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SIAM Journal on Imaging Sciences
Journal Prestige (SJR): 1.371  Citation Impact (citeScore): 3 Number of Followers: 7  Hybrid journal * Containing 2  Open Access article(s) in this issue *ISSN (Print) 1936-4954 Published by Society for Industrial and Applied Mathematics  [17 journals]
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- IML FISTA: A Multilevel Framework for Inexact and Inertial
Forward-Backward. Application to Image Restoration-
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Authors: Guillaume Lauga, Elisa Riccietti, Nelly Pustelnik, Paulo Gonçalves
Pages: 1347 - 1376
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-01T07:00:00Z
DOI: 10.1137/23M1582345
Issue No: Vol. 17, No. 3 (2024)
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- Imaging a Moving Point Source from Multifrequency Data Measured at One and
Sparse Observation Points (Part II): Near-Field Case in 3D-
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Authors: Guanqiu Ma, Hongxia Guo, Guanghui Hu
Pages: 1377 - 1414
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-03T07:00:00Z
DOI: 10.1137/23M162260X
Issue No: Vol. 17, No. 3 (2024)
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- Fast Certifiable Algorithm for the Absolute Pose Estimation of a Camera
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Authors: Mercedes Garcia-Salguero, Elijs Dima, André Mateus, Javier Gonzalez-Jimenez
Pages: 1415 - 1432
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-11T07:00:00Z
DOI: 10.1137/23M159994X
Issue No: Vol. 17, No. 3 (2024)
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- A Wasserstein-Type Distance for Gaussian Mixtures on Vector Bundles with
Applications to Shape Analysis-
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Authors: Michael Wilson, Tom Needham, Chiwoo Park, Suparteek Kundu, Anuj Srivastava
Pages: 1433 - 1466
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-11T07:00:00Z
DOI: 10.1137/23M1620363
Issue No: Vol. 17, No. 3 (2024)
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- Imaging of Atmospheric Dispersion Processes with Differential Absorption
Lidar-
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Authors: Robert Lung, Nick Polydorides
Pages: 1467 - 1510
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-12T07:00:00Z
DOI: 10.1137/23M1598404
Issue No: Vol. 17, No. 3 (2024)
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- PhaseNet: A Deep Learning Based Phase Reconstruction Method for
Ground-Based Astronomy-
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Authors: Dihan Zheng, Shiqi Tang, Roland Wagner, Ronny Ramlau, Chenglong Bao, Raymond H. Chan
Pages: 1511 - 1538
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1511-1538, September 2024.
Abstract.Ground-based astronomy utilizes modern telescopes to obtain information on the universe by analyzing recorded signals. Due to atmospheric turbulence, the reconstruction process requires solving a deconvolution problem with an unknown point spread function (PSF). The crucial step in PSF estimation is to obtain a high-resolution phase from low-resolution phase gradients, which is a challenging problem. In this paper, when multiple frames of low-resolution phase gradients are available, we introduce PhaseNet, a deep learning approach based on the Taylor frozen flow hypothesis. Our approach incorporates a data-driven residual regularization term, of which the gradient is parameterized by a network, into the Laplacian regularization based model. To solve the model, we unroll the Nesterov accelerated gradient algorithm so that the network can be efficiently and effectively trained. Finally, we evaluate the performance of PhaseNet under various atmospheric conditions and demonstrate its superiority over TV and Laplacian regularization based methods.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-15T07:00:00Z
DOI: 10.1137/23M1592377
Issue No: Vol. 17, No. 3 (2024)
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- Dynamic Image Reconstruction with Motion Priors in Application to Three
Dimensional Magnetic Particle Imaging-
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Authors: Christina Brandt, Tobias Kluth, Tobias Knopp, Lena Westen
Pages: 1539 - 1586
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1539-1586, September 2024.
