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Similar Journals
 SIAM Journal on Imaging SciencesJournal Prestige (SJR): 1.371 Citation Impact (citeScore): 3Number of Followers: 7     Hybrid journal   * Containing 1 Open Access article(s) in this issue * ISSN (Print) 1936-4954 Published by Society for Industrial and Applied Mathematics  [17 journals]
• Bar Code Decoding in a Camera-Based Scanner: Analysis and Algorithm

Pages: 1017 - 1040
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1017-1040, September 2022.
The goal of this work is to provide a foundational understanding of the resolution requirement for camera-based bar code scanning systems and to develop an effective algorithm for decoding. The main theoretical question we wish to address is how well resolved must the bar code image be for the system to be able to decode it. To facilitate the analysis we consider the UPC bar code, which is widely used in retail and commerce. We further assume that the pixels of the camera are aligned with the bar code, making the problem one-dimensional. We find that if the narrowest bar in the bar code is larger than two-thirds the size of the camera pixel, it is possible to uniquely determine the encoded message in the bar code. The result, which exploits the symbology of the UPC bar code, shows that under-resolved bar code images can be decoded. To further gain insight into the robustness of the decoding process, we perform numerical experiments.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-07-14T07:00:00Z
DOI: 10.1137/21M1449658
Issue No: Vol. 15, No. 3 (2022)

• Compactification of the Rigid Motions Group in Image Processing

Authors: Tamir Bendory, Ido Hadi, Nir Sharon
Pages: 1041 - 1078
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1041-1078, September 2022.
Image processing problems in general, and in particular in the field of single-particle cryo-electron microscopy, often require considering images up to their rotations and translations. Such problems were tackled successfully when considering images up to rotations only, using quantities which are invariant to the action of rotations on images. Extending these methods to cases where translations are involved is more complicated. Here we present a computationally feasible and theoretically sound approximate invariant to the action of rotations and translations on images. It allows one to approximately reduce image processing problems to similar problems over the sphere, a compact domain acted on by the group of three-dimensional rotations, a compact group. We show that this invariant is induced by a family of mappings deforming, and thereby compactifying, the group structure of rotations and translations of the plane, i.e., the group of rigid motions, into the group of three-dimensional rotations. Furthermore, we demonstrate its viability in two image processing tasks: multireference alignment and classification. To our knowledge, this is the first instance of a quantity that is either exactly or approximately invariant to rotations and translations of images that both rests on a sound theoretical foundation and is applicable in practice.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-07-14T07:00:00Z
DOI: 10.1137/21M1429448
Issue No: Vol. 15, No. 3 (2022)

• A Fast Averaged Kaczmarz Iteration with Convex Penalty for Inverse
Problems in Hilbert Spaces

Authors: Yuxin Xia, Wei Wang, Bo Han
Pages: 1079 - 1103
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1079-1103, September 2022.
In this paper, we focus on developing a fast Kaczmarz-type method to solve inverse problems that can be written as systems of linear or nonlinear equations in Hilbert spaces. In order to capture the special feature of solutions, we incorporate nonsmooth convex functions into the averaged Kaczmarz iteration, leading to a new Kaczmarz-type method. In addition, aimed at further accelerating our proposed method, the choice of the step size is carefully discussed. Under the similar assumptions of the Kaczmarz-type method, we prove that our method is a convergent regularization method as long as it is terminated by an appropriate stopping rule. Finally, detailed numerical studies are presented for the limited data problem in photoacoustic tomography and the parameter identification problems to show the effectiveness of our method.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-07-18T07:00:00Z
DOI: 10.1137/21M1445181
Issue No: Vol. 15, No. 3 (2022)

• Stable Image Reconstruction Using Transformed Total Variation Minimization

Authors: Limei Huo, Wengu Chen, Huanmin Ge, Michael K. Ng
Pages: 1104 - 1139
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1104-1139, September 2022.
Transformed $L_1$ (TL1) regularization has been shown to have comparable signal recovery capability with $L_1-L_2$ regularization and $L_1/L_2$ regularization, regardless of whether the measurement matrix satisfies the restricted isometry property (RIP). In the spirit of the TL1 method, we introduce a transformed total variation (TTV) minimization model to investigate robust image recovery from a certain number of noisy measurements by the proposed TTV minimization model in this paper. An optimal error bound, up to a logarithmic factor, of robust image recovery from compressed measurements via the TTV minimization model is established, and the RIP based condition is improved compared with total variation (TV) minimization. Numerical results of image reconstruction demonstrate our theoretical results and illustrate the efficiency of the TTV minimization model among state-of-the-art methods. Empirically, the error bound between the reconstructed image and the original image is shown to be better than that produced by TV minimization.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-07-19T07:00:00Z
DOI: 10.1137/21M1438566
Issue No: Vol. 15, No. 3 (2022)

