SIAM Journal on Imaging Sciences最新文献

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Image Segmentation Using Bayesian Inference for Convex Variant Mumford–Shah Variational Model 利用贝叶斯推理进行凸变孟福德-沙赫变分模型的图像分割

IF 2.1 3区数学 Q1 Mathematics

SIAM Journal on Imaging Sciences

Pub Date : 2024-01-30 DOI: 10.1137/23m1545379

Xu Xiao, Youwei Wen, Raymond Chan, Tieyong Zeng

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 248-272, March 2024.
Abstract. The Mumford–Shah model is a classical segmentation model, but its objective function is nonconvex. The smoothing and thresholding (SaT) approach is a convex variant of the Mumford–Shah model, which seeks a smoothed approximation solution to the Mumford–Shah model. The SaT approach separates the segmentation into two stages: first, a convex energy function is minimized to obtain a smoothed image; then, a thresholding technique is applied to segment the smoothed image. The energy function consists of three weighted terms and the weights are called the regularization parameters. Selecting appropriate regularization parameters is crucial to achieving effective segmentation results. Traditionally, the regularization parameters are chosen by trial-and-error, which is a very time-consuming procedure and is not practical in real applications. In this paper, we apply a Bayesian inference approach to infer the regularization parameters and estimate the smoothed image. We analyze the convex variant Mumford–Shah variational model from a statistical perspective and then construct a hierarchical Bayesian model. A mean field variational family is used to approximate the posterior distribution. The variational density of the smoothed image is assumed to have a Gaussian density, and the hyperparameters are assumed to have Gamma variational densities. All the parameters in the Gaussian density and Gamma densities are iteratively updated. Experimental results show that the proposed approach is capable of generating high-quality segmentation results. Although the proposed approach contains an inference step to estimate the regularization parameters, it requires less CPU running time to obtain the smoothed image than previous methods.

SIAM 影像科学杂志》，第 17 卷第 1 期，第 248-272 页，2024 年 3 月。摘要Mumford-Shah 模型是一种经典的分割模型，但其目标函数是非凸的。平滑和阈值（SaT）方法是 Mumford-Shah 模型的凸变体，它寻求 Mumford-Shah 模型的平滑近似解。SaT 方法将分割分为两个阶段：首先，最小化凸能函数以获得平滑图像；然后，应用阈值技术分割平滑图像。能量函数由三个加权项组成，加权项称为正则化参数。选择合适的正则化参数对于获得有效的分割结果至关重要。传统上，正则化参数是通过试错来选择的，这是一个非常耗时的过程，在实际应用中并不实用。在本文中，我们采用贝叶斯推理方法来推断正则化参数并估计平滑图像。我们从统计学角度分析了凸变体 Mumford-Shah 变分模型，然后构建了一个分层贝叶斯模型。我们使用均值场变异族来近似后验分布。平滑图像的变分密度假定为高斯密度，超参数假定为伽马变分密度。高斯密度和伽玛密度中的所有参数都会进行迭代更新。实验结果表明，所提出的方法能够生成高质量的分割结果。虽然建议的方法包含一个推理步骤来估计正则化参数，但与以前的方法相比，它获得平滑图像所需的 CPU 运行时间更短。

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引用次数: 0

Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruptions 稳健的张量 CUR 分解：利用稀疏破坏快速恢复低塔克等级张量

IF 2.1 3区数学 Q1 Mathematics

SIAM Journal on Imaging Sciences

Pub Date : 2024-01-25 DOI: 10.1137/23m1574282

HanQin Cai, Zehan Chao, Longxiu Huang, Deanna Needell

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 225-247, March 2024.
Abstract. We study the tensor robust principal component analysis (TRPCA) problem, a tensorial extension of matrix robust principal component analysis, which aims to split the given tensor into an underlying low-rank component and a sparse outlier component. This work proposes a fast algorithm, called robust tensor CUR decompositions (RTCUR), for large-scale nonconvex TRPCA problems under the Tucker rank setting. RTCUR is developed within a framework of alternating projections that projects between the set of low-rank tensors and the set of sparse tensors. We utilize the recently developed tensor CUR decomposition to substantially reduce the computational complexity in each projection. In addition, we develop four variants of RTCUR for different application settings. We demonstrate the effectiveness and computational advantages of RTCUR against state-of-the-art methods on both synthetic and real-world datasets.