Abstract.Various imaging modalities allow for time-dependent image reconstructions from measurements where its acquisition also has a time-dependent nature. Magnetic particle imaging (MPI) falls into this class of imaging modalities and it thus also provides a dynamic inverse problem. Without proper consideration of the dynamic behavior, motion artifacts in the reconstruction become an issue. More sophisticated methods need to be developed and applied to the reconstruction of the time-dependent sequences of images. In this context, we investigate the incorporation of motion priors in terms of certain flow-parameter-dependent PDEs in the reconstruction process of time-dependent three dimensional (3D) images in magnetic particle imaging. The present work comprises the method development for a general 3D+time setting for time-dependent linear forward operators, analytical investigation of necessary properties in the MPI forward operator, modeling aspects in dynamic MPI, and extensive numerical experiments on 3D+time imaging including simulated data as well as measurements from a rotation phantom and in vivo data from a mouse.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-15T07:00:00Z
DOI: 10.1137/23M1580401
Issue No: Vol. 17, No. 3 (2024)
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- Tight-Frame-Like Analysis-Sparse Recovery Using Nontight Sensing Matrices
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Authors: Kartheek Kumar Reddy Nareddy, Abijith Jagannath Kamath, Chandra Sekhar Seelamantula
Pages: 1587 - 1618
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-17T07:00:00Z
DOI: 10.1137/23M1625846
Issue No: Vol. 17, No. 3 (2024)
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- Localization of Point Scatterers via Sparse Optimization on Measures
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Authors: Giovanni S. Alberti, Romain Petit, Matteo Santacesaria
Pages: 1619 - 1649
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-23T07:00:00Z
DOI: 10.1137/24M1636265
Issue No: Vol. 17, No. 3 (2024)
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- Discrete Morphological Neural Networks
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Authors: Diego Marcondes, Junior Barrera
Pages: 1650 - 1689
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-25T07:00:00Z
DOI: 10.1137/23M1598477
Issue No: Vol. 17, No. 3 (2024)
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- Three-Stage Approach for 2D/3D Diffeomorphic Multimodality Image
Registration with Textural Control-
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Authors: Ke Chen, Huan Han
Pages: 1690 - 1728
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-26T07:00:00Z
DOI: 10.1137/23M1583971
Issue No: Vol. 17, No. 3 (2024)
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- Proximal Langevin Sampling with Inexact Proximal Mapping
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Authors: Matthias J. Ehrhardt, Lorenz Kuger, Carola-Bibiane Schönlieb
Pages: 1729 - 1760
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-07-30T07:00:00Z
DOI: 10.1137/23M1593565
Issue No: Vol. 17, No. 3 (2024)
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- Provably Convergent Plug-and-Play Quasi-Newton Methods
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Authors: Hong Ye Tan, Subhadip Mukherjee, Junqi Tang, Carola-Bibiane Schönlieb
Pages: 785 - 819
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 785-819, June 2024.
Abstract.Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM, with applications in inverse problems and imaging. Provable PnP methods are a subclass of PnP methods with convergence guarantees, such as fixed point convergence or convergence to critical points of some energy function. Many existing provable PnP methods impose heavy restrictions on the denoiser or fidelity function, such as nonexpansiveness or strict convexity, respectively. In this work, we propose a novel algorithmic approach incorporating quasi-Newton steps into a provable PnP framework based on proximal denoisers, resulting in greatly accelerated convergence while retaining light assumptions on the denoiser. By characterizing the denoiser as the proximal operator of a weakly convex function, we show that the fixed points of the proposed quasi-Newton PnP algorithm are critical points of a weakly convex function. Numerical experiments on image deblurring and super-resolution demonstrate 2–8x faster convergence as compared to other provable PnP methods with similar reconstruction quality.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-04-02T07:00:00Z
DOI: 10.1137/23M157185X
Issue No: Vol. 17, No. 2 (2024)
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- NF-ULA: Normalizing Flow-Based Unadjusted Langevin Algorithm for Imaging
Inverse Problems-
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Authors: Ziruo Cai, Junqi Tang, Subhadip Mukherjee, Jinglai Li, Carola-Bibiane Schönlieb, Xiaoqun Zhang
Pages: 820 - 860
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 820-860, June 2024.