• Model-Centric Data Manifold: The Data Through the Eyes of the Model

Authors: Luca Grementieri, Rita Fioresi
Pages: 1140 - 1156
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1140-1156, September 2022.
We show that deep ReLU neural network classifiers can see a low-dimensional Riemannian manifold structure on data. Such structure comes via the \sl local data matrix, a variation of the Fisher information matrix, where the role of the model parameters is taken by the data variables. We obtain a foliation of the data domain, and we show that the dataset on which the model is trained lies on a leaf, the \sl data leaf, whose dimension is bounded by the number of classification labels. We validate our results with some experiments with the MNIST dataset: paths on the data leaf connect valid images, while other leaves cover noisy images.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-07-28T07:00:00Z
DOI: 10.1137/21M1437056
Issue No: Vol. 15, No. 3 (2022)

• A Generalized Primal-Dual Algorithm with Improved Convergence Condition

Authors: Bingsheng He, Feng Ma, Shengjie Xu, Xiaoming Yuan
Pages: 1157 - 1183
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1157-1183, September 2022.
We generalize the well-known primal-dual algorithm proposed by Chambolle and Pock for saddle point problems and relax the condition for ensuring its convergence. The relaxed convergence-guaranteeing condition is effective for the generic convex setting of saddle point problems, and we show by the canonical convex programming problem with linear equality constraints that the relaxed condition is optimal. It also allows us to discern larger step sizes for the resulting subproblems, and thus provides a simple and universal way to improve numerical performance of the original primal-dual algorithm. In addition, we present a structure-exploring heuristic to further relax the convergence-guaranteeing condition for some specific saddle point problems, which could yield much larger step sizes and hence significantly better performance. Effectiveness of this heuristic is numerically illustrated by the classic assignment problem.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-07-28T07:00:00Z
DOI: 10.1137/21M1453463
Issue No: Vol. 15, No. 3 (2022)

• Compressive Learning for Patch-Based Image Denoising

Authors: Hui Shi, Yann Traonmilin, Jean-Francois Aujol
Pages: 1184 - 1212
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1184-1212, September 2022.
The expected patch log-likelihood algorithm (EPLL) and its extensions have shown good performances for image denoising. The prior model used by EPLL is usually a Gaussian mixture model (GMM) estimated from a database of image patches. Classical mixture model estimation methods face computational issues as the high dimensionality of the problem requires training on large datasets. In this work, we adapt a compressive statistical learning framework to carry out the GMM estimation. With this method, called sketching, we estimate models from a compressive representation (the sketch) of the training patches. The cost of estimating the prior from the sketch no longer depends on the number of items in the original large database. To accelerate further the estimation, we add another dimension reduction technique (low-rank modeling of the covariance matrices) to the compressing learning framework. To demonstrate the advantages of our method, we test it on real large-scale data. We show that we can produce denoising performances similar to performances obtained with models estimated from the original training database using GMM priors learned from the sketch with improved execution times.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-08-02T07:00:00Z
DOI: 10.1137/21M1459812
Issue No: Vol. 15, No. 3 (2022)

• Analysis for Full-Field Photoacoustic Tomography with Variable Sound Speed

Authors: Linh Nguyen, Markus Haltmeier, Richard Kowar, Ngoc Do
Pages: 1213 - 1228
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1213-1228, September 2022.
Photoacoustic tomography (PAT) is a noninvasive imaging modality that requires recovering the initial data of the wave equation from certain measurements of the solution outside the object. In the standard PAT measurement setup, the used data consist of time-dependent signals measured on an observation surface. In contrast, the measured data from the recently invented full-field detection technique provide the solution of the wave equation on a spatial domain at a single instant in time. While reconstruction using classical PAT data has been extensively studied, not much is known for the full-field PAT problem. In this paper, we build mathematical foundations of the latter problem for variable sound speed and settle its uniqueness and stability. Moreover, we introduce an exact inversion method using time-reversal and study its convergence. Our results demonstrate the suitability of both the full-field approach and the proposed time-reversal technique for high-resolution photoacoustic imaging.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-08-02T07:00:00Z
DOI: 10.1137/21M1463409
Issue No: Vol. 15, No. 3 (2022)

• High-Resolution, Quantitative Signal Subspace Imaging for Synthetic

Open Access Article

Authors: Arnold D. Kim, Chrysoula Tsogka
Pages: 1229 - 1252
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1229-1252, September 2022.
We consider synthetic aperture radar imaging of a region containing point-like targets. Measurements are the set of frequency responses to scattering by the targets taken over a collection of individual spatial locations along the flight path making up the synthetic aperture. Because signal subspace imaging methods do not work on these measurements directly, we rearrange the frequency response at each spatial location using the Prony method and obtain a matrix that is suitable for these methods. We arrange the set of these Prony matrices as one block-diagonal matrix and introduce a signal subspace imaging method for it. We show that this signal subspace method yields high-resolution and quantitative images provided that the signal-to-noise ratio is sufficiently high. We give a resolution analysis for this imaging method and validate this theory using numerical simulations. Additionally, we show that this imaging method is stable to random perturbations to the travel times and validate this theory with numerical simulations using the random travel time model for random media.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-08-04T07:00:00Z
DOI: 10.1137/21M1467109
Issue No: Vol. 15, No. 3 (2022)

• Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable
Approach for Continuous Markov Random Fields