SIAM 影像科学杂志》第 17 卷第 1 期第 225-247 页，2024 年 3 月。摘要。我们研究了张量鲁棒主成分分析（TRPCA）问题，它是矩阵鲁棒主成分分析的一个张量扩展，旨在将给定的张量分成底层低秩成分和稀疏离群成分。本研究针对塔克秩设置下的大规模非凸 TRPCA 问题，提出了一种名为鲁棒张量 CUR 分解（RTCUR）的快速算法。RTCUR 是在交替投影框架内开发的，交替投影在低秩张量集合和稀疏张量集合之间进行投影。我们利用最近开发的张量 CUR 分解技术，大大降低了每次投影的计算复杂度。此外，我们还针对不同的应用设置开发了四种 RTCUR 变体。我们在合成数据集和现实世界数据集上展示了 RTCUR 与最先进方法相比的有效性和计算优势。

引用次数: 0

Direct Imaging Methods for Reconstructing a Locally Rough Interface from Phaseless Total-Field Data or Phased Far-Field Data 根据无相全场数据或相位远场数据重建局部粗糙界面的直接成像方法

IF 2.1 3区数学 Q1 Mathematics

SIAM Journal on Imaging Sciences

Pub Date : 2024-01-24 DOI: 10.1137/23m1571393

Long Li, Jiansheng Yang, Bo Zhang, Haiwen Zhang

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 188-224, March 2024.
Abstract. This paper is concerned with the problem of inverse scattering of time-harmonic acoustic plane waves by a two-layered medium with a locally rough interface in two dimensions. A direct imaging method is proposed to reconstruct the locally rough interface from the phaseless total-field data measured on the upper half of the circle with a large radius at a fixed frequency or from the phased far-field data measured on the upper half of the unit circle at a fixed frequency. The presence of the locally rough interface poses challenges in the theoretical analysis of the imaging methods. To address these challenges, a technically involved asymptotic analysis is provided for the relevant oscillatory integrals involved in the imaging methods, based mainly on the techniques and results in our recent work [L. Li, J. Yang, B. Zhang, and H. Zhang, arXiv:2208.00456, 2022] on the uniform far-field asymptotics of the scattered field for acoustic scattering in a two-layered medium. Finally, extensive numerical experiments are conducted to demonstrate the feasibility and robustness of our imaging algorithms.

SIAM 影像科学杂志》第 17 卷第 1 期第 188-224 页，2024 年 3 月。摘要本文研究时谐平面波在二维具有局部粗糙界面的两层介质中的反向散射问题。本文提出了一种直接成像方法，可根据在固定频率下在大半径圆的上半部分测量的无相全场数据或在固定频率下在单位圆的上半部分测量的有相远场数据重建局部粗糙界面。局部粗糙界面的存在给成像方法的理论分析带来了挑战。为了应对这些挑战，主要基于我们最近关于两层介质中声散射的均匀远场渐近分析的工作[L. Li, J. Yang, B. Zhang, and H. Zhang, arXiv:2208.00456, 2022]中的技术和结果，对成像方法中涉及的相关振荡积分进行了渐近分析。最后，我们进行了大量的数值实验，以证明我们的成像算法的可行性和鲁棒性。

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引用次数: 0

Conductivity Imaging from Internal Measurements with Mixed Least-Squares Deep Neural Networks 利用混合最小二乘深度神经网络根据内部测量结果绘制电导率图像

IF 2.1 3区数学 Q1 Mathematics

SIAM Journal on Imaging Sciences

Pub Date : 2024-01-23 DOI: 10.1137/23m1562536

Bangti Jin, Xiyao Li, Qimeng Quan, Zhi Zhou

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 147-187, March 2024.
Abstract. In this work, we develop a novel approach using deep neural networks (DNNs) to reconstruct the conductivity distribution in elliptic problems from one measurement of the solution over the whole domain. The approach is based on a mixed reformulation of the governing equation and utilizes the standard least-squares objective, with DNNs as ansatz functions to approximate the conductivity and flux simultaneously. We provide a thorough analysis of the DNN approximations of the conductivity for both continuous and empirical losses, including rigorous error estimates that are explicit in terms of the noise level, various penalty parameters, and neural network architectural parameters (depth, width, and parameter bounds). We also provide multiple numerical experiments in two dimensions and multidimensions to illustrate distinct features of the approach, e.g., excellent stability with respect to data noise and capability of solving high-dimensional problems.