Abstract.Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach offers the ability to quantify the uncertainty in the solution. In recent years, data-driven techniques for solving inverse problems have also been remarkably successful, due to their superior representation ability. In this work, we incorporate data-based models into a class of Langevin-based sampling algorithms for Bayesian inference in imaging inverse problems. In particular, we introduce NF-ULA (normalizing flow-based unadjusted Langevin algorithm), which involves learning a normalizing flow (NF) as the image prior. We use NF to learn the prior because a tractable closed-form expression for the log prior enables the differentiation of it using autograd libraries. Our algorithm only requires a normalizing flow-based generative network, which can be pretrained independently of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness and nonasymptotic convergence of the resulting NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various image restoration problems such as image deblurring, image inpainting, and limited-angle X-ray computed tomography reconstruction. NF-ULA is found to perform better than competing methods for severely ill-posed inverse problems.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-04-08T07:00:00Z
DOI: 10.1137/23M1581807
Issue No: Vol. 17, No. 2 (2024)
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- Sliding at First-Order: Higher-Order Momentum Distributions for
Discontinuous Image Registration-
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Authors: Lili Bao, Jiahao Lu, Shihui Ying, Stefan Sommer
Pages: 861 - 887
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 861-887, June 2024.
Abstract.In this paper, we propose a new approach to deformable image registration that captures sliding motions. The large deformation diffeomorphic metric mapping (LDDMM) registration method faces challenges in representing sliding motion since it per construction generates smooth warps. To address this issue, we extend LDDMM by incorporating both zeroth- and first-order momenta with a nondifferentiable kernel. This allows us to represent both discontinuous deformation at switching boundaries and diffeomorphic deformation in homogeneous regions. We provide a mathematical analysis of the proposed deformation model from the viewpoint of discontinuous systems. To evaluate our approach, we conduct experiments on both artificial images and the publicly available DIR-Lab 4DCT dataset. Results show the effectiveness of our approach in capturing plausible sliding motion.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-04-08T07:00:00Z
DOI: 10.1137/23M1558665
Issue No: Vol. 17, No. 2 (2024)
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- Exploring Structural Sparsity of Coil Images from 3-Dimensional
Directional Tight Framelets for SENSE Reconstruction-
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Authors: Yanran Li, Raymond H. Chan, Lixin Shen, Xiaosheng Zhuang, Risheng Wu, Yijun Huang, Junwei Liu
Pages: 888 - 916
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 888-916, June 2024.
Abstract. Each coil image in a parallel magnetic resonance imaging (pMRI) system is an imaging slice modulated by the corresponding coil sensitivity. These coil images, structurally similar to each other, are stacked together as 3-dimensional (3D) image data, and their sparsity property can be explored via 3D directional Haar tight framelets. The features of the 3D image data from the 3D framelet systems are utilized to regularize sensitivity encoding (SENSE) pMRI reconstruction. Accordingly, a so-called SENSE3d algorithm is proposed to reconstruct images of high quality from the sampled [math]-space data with a high acceleration rate by decoupling effects of the desired image (slice) and sensitivity maps. Since both the imaging slice and sensitivity maps are unknown, this algorithm repeatedly performs a slice step followed by a sensitivity step by using updated estimations of the desired image and the sensitivity maps. In the slice step, for the given sensitivity maps, the estimation of the desired image is viewed as the solution to a convex optimization problem regularized by the sparsity of its 3D framelet coefficients of coil images. This optimization problem, involving data from the complex field, is solved by a primal-dual three-operator splitting (PD3O) method. In the sensitivity step, the estimation of sensitivity maps is modeled as the solution to a Tikhonov-type optimization problem that favors the smoothness of the sensitivity maps. This corresponding problem is nonconvex and could be solved by a forward-backward splitting method. Experiments on real phantoms and in vivo data show that the proposed SENSE3d algorithm can explore the sparsity property of the imaging slices and efficiently produce reconstructed images of high quality with reduced aliasing artifacts caused by high acceleration rate, additive noise, and the inaccurate estimation of each coil sensitivity. To provide a comprehensive picture of the overall performance of our SENSE3d model, we provide the quantitative index (HaarPSI) and comparisons to some deep learning methods such as VarNet and fastMRI-UNet.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-04-11T07:00:00Z
DOI: 10.1137/23M1571150
Issue No: Vol. 17, No. 2 (2024)
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- Generalized Nonconvex Hyperspectral Anomaly Detection via Background
Representation Learning with Dictionary Constraint-
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Authors: Quan Yu, Minru Bai
Pages: 917 - 950
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 917-950, June 2024.