Authors: Hartmut Bauermeister, Emanuel Laude, Thomas Möllenhoff, Michael Moeller, Daniel Cremers
Pages: 1253 - 1281
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1253-1281, January 2022.
Dual decomposition approaches in nonconvex optimization may suffer from a duality gap. This poses a challenge when applying them directly to nonconvex problems such as MAP-inference in a Markov random field with continuous state spaces. To eliminate such gaps, this paper considers a reformulation of the original nonconvex task in the space of measures. This infinite-dimensional reformulation is then approximated by a semi-infinite one, which is obtained via a piecewise polynomial discretization in the dual. We provide a geometric intuition behind the primal problem induced by the dual discretization and draw connections to optimization over moment spaces. In contrast to existing discretizations which suffer from a grid bias, we show that a piecewise polynomial discretization better preserves the continuous nature of our problem. Invoking results from optimal transport theory and convex algebraic geometry we reduce the semi-infinite program to a finite one and provide a practical implementation based on semidefinite programming. We show, experimentally and in theory, that the approach successfully reduces the duality gap. To showcase the scalability of our approach, we apply it to the stereo matching problem between two images.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-08-11T07:00:00Z
DOI: 10.1137/21M1433241
Issue No: Vol. 15, No. 3 (2022)

• The Whole and the Parts: The Minimum Description Length Principle and the
A-Contrario Framework

Authors: Rafael Grompone von Gioi, Ignacio Ramírez Paulino, Gregory Randall
Pages: 1282 - 1313
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1282-1313, January 2022.
This work explores the connections between the minimum description length (MDL) principle as developed by Rissanen, and the a-contrario framework for structure detection proposed by Desolneux, Moisan, and Morel. The MDL principle focuses on the best interpretation for the whole data while the a-contrario approach concentrates on detecting parts of the data with anomalous statistics. Although framed in different theoretical formalisms, we show that both methodologies share many common concepts and tools in their machinery and yield very similar formulations in a number of interesting scenarios ranging from simple toy examples to practical applications such as polygonal approximation of curves and line segment detection in images. We also formulate the conditions under which both approaches are formally equivalent.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-08-11T07:00:00Z
DOI: 10.1137/21M145745X
Issue No: Vol. 15, No. 3 (2022)

• Image Segmentation with Adaptive Spatial Priors from Joint Registration

Authors: Haifeng Li, Weihong Guo, Jun Liu, Li Cui, Dongxing Xie
Pages: 1314 - 1344
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 3, Page 1314-1344, January 2022.
Image segmentation is a crucial but challenging task that has many applications. In medical imaging, for instance, intensity inhomogeneity and noise are common. In thigh muscle images, different muscles are closely packed together and there are often no clear boundaries between them. Intensity based segmentation models cannot separate one muscle from another. To solve such problems, in this work we present a segmentation model with adaptive spatial priors from joint registration. This model combines segmentation and registration in a unified framework to leverage their positive mutual influence. The segmentation is based on a modified Gaussian mixture model, which integrates intensity inhomogeneity and spatial smoothness. The registration plays the role of providing a shape prior. We adopt a modified sum of squared difference fidelity term and Tikhonov regularity term for registration and also utilize a Gaussian pyramid and parametric method for robustness. The connection between segmentation and registration is guaranteed by the cross entropy metric that aims to make the segmentation map (from segmentation) and deformed atlas (from registration) as similar as possible. This joint framework is implemented within a constraint optimization framework, which leads to an efficient algorithm. We evaluate our proposed model on synthetic and thigh muscle MR images. Numerical results show the improvement as compared to segmentation and registration performed separately and other joint models.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-08-11T07:00:00Z
DOI: 10.1137/21M1444874
Issue No: Vol. 15, No. 3 (2022)

• Compressive Imaging Through Optical Fiber with Partial Speckle Scanning

Authors: Stéphanie Guérit, Siddharth Sivankutty, John Lee, Hervé Rigneault, Laurent Jacques
Pages: 387 - 423
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 387-423, June 2022.
Fluorescence imaging through ultrathin fibers is a promising approach to obtain high-resolution imaging with molecular specificity at depths much larger than the scattering mean-free paths of biological tissues. Such imaging techniques, generally termed lensless endoscopy, rely upon the wavefront control at the distal end of a fiber to coherently combine multiple spatial modes of a multicore (MCF) or multimode fiber (MMF). Typically, a spatial light modulator (SLM) is employed to combine hundreds of modes by phase-matching to generate a high-intensity focal spot. This spot is subsequently scanned across the sample to obtain an image. We propose here a novel scanning scheme, partial speckle scanning (PSS), inspired by compressive sensing theory, that avoids the use of an SLM to perform fluorescent imaging with optical fibers with reduced acquisition time. Such a strategy avoids photo-bleaching while keeping high reconstruction quality. We develop our approach on two key properties of the MCF: (i) the ability to easily generate speckles, and (ii) the memory effect that allows one to use fast scan mirrors to shift light patterns. First, we show that speckles are subexponential random fields. Despite their granular structure, an appropriate choice of the reconstruction parameters makes them good candidates to build efficient sensing matrices. Then, we numerically validate our approach and apply it on experimental data. The proposed sensing technique outperforms conventional raster scanning: higher reconstruction quality is achieved with far fewer observations. For a fixed reconstruction quality, our speckle scanning approach is faster than compressive sensing schemes which require changing the speckle pattern for each observation.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-04-04T07:00:00Z
DOI: 10.1137/21M1407586
Issue No: Vol. 15, No. 2 (2022)