SIAM 影像科学杂志》第 17 卷第 1 期第 147-187 页，2024 年 3 月。摘要在这项工作中，我们利用深度神经网络（DNN）开发了一种新方法，通过对整个域的解的一次测量来重建椭圆问题中的电导率分布。该方法基于对控制方程的混合重述，并利用标准最小二乘法目标，以 DNNs 作为解析函数，同时逼近电导率和通量。我们对连续损失和经验损失的 DNN 近似电导率进行了全面分析，包括严格的误差估计，这些误差估计明确反映了噪声水平、各种惩罚参数和神经网络架构参数（深度、宽度和参数边界）。我们还提供了两个维度和多个维度的多个数值实验，以说明该方法的显著特点，如对数据噪声的出色稳定性和解决高维问题的能力。

引用次数: 0

Polynomial Preconditioners for Regularized Linear Inverse Problems 正则化线性逆问题的多项式预调器

IF 2.1 3区数学 Q1 Mathematics

SIAM Journal on Imaging Sciences

Pub Date : 2024-01-22 DOI: 10.1137/22m1530355

Siddharth S. Iyer, Frank Ong, Xiaozhi Cao, Congyu Liao, Luca Daniel, Jonathan I. Tamir, Kawin Setsompop

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 116-146, March 2024.
Abstract. This work aims to accelerate the convergence of proximal gradient methods used to solve regularized linear inverse problems. This is achieved by designing a polynomial-based preconditioner that targets the eigenvalue spectrum of the normal operator derived from the linear operator. The preconditioner does not assume any explicit structure on the linear function and thus can be deployed in diverse applications of interest. The efficacy of the preconditioner is validated on three different Magnetic Resonance Imaging applications, where it is seen to achieve faster iterative convergence (around [math] faster, depending on the application of interest) while achieving similar reconstruction quality.

SIAM 影像科学杂志》，第 17 卷第 1 期，第 116-146 页，2024 年 3 月。摘要本研究旨在加速用于解决正则化线性逆问题的近似梯度法的收敛速度。这是通过设计一种基于多项式的预处理器来实现的，该预处理器以线性算子导出的正则算子的特征值谱为目标。该预处理器不假定线性函数有任何显式结构，因此可用于各种相关应用。我们在三种不同的磁共振成像应用中验证了该预调器的功效，发现它能实现更快的迭代收敛（根据感兴趣的应用，大约快[数学]），同时获得相似的重建质量。

引用次数: 0

Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms 学习弱凸正则，实现可收敛的图像重建算法

IF 2.1 3区数学 Q1 Mathematics

SIAM Journal on Imaging Sciences

Pub Date : 2024-01-18 DOI: 10.1137/23m1565243

Alexis Goujon, Sebastian Neumayer, Michael Unser

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 91-115, March 2024.
Abstract.We propose to learn non-convex regularizers with a prescribed upper bound on their weak-convexity modulus. Such regularizers give rise to variational denoisers that minimize a convex energy. They rely on few parameters (less than 15,000) and offer a signal-processing interpretation as they mimic handcrafted sparsity-promoting regularizers. Through numerical experiments, we show that such denoisers outperform convex-regularization methods as well as the popular BM3D denoiser. Additionally, the learned regularizer can be deployed to solve inverse problems with iterative schemes that provably converge. For both CT and MRI reconstruction, the regularizer generalizes well and offers an excellent tradeoff between performance, number of parameters, guarantees, and interpretability when compared to other data-driven approaches.

SIAM 影像科学期刊》第 17 卷第 1 期第 91-115 页，2024 年 3 月。摘要.我们建议学习非凸正则，并为其弱凸模设定上限。这种正则化器可以产生最小化凸能的变分去噪器。它们依赖于很少的参数（少于 15000 个），并提供了一种信号处理解释，因为它们模仿了手工制作的促进稀疏性的正则器。通过数值实验，我们发现这种去噪器的性能优于凸正则化方法和流行的 BM3D 去噪器。此外，学习到的正则化器可用于解决逆问题，其迭代方案可证明收敛。对于 CT 和 MRI 重建，正则化器都有很好的通用性，与其他数据驱动方法相比，它在性能、参数数量、保证和可解释性之间实现了很好的权衡。

引用次数: 0

Identification of Sparsely Representable Diffusion Parameters in Elliptic Problems 椭圆问题中可稀疏表示的扩散参数的识别

