application, we consider a sequence of populations of particle shapes arising from a manufacturing process and utilize the Wasserstein-type distance to perform change-point detection.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141609696","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}
Mercedes Garcia-Salguero, Elijs Dima, André Mateus, Javier Gonzalez-Jimenez
SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1415-1432, September 2024. Abstract.Estimating the absolute pose of a camera given a set of [math] points and their observations is known as the resectioning or Perspective-n-Point (PnP) problem. It is at the core of most computer vision applications and it can be stated as an instance of three-dimensional registration with point-line distances, making the error quadratic in the unknown pose. The PnP problem, though, is nonconvex due to the constraints associated with the rotation, and iterative algorithms may get trapped into any suboptimal solutions without notice. This work proposes an efficient certification algorithm for central and noncentral cameras that either confirms the optimality of a solution or is inconclusive. We exploit different sets of constraints for the rotation to assess their performance in terms of certification. Two of the formulations lack the Linear Independence Constraint Qualification (LICQ) while one of them has more constraints than variables. This hinders the usage of the “standard” procedure which estimates the Lagrange multipliers in closed-form. To overcome that, we formulate the certification as an eigenvalue optimization and solve it through a line-search method. Our evaluation on synthetic and real data shows that minimal formulations certify most solutions (more than [math] on real data) whereas redundant formulations are able to certify all of them and even random problem instances. The proposed algorithm runs in microseconds for all these formulations.
{"title":"Fast Certifiable Algorithm for the Absolute Pose Estimation of a Camera","authors":"Mercedes Garcia-Salguero, Elijs Dima, André Mateus, Javier Gonzalez-Jimenez","doi":"10.1137/23m159994x","DOIUrl":"https://doi.org/10.1137/23m159994x","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1415-1432, September 2024. <br/> Abstract.Estimating the absolute pose of a camera given a set of [math] points and their observations is known as the resectioning or Perspective-n-Point (PnP) problem. It is at the core of most computer vision applications and it can be stated as an instance of three-dimensional registration with point-line distances, making the error quadratic in the unknown pose. The PnP problem, though, is nonconvex due to the constraints associated with the rotation, and iterative algorithms may get trapped into any suboptimal solutions without notice. This work proposes an efficient certification algorithm for central and noncentral cameras that either confirms the optimality of a solution or is inconclusive. We exploit different sets of constraints for the rotation to assess their performance in terms of certification. Two of the formulations lack the Linear Independence Constraint Qualification (LICQ) while one of them has more constraints than variables. This hinders the usage of the “standard” procedure which estimates the Lagrange multipliers in closed-form. To overcome that, we formulate the certification as an eigenvalue optimization and solve it through a line-search method. Our evaluation on synthetic and real data shows that minimal formulations certify most solutions (more than [math] on real data) whereas redundant formulations are able to certify all of them and even random problem instances. The proposed algorithm runs in microseconds for all these formulations.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141609720","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 1377-1414, September 2024. Abstract.In this paper, we introduce a frequency-domain approach to extract information on the trajectory of a moving point source. The method hinges on the analysis of multifrequency near-field data recorded at one and sparse observation points in three dimensions. The radiating period of the moving point source is supposed to be supported on the real axis and a priori known. In contrast to inverse stationary source problems, one needs to classify observable and non-observable measurement positions. The analogues of these concepts in the far-field regime were first proposed in the authors’ previous paper [SIAM J. Imaging Sci., 16 (2023), pp. 1535–1571]. In this paper we shall derive the observable and non-observable measurement positions for straight and circular motions in [math]. In the near-field case, we verify that the smallest annular region centered at an observable position that contains the trajectory can be imaged for an admissible class of orbit functions. Using the data from sparse observable positions, it is possible to reconstruct the [math]-convex domain of the trajectory. Intensive 3D numerical tests with synthetic data are performed to show effectiveness and feasibility of this new algorithm.
{"title":"Imaging a Moving Point Source from Multifrequency Data Measured at One and Sparse Observation Points (Part II): Near-Field Case in 3D","authors":"Guanqiu Ma, Hongxia Guo, Guanghui Hu","doi":"10.1137/23m162260x","DOIUrl":"https://doi.org/10.1137/23m162260x","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1377-1414, September 2024. <br/> Abstract.In this paper, we introduce a frequency-domain approach to extract information on the trajectory of a moving point source. The method hinges on the analysis of multifrequency near-field data recorded at one and sparse observation points in three dimensions. The radiating period of the moving point source is supposed to be supported on the real axis and a priori known. In contrast to inverse stationary source problems, one needs to classify observable and non-observable measurement positions. The analogues of these concepts in the far-field regime were first proposed in the authors’ previous paper [SIAM J. Imaging Sci., 16 (2023), pp. 1535–1571]. In this paper we shall derive the observable and non-observable measurement positions for straight and circular motions in [math]. In the near-field case, we verify that the smallest annular region centered at an observable position that contains the trajectory can be imaged for an admissible class of orbit functions. Using the data from sparse observable positions, it is possible to reconstruct the [math]-convex domain of the trajectory. Intensive 3D numerical tests with synthetic data are performed to show effectiveness and feasibility of this new algorithm.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551083","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}
Guillaume Lauga, Elisa Riccietti, Nelly Pustelnik, Paulo Gonçalves
SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1347-1376, September 2024. Abstract. This paper presents a multilevel framework for inertial and inexact proximal algorithms that encompasses multilevel versions of classical algorithms such as forward-backward and FISTA. The methods are supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We propose a particular instance of IML (Inexact MultiLevel) FISTA, based on the use of the Moreau envelope to build efficient and useful coarse corrections, fully adapted to solve problems in image restoration. Such a construction is derived for a broad class of composite optimization problems with proximable functions. We evaluate our approach on several image reconstruction problems, and we show that it considerably accelerates the convergence of the corresponding one-level (i.e., standard) version of the methods for large-scale images.