• Efficient Boosted DC Algorithm for Nonconvex Image Restoration with Rician
Noise

Authors: Tingting Wu, Xiaoyu Gu, Zeyu Li, Zhi Li, Jianwei Niu, Tieyong Zeng
Pages: 424 - 454
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 424-454, June 2022.
Image deblurring under Rician noise has attracted considerable attention in imaging science. Frequently appearing in medical imaging, Rician noise leads to an interesting nonconvex optimization problem, termed as the MAP-Rician model, which is based on the Maximum a Posteriori (MAP) estimation approach. As the MAP-Rician model is deeply rooted in Bayesian analysis, we want to understand its mathematical analysis carefully. Moreover, one needs to properly select a suitable algorithm for tackling this nonconvex problem to get the best performance. This paper investigates both issues. Indeed, we first present a theoretical result about the existence of a minimizer for the MAP-Rician model under mild conditions. Next, we aim to adopt an efficient boosted difference of convex functions algorithm (BDCA) to handle this challenging problem. Basically, BDCA combines the classical difference of convex functions algorithm (DCA) with a backtracking line search, which utilizes the point generated by DCA to define a search direction. In particular, we apply a smoothing scheme to handle the nonsmooth total variation (TV) regularization term in the discrete MAP-Rician model. Theoretically, using the Kurdyka--Lojasiewicz (KL) property, the convergence of the numerical algorithm can be guaranteed. We also prove that the sequence generated by the proposed algorithm converges to a stationary point with the objective function values decreasing monotonically. Numerical simulations are then reported to clearly illustrate that our BDCA approach outperforms some state-of-the-art methods for both medical and natural images in terms of image recovery capability and CPU-time cost.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-04-04T07:00:00Z
DOI: 10.1137/21M1421660
Issue No: Vol. 15, No. 2 (2022)

• The Modulo Radon Transform: Theory, Algorithms, and Applications

Authors: Matthias Beckmann, Ayush Bhandari, Felix Krahmer
Pages: 455 - 490
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 455-490, June 2022.
Recently, experiments have been reported where researchers were able to perform high dynamic range (HDR) tomography in a heuristic fashion, by fusing multiple tomographic projections. This approach to HDR tomography has been inspired by HDR photography and inherits the same disadvantages. Taking a computational imaging approach to the HDR tomography problem, we here suggest a new model based on the modulo Radon transform (MRT), which we rigorously introduce and analyze. By harnessing a joint design between hardware and algorithms, we present a single-shot HDR tomography approach, which to our knowledge, is the only approach that is backed by mathematical guarantees. On the hardware front, instead of recording the Radon transform projections that may potentially saturate, we propose to measure modulo values of the same. This ensures that the HDR measurements are folded into a lower dynamic range. On the algorithmic front, our recovery algorithms reconstruct the HDR images from folded measurements. Beyond mathematical aspects such as injectivity and inversion of the MRT for different scenarios including band-limited and approximately compactly supported images, we also provide a first proof-of-concept demonstration. To do so, we implement MRT by experimentally folding tomographic measurements available as an open source dataset using our custom designed modulo hardware. Our reconstruction clearly shows the advantages of our approach for experimental data. In this way, our MRT based solution paves a path for HDR acquisition in a number of related imaging problems.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-04-14T07:00:00Z
DOI: 10.1137/21M1424615
Issue No: Vol. 15, No. 2 (2022)

• A Spectral Estimation Framework for Phase Retrieval via Bregman Divergence
Minimization

Authors: Bariscan Yonel, Birsen Yazici
Pages: 491 - 520
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 491-520, June 2022.
In this paper, we develop a novel framework to guide the design of initial estimates for phase retrieval given measurements realized from an arbitrary forward model. Particularly, we propose a general formalism for spectral initialization as an approximate Bregman loss minimization procedure in the range of the lifted forward model, such that a search over rank-1, positive semidefinite matrices is tractable by the synthesis of a quadratic form using elementwise processing of the intensity-only measurements. Via the Bregman loss approach we transcend the Euclidean sense alignment based similarity measure between phaseless measurements that is inherent in the state-of-the art techniques in the literature, in favor of information theory inspired divergence metrics over the positive reals. We derive spectral methods that perform approximate minimization of Kullback--Leibler and Itakura--Saito divergences over phaseless measurements by using elementwise sample processing functions which are designed under a minimal distortion principle. Our formulation relates and extends existing results on model dependent design of optimal sample processing functions in the literature to a model independent sense of metric-based optimality. Numerical simulations confirm the effectiveness of our approach in imaging problems using synthetic and real data sets.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-04-14T07:00:00Z
DOI: 10.1137/20M1388061
Issue No: Vol. 15, No. 2 (2022)