IF 2.1 3区数学 Q1 Mathematics

SIAM Journal on Imaging Sciences

Pub Date : 2024-01-17 DOI: 10.1137/23m1565346

Luzia N. Felber, Helmut Harbrecht, Marc Schmidlin

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 61-90, March 2024.
Abstract. We consider the task of estimating the unknown diffusion parameter in an elliptic PDE as a model problem to develop and test the effectiveness and robustness to noise of reconstruction schemes with sparsity regularization. To this end, the model problem is recast as a nonlinear infinite dimensional optimization problem, where the logarithm of the unknown diffusion parameter is modeled using a linear combination of the elements of a dictionary, i.e., a known bounded sequence of [math] functions, with unknown coefficients that form a sequence in [math]. We show that the regularization of this nonlinear optimization problem using a weighted [math]-norm has minimizers that are finitely supported. We then propose modifications of well-known algorithms (ISTA and FISTA) to find a minimizer of this weighted [math]-norm regularized nonlinear optimization problem that accounts for the fact that in general the smooth part of the functional being optimized is a functional only defined over [math]. We also introduce semismooth methods (ASISTA and FASISTA) for finding a minimizer, which locally uses Gauss–Newton type surrogate models that additionally are stabilized by means of a Levenberg–Marquardt type approach. Our numerical examples show that the regularization with the weighted [math]-norm indeed does make the estimation more robust with respect to noise. Moreover, the numerical examples also demonstrate that the ASISTA and FASISTA methods are quite efficient, outperforming both ISTA and FISTA.

SIAM 影像科学杂志》第 17 卷第 1 期第 61-90 页，2024 年 3 月。摘要。我们将估计椭圆 PDE 中未知扩散参数的任务视为一个模型问题，以开发和测试稀疏正则化重建方案的有效性和对噪声的鲁棒性。为此，该模型问题被重构为一个非线性无限维优化问题，其中未知扩散参数的对数使用字典元素的线性组合来建模，即已知有界的[math]函数序列，其未知系数在[math]中形成一个序列。我们证明，使用加权[math]正则对这一非线性优化问题进行正则化，其最小值是有限支持的。然后，我们提出了对著名算法（ISTA 和 FISTA）的修改，以找到这个加权[math]正则化非线性优化问题的最小值，该算法考虑到了这样一个事实，即一般情况下，被优化函数的光滑部分是一个仅定义在[math]上的函数。我们还介绍了寻找最小值的半光滑方法（ASISTA 和 FASISTA），该方法局部使用高斯-牛顿类型的代用模型，并通过 Levenberg-Marquardt 类型的方法对其进行稳定。我们的数值示例表明，使用加权[数学]正则的正则化方法确实能使估计结果对噪声更加稳健。此外，数值示例还表明，ASISTA 和 FASISTA 方法相当高效，优于 ISTA 和 FISTA 方法。

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引用次数: 0

Learning Sparsity-Promoting Regularizers Using Bilevel Optimization 利用双层优化学习稀疏性促进正则表达式

IF 2.1 3区数学 Q1 Mathematics

SIAM Journal on Imaging Sciences

Pub Date : 2024-01-10 DOI: 10.1137/22m1506547

Avrajit Ghosh, Michael McCann, Madeline Mitchell, Saiprasad Ravishankar

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 31-60, March 2024.
Abstract. We present a gradient-based heuristic method for supervised learning of sparsity-promoting regularizers for denoising signals and images. Sparsity-promoting regularization is a key ingredient in solving modern signal reconstruction problems; however, the operators underlying these regularizers are usually either designed by hand or learned from data in an unsupervised way. The recent success of supervised learning (e.g., with convolutional neural networks) in solving image reconstruction problems suggests that it could be a fruitful approach to designing regularizers. Towards this end, we propose to denoise signals using a variational formulation with a parametric, sparsity-promoting regularizer, where the parameters of the regularizer are learned to minimize the mean squared error of reconstructions on a training set of ground truth image and measurement pairs. Training involves solving a challenging bilevel optimization problem; we derive an expression for the gradient of the training loss using the closed-form solution of the denoising problem and provide an accompanying gradient descent algorithm to minimize it. Our experiments with structured 1D signals and natural images indicate that the proposed method can learn an operator that outperforms well-known regularizers (total variation, DCT-sparsity, and unsupervised dictionary learning) and collaborative filtering for denoising.