{"title":"IML FISTA: A Multilevel Framework for Inexact and Inertial Forward-Backward. Application to Image Restoration","authors":"Guillaume Lauga, Elisa Riccietti, Nelly Pustelnik, Paulo Gonçalves","doi":"10.1137/23m1582345","DOIUrl":"https://doi.org/10.1137/23m1582345","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1347-1376, September 2024. <br/> Abstract. This paper presents a multilevel framework for inertial and inexact proximal algorithms that encompasses multilevel versions of classical algorithms such as forward-backward and FISTA. The methods are supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We propose a particular instance of IML (Inexact MultiLevel) FISTA, based on the use of the Moreau envelope to build efficient and useful coarse corrections, fully adapted to solve problems in image restoration. Such a construction is derived for a broad class of composite optimization problems with proximable functions. We evaluate our approach on several image reconstruction problems, and we show that it considerably accelerates the convergence of the corresponding one-level (i.e., standard) version of the methods for large-scale images.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503736","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 1314-1346, June 2024. Abstract.Neural networks allow solving many ill-posed inverse problems with unprecedented performance. Physics informed approaches already progressively replace carefully hand-crafted reconstruction algorithms in real applications. However, these networks suffer from a major defect: when trained on a given forward operator, they do not generalize well to a different one. The aim of this paper is twofold. First, we show through various applications that training the network with a family of forward operators allows solving the adaptivity problem without compromising the reconstruction quality significantly. Second, we illustrate that this training procedure allows tackling challenging blind inverse problems. Our experiments include partial Fourier sampling problems arising in magnetic resonance imaging with sensitivity estimation and off-resonance effects, computerized tomography with a tilted geometry, and image deblurring with Fresnel diffraction kernels.
{"title":"Training Adaptive Reconstruction Networks for Blind Inverse Problems","authors":"Alban Gossard, Pierre Weiss","doi":"10.1137/23m1545628","DOIUrl":"https://doi.org/10.1137/23m1545628","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 1314-1346, June 2024. <br/> Abstract.Neural networks allow solving many ill-posed inverse problems with unprecedented performance. Physics informed approaches already progressively replace carefully hand-crafted reconstruction algorithms in real applications. However, these networks suffer from a major defect: when trained on a given forward operator, they do not generalize well to a different one. The aim of this paper is twofold. First, we show through various applications that training the network with a family of forward operators allows solving the adaptivity problem without compromising the reconstruction quality significantly. Second, we illustrate that this training procedure allows tackling challenging blind inverse problems. Our experiments include partial Fourier sampling problems arising in magnetic resonance imaging with sensitivity estimation and off-resonance effects, computerized tomography with a tilted geometry, and image deblurring with Fresnel diffraction kernels.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503737","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 1255-1283, June 2024. Abstract.Image processing on surfaces has gotten increasing interest in recent years, and denoising is a basic problem in image processing. In this paper, we extend non-Lipschitz variational methods for 2D image denoising, including TV[math], to image denoising on surfaces. We establish a lower bound for nonzero gradients of the recovered image, implying the advantage of the models in recovering piecewise constant images. A new iteratively reweighted least squares algorithm with the thresholding and support shrinking strategy is proposed. The global convergence of the algorithm is established under the assumption that the object function is a Kurdyka–Łojasiewicz function. Numerical examples are given to show good performance of the algorithm.
{"title":"Non-Lipschitz Variational Models and their Iteratively Reweighted Least Squares Algorithms for Image Denoising on Surfaces","authors":"Yuan Liu, Chunlin Wu, Chao Zeng","doi":"10.1137/23m159439x","DOIUrl":"https://doi.org/10.1137/23m159439x","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 1255-1283, June 2024. <br/> Abstract.Image processing on surfaces has gotten increasing interest in recent years, and denoising is a basic problem in image processing. In this paper, we extend non-Lipschitz variational methods for 2D image denoising, including TV[math], to image denoising on surfaces. We establish a lower bound for nonzero gradients of the recovered image, implying the advantage of the models in recovering piecewise constant images. A new iteratively reweighted least squares algorithm with the thresholding and support shrinking strategy is proposed. The global convergence of the algorithm is established under the assumption that the object function is a Kurdyka–Łojasiewicz function. Numerical examples are given to show good performance of the algorithm.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503738","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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