• Variational Rician Noise Removal via Splitting on Spheres

Authors: Zhifang Liu, Huibin Chang, Yuping Duan
Pages: 521 - 549
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 521-549, June 2022.
We propose a novel variational method for Rician noise removal in magnitude-based magnetic resonance (MR) imaging. We first explore the link between the Gaussian noise removal for complex images and the Rician noise removal for magnitude images. Then we establish the constraint optimization model via signal-noise splitting, consisting of a total variation regularizer, two quadratic terms, and a constraint on the field of spheres. Specifically, this constraint represents the forward model of calculating the magnitude of complex images corrupted by Gaussian noises. Namely, the proposed model is completely different from the existing maximum a posteriori based methods, which inevitably involved the sophisticated Bessel function causing high computation costs. It is further efficiently solved by the alternating direction method of multipliers with convergence guarantee. Numerical comparisons with existing variational methods show that the proposed method produces comparable results in terms of image quality, but saves about 50% of overall computational cost on average.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-05-16T07:00:00Z
DOI: 10.1137/21M1452792
Issue No: Vol. 15, No. 2 (2022)

• Distributed Stochastic Inertial-Accelerated Methods with Delayed
Derivatives for Nonconvex Problems

Authors: Yangyang Xu, Yibo Xu, Yonggui Yan, Jie Chen
Pages: 550 - 590
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 550-590, June 2022.
Stochastic gradient methods (SGMs) are predominant approaches for solving stochastic optimization. On smooth nonconvex problems, a few acceleration techniques have been applied to improve the convergence rate of SGMs. However, little exploration has been made on applying a certain acceleration technique to a stochastic subgradient method (SsGM) for nonsmooth nonconvex problems. In addition, few efforts have been made to analyze an (accelerated) SsGM with delayed derivatives. The information delay naturally happens in a distributed system, where computing workers do not coordinate with each other. In this paper, we propose an inertial proximal SsGM for solving nonsmooth nonconvex stochastic optimization problems. The proposed method can have guaranteed convergence even with delayed derivative information in a distributed environment. Convergence rate results are established for three classes of nonconvex problems: weakly convex nonsmooth problems with a convex regularizer, composite nonconvex problems with a nonsmooth convex regularizer, and smooth nonconvex problems. For each problem class, the convergence rate is $O(1/K^{\frac{1}{2}})$ in the expected value of the gradient norm square, for $K$ iterations. In a distributed environment, the convergence rate of the proposed method will be slowed down by the information delay. Nevertheless, the slow-down effect will decay with the number of iterations for the latter two problem classes. We test the proposed method on three applications. The numerical results clearly demonstrate the advantages of using the inertial-based acceleration. Furthermore, we observe higher parallelization speed-up in asynchronous updates over the synchronous counterpart, though the former uses delayed derivatives. Our source code is available at https://github.com/RPI-OPT/Inertial-SsGM.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-05-19T07:00:00Z
DOI: 10.1137/21M1435719
Issue No: Vol. 15, No. 2 (2022)

• SISAL Revisited

Authors: Chujun Huang, Mingjie Shao, Wing-Kin Ma, Anthony Man-Cho So
Pages: 591 - 624
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 591-624, June 2022.
Simplex identification via split augmented Lagrangian (SISAL) is a popularly used algorithm in blind unmixing of hyperspectral images. Developed by José M. Bioucas-Dias in 2009, the algorithm is fundamentally relevant to tackling simplex-structured matrix factorization and, by extension, nonnegative matrix factorization, which have many applications under their umbrellas. In this article, we revisit SISAL and provide new meanings to this quintessential algorithm. The formulation of SISAL was motivated from a geometric perspective, with no noise. We show that SISAL can be explained as an approximation scheme from a probabilistic simplex component analysis framework, which is statistical and is principally more powerful in accommodating the presence of noise. The algorithm for SISAL was designed based on a successive convex approximation method, with a focus on practical utility. It was not known, by analyses, whether the SISAL algorithm has any kind of guarantee of convergence to a stationary point. By establishing associations between the SISAL algorithm and a line search--based proximal gradient method, we confirm that SISAL can indeed guarantee convergence to a stationary point. Our re-explanation of SISAL also reveals new formulations and algorithms. The performance of these new possibilities is demonstrated by numerical experiments.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-05-19T07:00:00Z
DOI: 10.1137/21M1430613
Issue No: Vol. 15, No. 2 (2022)