Abstract. Anomaly detection in the hyperspectral images, which aims to separate interesting sparse anomalies from backgrounds, is a significant topic in remote sensing. In this paper, we propose a generalized nonconvex background representation learning with dictionary constraint (GNBRL) model for hyperspectral anomaly detection. Unlike existing methods that use a specific nonconvex function for a low rank term, GNBRL uses a class of nonconvex functions for both low rank and sparse terms simultaneously, which can better capture the low rank structure of the background and the sparsity of the anomaly. In addition, GNBRL simultaneously learns the dictionary and anomaly tensor in a unified framework by imposing a three-dimensional correlated total variation constraint on the dictionary tensor to enhance the quality of representation. An extrapolated linearized alternating direction method of multipliers (ELADMM) algorithm is then developed to solve the proposed GNBRL model. Finally, a novel coarse to fine two-stage framework is proposed to enhance the GNBRL model by exploiting the nonlocal similarity of the hyperspectral data. Theoretically, we establish an error bound for the GNBRL model and show that this error bound can be superior to those of similar models based on Tucker rank. We prove that the sequence generated by the proposed ELADMM algorithm converges to a Karush–Kuhn–Tucker point of the GNBRL model. This is a challenging task due to the nonconvexity of the objective function. Experiments on hyperspectral image datasets demonstrate that our proposed method outperforms several state-of-the-art methods in terms of detection accuracy.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-04-12T07:00:00Z
DOI: 10.1137/23M157363X
Issue No: Vol. 17, No. 2 (2024)
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- Weighted Spectral Filters for Kernel Interpolation on Spheres: Estimates
of Prediction Accuracy for Noisy Data-
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Authors: Xiaotong Liu, Jinxin Wang, Di Wang, Shao-Bo Lin
Pages: 951 - 983
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 951-983, June 2024.
Abstract.Spherical radial-basis-based kernel interpolation abounds in image sciences, including geophysical image reconstruction, climate trends description, and image rendering, due to its excellent spatial localization property and perfect approximation performance. However, in dealing with noisy data, kernel interpolation frequently behaves not so well due to the large condition number of the kernel matrix and instability of the interpolation process. In this paper, we introduce a weighted spectral filter approach to reduce the condition number of the kernel matrix and then stabilize kernel interpolation. The main building blocks of the proposed method are the well-developed spherical positive quadrature rules and high-pass spectral filters. Using a recently developed integral operator approach for spherical data analysis, we theoretically demonstrate that the proposed weighted spectral filter approach succeeds in breaking through the bottleneck of kernel interpolation, especially in fitting noisy data. We provide optimal approximation rates of the new method to show that our approach does not compromise the predicting accuracy. Furthermore, we conduct both toy simulations and two real-world data experiments with synthetically added noise in geophysical image reconstruction and climate image processing to verify our theoretical assertions and show the feasibility of the weighted spectral filter approach.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-05-20T07:00:00Z
DOI: 10.1137/23M1585350
Issue No: Vol. 17, No. 2 (2024)
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- Imaging with Thermal Noise Induced Currents
Open Access Article-
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Authors: Trent DeGiovanni, Fernando Guevara Vasquez, China Mauck
Pages: 984 - 1006
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 984-1006, June 2024.
Abstract.We use thermal noise induced currents to image the real and imaginary parts of the conductivity of a body. Covariances of the thermal noise currents measured at a few electrodes are shown to be related to a deterministic problem. We use the covariances obtained while selectively heating the body to recover the real power density in the body under known boundary conditions and at a known frequency. The resulting inverse problem is related to acousto-electric tomography, but where the conductivity is complex and only the real power is measured. We study the local solvability of this problem by determining where its linearization is elliptic. Numerical experiments illustrating this inverse problem are included.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-05-21T07:00:00Z
DOI: 10.1137/23M1571630
Issue No: Vol. 17, No. 2 (2024)
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- Assembling a Learnable Mumford–Shah Type Model with Multigrid
Technique for Image Segmentation-
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Authors: Junying Meng, Weihong Guo, Jun Liu, Mingrui Yang
Pages: 1007 - 1039
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 1007-1039, June 2024.
Abstract.The classical Mumford–Shah (MS) model has been successful in some medical image segmentation tasks, providing segmentation results with smooth boundaries of objects. However, the MS model, which operates at the pixel level of the images, faces challenges when dealing with medical images with low contrast or unclear edges. In this paper, we begin by using a feature extractor to capture high-dimensional deep features that contain more comprehensive semantic information than pixel-level data alone. Inspired by the MS model, we develop a variational model that incorporates threshold dynamics (TD) regularization for segmenting each feature. We obtain the final segmentation result for the original image by assembling segmentation results of all the features. This process results in MS-MGNet, a lightweight trainable segmentation network with a similar architecture to many encoder–decoder networks. The intermediate layers of MS-MGNet are designed by unrolling the numerical scheme based on the multigrid method for solving the variational model. We provide interpretability for the encoder–decoder architecture by elucidating the roles of each layer and offering explanations of the underlying mathematical models. By incorporating the TD regularizer, we integrate spatial priors from the variational models into the network architecture, resulting in better segmentation results with smoother edges and a certain robustness to noise. Compared to some relevant methods, experimental results on the selected data sets with low contrast or unclear edges show that the proposed method can achieve better segmentation performance with fewer parameters, even when trained on smaller data sets.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-05-22T07:00:00Z
DOI: 10.1137/23M1577663
Issue No: Vol. 17, No. 2 (2024)
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- Total Generalized Variation on a Tree
Open Access Article-
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Authors: Muhamed Kuric, Jan Ahmetspahic, Thomas Pock
Pages: 1040 - 1077
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 1040-1077, June 2024.
Abstract.We consider a class of optimization problems defined over trees with unary cost terms and shifted pairwise cost terms. These problems arise when considering block coordinate descent (BCD) approaches for solving inverse problems with total generalized variation (TGV) regularizers or their nonconvex generalizations. We introduce a linear-time reduction that transforms the shifted problems into their nonshifted counterparts. However, combining existing continuous dynamic programming (DP) algorithms with the reduction does not lead to BCD iterations that compute TGV-like solutions. This problem can be overcome by considering a box-constrained modification of the subproblems or smoothing the cost terms of the TGV regularized problem. The former leads to shifted and box-constrained subproblems, for which we propose a linear-time reduction to their unconstrained counterpart. The latter naturally leads to problems with smooth unary and pairwise cost terms. With this in mind, we propose two novel continuous DP algorithms that can solve (convex and nonconvex) problems with piecewise quadratic unary and pairwise cost terms. We prove that the algorithm for the convex case has quadratic worst-case time and memory complexity, while the algorithm for the nonconvex case has exponential time and memory complexity, but works well in practice for smooth truncated total variation pairwise costs. Finally, we demonstrate the applicability of the proposed algorithms for solving inverse problems with first-order and higher-order regularizers.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-05-30T07:00:00Z
DOI: 10.1137/23M1556915
Issue No: Vol. 17, No. 2 (2024)
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- Accelerated Bayesian Imaging by Relaxed Proximal-Point Langevin Sampling
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Authors: Teresa Klatzer, Paul Dobson, Yoann Altmann, Marcelo Pereyra, Jesus Maria Sanz-Serna, Konstantinos C. Zygalakis
Pages: 1078 - 1117
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 1078-1117, June 2024.
Abstract.This paper presents a new accelerated proximal Markov chain Monte Carlo methodology to perform Bayesian inference in imaging inverse problems with an underlying convex geometry. The proposed strategy takes the form of a stochastic relaxed proximal-point iteration that admits two complementary interpretations. For models that are smooth or regularized by Moreau–Yosida smoothing, the algorithm is equivalent to an implicit midpoint discretization of an overdamped Langevin diffusion targeting the posterior distribution of interest. This discretization is asymptotically unbiased for Gaussian targets and shown to converge in an accelerated manner for any target that is [math]-strongly log-concave (i.e., requiring in the order of [math] iterations to converge, similar to accelerated optimization schemes), comparing favorably to Pereyra, Vargas Mieles, and Zygalakis [SIAM J. Imaging Sci., 13 (2020), pp. 905–935], which is only provably accelerated for Gaussian targets and has bias. For models that are not smooth, the algorithm is equivalent to a Leimkuhler–Matthews discretization of a Langevin diffusion targeting a Moreau–Yosida approximation of the posterior distribution of interest and hence achieves a significantly lower bias than conventional unadjusted Langevin strategies based on the Euler–Maruyama discretization. For targets that are [math]-strongly log-concave, the provided nonasymptotic convergence analysis also identifies the optimal time step, which maximizes the convergence speed. The proposed methodology is demonstrated through a range of experiments related to image deconvolution with Gaussian and Poisson noise with assumption-driven and data-driven convex priors. Source codes for the numerical experiments of this paper are available from https://github.com/MI2G/accelerated-langevin-imla.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-06-03T07:00:00Z
DOI: 10.1137/23M1594832
Issue No: Vol. 17, No. 2 (2024)
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- Stochastic Variance Reduced Gradient for Affine Rank Minimization Problem
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Authors: Ningning Han, Juan Nie, Jian Lu, Michael K. Ng
Pages: 1118 - 1144
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 1118-1144, June 2024.