• Robust Tensor Completion: Equivalent Surrogates, Error Bounds, and
Algorithms

Authors: Xueying Zhao, Minru Bai, Defeng Sun, Libin Zheng
Pages: 625 - 669
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 625-669, June 2022.
Robust low-rank tensor completion (RTC) problems have received considerable attention in recent years such as in signal processing and computer vision. In this paper, we focus on the bound constrained RTC problem for third-order tensors which recovers a low-rank tensor from partial observations corrupted by impulse noise. A widely used convex relaxation of this problem is to minimize the tensor nuclear norm for low rank and the $\ell_1$-norm for sparsity. However, it may result in biased solutions. To handle this issue, we propose a nonconvex model with a novel nonconvex tensor rank surrogate function and a novel nonconvex sparsity measure for RTC problems under limited sample constraints and two bound constraints, where these two nonconvex terms have a difference of convex functions structure. Then, a proximal majorization-minimization (PMM) algorithm is developed to solve the proposed model and this algorithm consists of solving a series of convex subproblems with an initial estimator to generate a new estimator which is used for the next subproblem. Theoretically, for this new estimator, we establish a recovery error bound for its recoverability and give the theoretical guarantee that lower error bounds can be obtained when a reasonable initial estimator is available. Then, by using the Kurdyka--Ł ojasiewicz property exhibited in the resulting problem, we show that the sequence generated by the PMM algorithm globally converges to a critical point of the problem. Extensive numerical experiments including color images and multispectral images show the high efficiency of the proposed model.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-05-24T07:00:00Z
DOI: 10.1137/21M1429539
Issue No: Vol. 15, No. 2 (2022)

• Steerable Near-Quadrature Filter Pairs in Three Dimensions

Authors: Tommy M. Tang, Hemant D. Tagare
Pages: 670 - 700
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 670-700, June 2022.
Steerable filter pairs that are near quadrature have many image processing applications. This paper proposes a new methodology for designing such filters. The key idea is to design steerable filters by minimizing a departure-from-quadrature function. These minimizing filter pairs are almost exactly in quadrature. The polar part of the filters is nonnegative, monotonic, and highly focused around an axis, and asymptotically the filters achieve exact quadrature. These results are established by exploiting a relation between the filters and generalized Hilbert matrices. These near-quadrature filters closely approximate three dimensional Gabor filters. We experimentally verify the asymptotic mathematical results and further demonstrate the use of these filter pairs by efficient calculation of local Fourier shell correlation of cryogenic electron microscopy.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-05-26T07:00:00Z
DOI: 10.1137/21M143529X
Issue No: Vol. 15, No. 2 (2022)

• Bayesian Imaging Using Plug & Play Priors: When Langevin Meets Tweedie

Authors: Rémi Laumont, Valentin De Bortoli, Andrés Almansa, Julie Delon, Alain Durmus, Marcelo Pereyra
Pages: 701 - 737
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 701-737, June 2022.
Since the seminal work of Venkatakrishnan, Bouman, and Wohlberg [Proceedings of the Global Conference on Signal and Information Processing, IEEE, 2013, pp. 945--948] in 2013, Plug & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive estimators for inverse problems in imaging by combining an explicit likelihood function with a prior that is implicitly defined by an image denoising algorithm. In the case of optimization schemes, some recent works guarantee the convergence to a fixed point, albeit not necessarily a maximum a posteriori Bayesian estimate. In the case of Monte Carlo sampling schemes for general Bayesian computation, to the best of our knowledge there is no known proof of convergence. Algorithm convergence issues aside, there are important open questions regarding whether the underlying Bayesian models and estimators are well defined, are well posed, and have the basic regularity properties required to support efficient Bayesian computation schemes. This paper develops theory for Bayesian analysis and computation with PnP priors. We introduce PnP-ULA (Plug & Play unadjusted Langevin algorithm) for Monte Carlo sampling and minimum mean square error estimation. Using recent results on the quantitative convergence of Markov chains, we establish detailed convergence guarantees for this algorithm under realistic assumptions on the denoising operators used, with special attention to denoisers based on deep neural networks. We also show that these algorithms approximately target a decision-theoretically optimal Bayesian model that is well posed and meaningful from a frequentist viewpoint. PnP-ULA is demonstrated on several canonical problems such as image deblurring and inpainting, where it is used for point estimation as well as for uncertainty visualization and quantification.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-05-31T07:00:00Z
DOI: 10.1137/21M1406349
Issue No: Vol. 15, No. 2 (2022)

• Scheduled Restart Momentum for Accelerated Stochastic Gradient Descent

Authors: Bao Wang, Tan Nguyen, Tao Sun, Andrea L. Bertozzi, Richard G. Baraniuk, Stanley J. Osher
Pages: 738 - 761
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 738-761, June 2022.
Stochastic gradient descent (SGD) algorithms, with constant momentum and its variants such as Adam, are the optimization methods of choice for training deep neural networks (DNNs). There is great interest in speeding up the convergence of these methods due to their high computational expense. Nesterov accelerated gradient with a time-varying momentum (NAG) improves the convergence rate of gradient descent for convex optimization using a specially designed momentum; however, it accumulates error when the stochastic gradient is used, slowing convergence at best and diverging at worst. In this paper, we propose scheduled restart SGD (SRSGD), a new NAG-style scheme for training DNNs. SRSGD replaces the constant momentum in SGD by the increasing momentum in NAG but stabilizes the iterations by resetting the momentum to zero according to a schedule. Using a variety of models and benchmarks for image classification, we demonstrate that, in training DNNs, SRSGD significantly improves convergence and generalization; for instance, in training ResNet-200 for ImageNet classification, SRSGD achieves an error rate of 20.93% versus the benchmark of 22.13%. These improvements become more significant as the network grows deeper. Furthermore, on both CIFAR and ImageNet, SRSGD reaches similar or even better error rates with significantly fewer training epochs compared to the SGD baseline. Our implementation of SRSGD is available at https://github.com/minhtannguyen/SRSGD.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-05-31T07:00:00Z
DOI: 10.1137/21M1453311
Issue No: Vol. 15, No. 2 (2022)

• A PDE-Based Method for Shape Registration

Authors: Esten Nicolai Wøien, Markus Grasmair
Pages: 762 - 796
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 762-796, June 2022.
In the square root velocity framework and similar approaches, the computation of shape space distances and the registration of curves requires the solution of a nonconvex variational problem. In this paper, we present a new PDE-based method for solving this problem numerically. The method is constructed from numerical approximation of the Hamilton--Jacobi--Bellman equation for the variational problem and has quadratic complexity and global convergence for the distance estimate. In conjunction, we propose a backtracking scheme for approximating solutions of the registration problem, which additionally can be used to compute shape space geodesics. The methods have linear numerical convergence and improved efficiency compared previous global solvers.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-06-07T07:00:00Z
DOI: 10.1137/21M1408932
Issue No: Vol. 15, No. 2 (2022)

• An Optimal Bayesian Estimator for Absorption Coefficient in Diffuse
Optical Tomography

Authors: Anuj Abhishek, Thilo Strauss, Taufiquar Khan
Pages: 797 - 821
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 797-821, June 2022.
Diffuse Optical Tomography (DOT) is a well-known imaging technique for detecting the optical properties of an object in order to detect anomalies, such as diffusive or absorptive targets. More specifically, DOT has many applications in medical imaging including breast cancer screening. It is an affordable and noninvasive method to recover the optical properties of a body using photon density measurements from applying laser sources placed at its surface. Mathematically, the reconstruction of the internal absorption or scattering is a severely ill-posed inverse problem and yields a poor quality image reconstruction. Studying coefficient inverse problems in a stochastic setting has increasingly gained in prominence in the past couple of decades. In this work, we will show convergence and optimality for a Bayesian estimator for the absorption coefficient built from the noisy data obtained in a simplified DOT Model. We establish the rate of convergence of such an estimator in the supremum norm loss and show that it is optimal. We also present numerical experiments in support of our theoretical findings.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-06-16T07:00:00Z
DOI: 10.1137/21M1462842
Issue No: Vol. 15, No. 2 (2022)

• Solving Inverse Problems by Joint Posterior Maximization with Autoencoding
Prior

Authors: Mario González, Andrés Almansa, Pauline Tan
Pages: 822 - 859
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 822-859, June 2022.
In this work we address the problem of solving ill-posed inverse problems in imaging where the prior is a variational autoencoder (VAE). Specifically we consider the decoupled case where the prior is trained once and can be reused for many different log-concave degradation models without retraining. Whereas previous MAP-based approaches to this problem lead to highly nonconvex optimization algorithms, our approach computes the joint (space-latent) MAP that naturally leads to alternate optimization algorithms and to the use of a stochastic encoder to accelerate computations. The resulting technique (JPMAP) performs joint posterior maximization using an autoencoding prior. We show theoretical and experimental evidence that the proposed objective function is quite close to biconvex. Indeed it satisfies a weak biconvexity property which is sufficient to guarantee that our optimization scheme converges to a stationary point. We also highlight the importance of correctly training the VAE using a denoising criterion, in order to ensure that the encoder generalizes well to out-of-distribution images, without affecting the quality of the generative model. This simple modification is key to providing robustness to the whole procedure. Finally we show how our joint MAP methodology relates to more common MAP approaches, and we propose a continuation scheme that makes use of our JPMAP algorithm to provide more robust MAP estimates. Experimental results also show the higher quality of the solutions obtained by our JPMAP approach with respect to other nonconvex MAP approaches which more often get stuck in spurious local optima.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-06-23T07:00:00Z
DOI: 10.1137/21M140225X
Issue No: Vol. 15, No. 2 (2022)

• Imaging Anisotropic Conductivities from Current Densities

Authors: Huan Liu, Bangti Jin, Xiliang Lu
Pages: 860 - 891
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 860-891, June 2022.
In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(\Omega)^{d,d}$ penalty, which is then discretized by the standard Galerkin finite element method. We establish the continuity and differentiability of the forward map with respect to the conductivity tensor in the $L^p(\Omega)^{d,d}$-norms, the existence of minimizers and optimality systems of the regularized formulation using the concept of H-convergence. Further, we provide a detailed analysis of the discretized problem, especially the convergence of the discrete approximations with respect to the mesh size, using the discrete counterpart of H-convergence. In addition, we develop a projected Newton algorithm for solving the first-order optimality system. We present extensive two-dimensional numerical examples to show the efficiency of the proposed method.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-06-23T07:00:00Z
DOI: 10.1137/21M1437810
Issue No: Vol. 15, No. 2 (2022)