Abstract.In this paper, we develop an efficient stochastic variance reduced gradient descent algorithm to solve the affine rank minimization problem consisting of finding a matrix of minimum rank from linear measurements. The proposed algorithm as a stochastic gradient descent strategy enjoys a more favorable complexity than that using full gradients. It also reduces the variance of the stochastic gradient at each iteration and accelerates the rate of convergence. We prove that the proposed algorithm converges linearly in expectation to the solution under a restricted isometry condition. Numerical experimental results demonstrate that the proposed algorithm has a clear advantageous balance of efficiency, adaptivity, and accuracy compared with other state-of-the-art algorithms.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-06-04T07:00:00Z
DOI: 10.1137/23M1555387
Issue No: Vol. 17, No. 2 (2024)
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- Extrapolated Plug-and-Play Three-Operator Splitting Methods for Nonconvex
Optimization with Applications to Image Restoration-
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Authors: Zhongming Wu, Chaoyan Huang, Tieyong Zeng
Pages: 1145 - 1181
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 1145-1181, June 2024.
Abstract.This paper investigates the convergence properties and applications of the three-operator splitting method, also known as the Davis–Yin splitting (DYS) method, integrated with extrapolation and plug-and-play (PnP) denoiser within a nonconvex framework. We first propose an extrapolated DYS method to effectively solve a class of structural nonconvex optimization problems that involve minimizing the sum of three possibly nonconvex functions. Our approach provides an algorithmic framework that encompasses both extrapolated forward–backward splitting and extrapolated Douglas–Rachford splitting methods. To establish the convergence of the proposed method, we rigorously analyze its behavior based on the Kurdyka–Łojasiewicz property, subject to some tight parameter conditions. Moreover, we introduce two extrapolated PnP-DYS methods with convergence guarantee, where the traditional regularization step is replaced by a gradient step–based denoiser. This denoiser is designed using a differentiable neural network and can be reformulated as the proximal operator of a specific nonconvex functional. We conduct extensive experiments on image deblurring and image superresolution problems, where our numerical results showcase the advantage of the extrapolation strategy and the superior performance of the learning-based model that incorporates the PnP denoiser in terms of achieving high-quality recovery images.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-06-13T07:00:00Z
DOI: 10.1137/23M1611166
Issue No: Vol. 17, No. 2 (2024)
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- Stable Local-Smooth Principal Component Pursuit
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Authors: Jiangjun Peng, Hailin Wang, Xiangyong Cao, Xixi Jia, Hongying Zhang, Deyu Meng
Pages: 1182 - 1205
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 1182-1205, June 2024.
Abstract.Recently, the CTV-RPCA model proposed the first recoverable theory for separating low-rank and local-smooth matrices and sparse matrices based on the correlated total variation (CTV) regularizer. However, the CTV-RPCA model ignores the influence of noise, which makes the model unable to effectively extract low-rank and local-smooth principal components under noisy circumstances. To alleviate this issue, this article extends the CTV-RPCA model by considering the influence of noise and proposes two robust models with parameter adaptive adjustment, i.e., Stable Principal Component Pursuit based on CTV (CTV-SPCP) and Square Root Principal Component Pursuit based on CTV (CTV-[math]). Furthermore, we present a statistical recoverable error bound for the proposed models, which allows us to know the relationship between the solution of the proposed models and the ground-truth. It is worth mentioning that, in the absence of noise, our theory degenerates back to the exact recoverable theory of the CTV-RPCA model. Finally, we develop the effective algorithms with the strict convergence guarantees. Extensive experiments adequately validate the theoretical assertions and also demonstrate the superiority of the proposed models over many state-of-the-art methods on various typical applications, including video foreground extraction, multispectral image denoising, and hyperspectral image denoising. The source code is released at https://github.com/andrew-pengjj/CTV-SPCP.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-06-17T07:00:00Z
DOI: 10.1137/23M1580164
Issue No: Vol. 17, No. 2 (2024)
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- Marginal Likelihood Estimation in Semiblind Image Deconvolution: A
Stochastic Approximation Approach-
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Authors: Charlesquin Kemajou Mbakam, Marcelo Pereyra, Jean-François Giovannelli
Pages: 1206 - 1254
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 1206-1254, June 2024.