• Bayesian Imaging with Data-Driven Priors Encoded by Neural Networks

Authors: Matthew Holden, Marcelo Pereyra, Konstantinos C. Zygalakis
Pages: 892 - 924
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 892-924, June 2022.
This paper proposes a new methodology for performing Bayesian inference in imaging inverse problems where the prior knowledge is available in the form of training data. Following the manifold hypothesis, we adopt a data-driven prior that is supported on a submanifold of the ambient space, which we can learn from the training data using a generative model, such as a variational autoencoder or generative adversarial network. We establish the existence and well-posedness of the associated posterior distribution and posterior moments under easily verifiable conditions, providing a rigorous underpinning for Bayesian estimators and uncertainty quantification analyses. Bayesian computation is performed using a parallel tempered version of the pCN algorithm on the manifold, which is shown to be ergodic and robust to the nonconvex nature of these data-driven models. In addition to point estimators and uncertainty quantification analyses, we derive a model misspecification test to automatically detect situations where the data-driven prior is unreliable, and we explain how to identify the dimension of the latent space directly from the training data. The proposed approach is illustrated with a range of experiments with the MNIST dataset and is compared with some variational and message passing image reconstruction approaches from the state of the art that also use data-driven regularization. A model accuracy analysis suggests that the Bayesian probabilities reported by the proposed data-driven models are also accurate under a frequentist definition of probability, suggesting that the learnt prior is close to the true marginal distribution of the unknown image.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-06-27T07:00:00Z
DOI: 10.1137/21M1406313
Issue No: Vol. 15, No. 2 (2022)

• A Splitting Scheme for Flip-Free Distortion Energies

Authors: Oded Stein, Jiajin Li, Justin Solomon
Pages: 925 - 959
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 925-959, June 2022.
We introduce a robust optimization method for flip-free distortion energies used, for example, in parametrization, deformation, and volume correspondence. This method can minimize a variety of distortion energies, such as the symmetric Dirichlet energy and our new symmetric gradient energy. We identify and exploit the special structure of distortion energies to employ an operator splitting technique, leading us to propose a novel alternating direction method of multipliers (ADMM) algorithm to deal with the nonconvex, nonsmooth nature of distortion energies. The scheme results in an efficient method where the global step involves a single matrix multiplication and the local steps are closed-form per-triangle/per-tetrahedron expressions that are highly parallelizable. The resulting general-purpose optimization algorithm exhibits robustness to flipped triangles and tetrahedra in initial data as well as during the optimization. We establish the convergence of our proposed algorithm under certain conditions and demonstrate applications to parametrization, deformation, and volume correspondence.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-06-28T07:00:00Z
DOI: 10.1137/21M1433058
Issue No: Vol. 15, No. 2 (2022)

• A Unifying Framework for $n$-Dimensional Quasi-Conformal Mappings

Authors: Daoping Zhang, Gary P. T. Choi, Jianping Zhang, Lok Ming Lui
Pages: 960 - 988
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 960-988, June 2022.
With the advancement of computer technology, there is a surge of interest in effective mapping methods for objects in higher-dimensional spaces. To establish a one-to-one correspondence between objects, higher-dimensional quasi-conformal theory can be utilized for ensuring the bijectivity of the mappings. In addition, it is often desirable for the mappings to satisfy certain prescribed geometric constraints and possess low distortion in conformality or volume. In this work, we develop a unifying framework for computing $n$-dimensional quasi-conformal mappings. More specifically, we propose a variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch, and volume prior information to handle a large variety of deformation problems. We further prove the existence of a minimizer for the proposed model and devise efficient numerical methods to solve the optimization problem. We demonstrate the effectiveness of the proposed framework using various experiments in two and three dimensions, with applications to medical image registration, adaptive remeshing, and shape modeling.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-06-28T07:00:00Z
DOI: 10.1137/21M1457497
Issue No: Vol. 15, No. 2 (2022)

• Linear Convergence of Randomized Kaczmarz Method for Solving
Complex-Valued Phaseless Equations

Authors: Meng Huang, Yang Wang
Pages: 989 - 1016
Abstract: SIAM Journal on Imaging Sciences, Volume 15, Issue 2, Page 989-1016, June 2022.
A randomized Kaczmarz method was recently proposed for phase retrieval, which has been shown numerically to exhibit empirical performance over other state-of-the-art phase retrieval algorithms both in terms of the sampling complexity and computation time. While the rate of convergence has been well studied in the real case where the signals and measurement vectors are all real-valued, there is no guarantee for the convergence in the complex case. In fact, the linear convergence of the randomized Kaczmarz method for phase retrieval in the complex setting is left as a conjecture by Tan and Vershynin [Inf. Inference, 8 (2019), pp. 97--123]. In this paper, we provide the first theoretical guarantees for it. We show that for random measurements ${a}_j \in \mathbb{C}^n,\, j=1,\ldots,m,$ which are drawn independently and uniformly from the complex unit sphere, or equivalently are independent complex Gaussian random vectors, when $m\ge Cn$ for some universal positive constant $C$, the randomized Kaczmarz scheme with a good initialization converges linearly to the target solution (up to a global phase) in expectation with high probability. This gives a positive answer to that conjecture.
Citation: SIAM Journal on Imaging Sciences
PubDate: 2022-06-30T07:00:00Z
DOI: 10.1137/21M1450537
Issue No: Vol. 15, No. 2 (2022)

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