Abstract.This paper presents a novel stochastic optimization methodology to perform empirical Bayesian inference in semi-blind image deconvolution problems. Given a blurred image and a parametric class of possible operators, the proposed optimization approach automatically calibrates the parameters of the blur model by maximum marginal likelihood estimation, followed by (non-blind) image deconvolution by maximum a posteriori estimation conditionally to the estimated model parameters. In addition to the blur model, the proposed approach also automatically calibrates the noise level as well as any regularization parameters. The marginal likelihood of the blur, noise, and regularization parameters is generally computationally intractable, as it requires calculating several integrals over the entire solution space. Our approach addresses this difficulty by using a stochastic approximation proximal gradient optimization scheme, which iteratively solves such integrals by using a Moreau–Yosida regularized unadjusted Langevin Markov chain Monte Carlo algorithm. This optimization strategy can be easily and efficiently applied to any model that is log-concave and by using the same gradient and proximal operators that are required to compute the maximum a posteriori solution by convex optimization. We provide convergence guarantees for the proposed optimization scheme under realistic and easily verifiable conditions and subsequently demonstrate the effectiveness of the approach with a series of deconvolution experiments and comparisons with alternative strategies from the state of the art
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-06-19T07:00:00Z
DOI: 10.1137/23M1584496
Issue No: Vol. 17, No. 2 (2024)
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- Non-Lipschitz Variational Models and their Iteratively Reweighted Least
Squares Algorithms for Image Denoising on Surfaces-
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Authors: Yuan Liu, Chunlin Wu, Chao Zeng
Pages: 1255 - 1283
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-06-20T07:00:00Z
DOI: 10.1137/23M159439X
Issue No: Vol. 17, No. 2 (2024)
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- Riesz Feature Representation: Scale Equivariant Scattering Network for
Classification Tasks-
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Authors: Tin Barisin, Jesus Angulo, Katja Schladitz, Claudia Redenbach
Pages: 1284 - 1313
Abstract: SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 1284-1313, June 2024.
Abstract. Scattering networks yield powerful and robust hierarchical image descriptors which do not require lengthy training and which work well with very few training data. However, they rely on sampling the scale dimension. Hence, they become sensitive to scale variations and are unable to generalize to unseen scales. In this work, we define an alternative feature representation based on the Riesz transform. We detail and analyze the mathematical foundations behind this representation. In particular, it inherits scale equivariance from the Riesz transform and completely avoids sampling of the scale dimension. Additionally, the number of features in the representation is reduced by a factor four compared to scattering networks. Nevertheless, our representation performs comparably well for texture classification with an interesting addition: scale equivariance. Our method yields very good performance when dealing with scales outside of those covered by the training dataset. The usefulness of the equivariance property is demonstrated on the digit classification task, where accuracy remains stable even for scales four times larger than the one chosen for training. As a second example, we consider classification of textures. Finally, we show how this representation can be used to build hybrid deep learning methods that are more stable to scale variations than standard deep networks.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-06-20T07:00:00Z
DOI: 10.1137/23M1584836
Issue No: Vol. 17, No. 2 (2024)
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- Training Adaptive Reconstruction Networks for Blind Inverse Problems
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Authors: Alban Gossard, Pierre Weiss
Pages: 1314 - 1346
Abstract: 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.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2024-06-21T07:00:00Z
DOI: 10.1137/23M1545628
Issue No: Vol. 17, No. 2 (2024)
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