SIAM 影像科学杂志》，第 17 卷第 1 期，第 31-60 页，2024 年 3 月。摘要我们提出了一种基于梯度的启发式方法，用于监督学习用于去噪信号和图像的稀疏性促进正则化。稀疏性促进正则化是解决现代信号重建问题的关键要素；然而，这些正则化的基本算子通常要么是手工设计的，要么是以无监督的方式从数据中学习的。最近，监督学习（如卷积神经网络）在解决图像重建问题方面取得了成功，这表明监督学习可能是设计正则化器的一种富有成效的方法。为此，我们建议使用参数化、稀疏性促进正则器的变分公式对信号进行去噪，其中正则器参数的学习是为了最小化在一组基本真实图像和测量对的训练集上重建的均方误差。训练涉及解决一个具有挑战性的双层优化问题；我们利用去噪问题的闭式解推导出了训练损失梯度表达式，并提供了一种相应的梯度下降算法来最小化训练损失。我们用结构化一维信号和自然图像进行的实验表明，所提出的方法可以学习一种算子，其性能优于众所周知的正则化器（总变异、DCT-稀疏性和无监督字典学习）以及用于去噪的协同过滤。

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引用次数: 0

A Variational Model for Nonuniform Low-Light Image Enhancement 非均匀弱光图像增强的变量模型

IF 2.1 3区数学 Q1 Mathematics

SIAM Journal on Imaging Sciences

Pub Date : 2024-01-04 DOI: 10.1137/22m1543161

Fan Jia, Shen Mao, Xue-Cheng Tai, Tieyong Zeng

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 1-30, March 2024.
Abstract. Low-light image enhancement plays an important role in computer vision applications, which is a fundamental low-level task and can affect high-level computer vision tasks. To solve this ill-posed problem, a lot of methods have been proposed to enhance low-light images. However, their performance degrades significantly under nonuniform lighting conditions. Due to the rapid variation of illuminance in different regions in natural images, it is challenging to enhance low-light parts and retain normal-light parts simultaneously in the same image. Commonly, either the low-light parts are underenhanced or the normal-light parts are overenhanced, accompanied by color distortion and artifacts. To overcome this problem, we propose a simple and effective Retinex-based model with reflectance map reweighting for images under nonuniform lighting conditions. An alternating proximal gradient (APG) algorithm is proposed to solve the proposed model, in which the illumination map, the reflectance map, and the weighting map are updated iteratively. To make our model applicable to a wide range of light conditions, we design an initialization scheme for the weighting map. A theoretical analysis of the existence of the solution to our model and the convergence of the APG algorithm are also established. A series of experiments on real-world low-light images are conducted, which demonstrate the effectiveness of our method.

SIAM 影像科学杂志》第 17 卷第 1 期第 1-30 页，2024 年 3 月。摘要低照度图像增强在计算机视觉应用中起着重要作用，它是一项基本的低级任务，并会影响高级计算机视觉任务。为了解决这一难题，人们提出了很多增强低照度图像的方法。然而，在非均匀光照条件下，这些方法的性能会明显下降。由于自然图像中不同区域照度的快速变化，在同一图像中同时增强低照度部分和保留正常照度部分具有挑战性。通常情况下，要么低照度部分增强不足，要么正常照度部分增强过度，并伴随着色彩失真和伪影。为了克服这一问题，我们提出了一种简单有效的基于 Retinex 的模型，并对非均匀光照条件下的图像进行反射图再加权。我们提出了一种交替近似梯度（APG）算法来求解所提出的模型，在该算法中，照明图、反射图和加权图会进行迭代更新。为了使我们的模型适用于各种光照条件，我们为加权图设计了一个初始化方案。我们还对模型解的存在性和 APG 算法的收敛性进行了理论分析。我们在真实世界的弱光图像上进行了一系列实验，证明了我们方法的有效性。

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引用次数: 0

Self-Supervised Deep Learning for Image Reconstruction: A Langevin Monte Carlo Approach 自监督深度学习用于图像重建:一种Langevin Monte Carlo方法

IF 2.1 3区数学 Q1 Mathematics

SIAM Journal on Imaging Sciences

Pub Date : 2023-11-30 DOI: 10.1137/23m1548025

Ji Li, Weixi Wang, Hui Ji

SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2247-2284, December 2023.
Abstract. Deep learning has proved to be a powerful tool for solving inverse problems in imaging, and most of the related work is based on supervised learning. In many applications, collecting truth images is a challenging and costly task, and the prerequisite of having a training dataset of truth images limits its applicability. This paper proposes a self-supervised deep learning method for solving inverse imaging problems that does not require any training samples. The proposed approach is built on a reparametrization of latent images using a convolutional neural network, and the reconstruction is motivated by approximating the minimum mean square error estimate of the latent image using a Langevin dynamics–based Monte Carlo (MC) method. To efficiently sample the network weights in the context of image reconstruction, we propose a Langevin MC scheme called Adam-LD, inspired by the well-known optimizer in deep learning, Adam. The proposed method is applied to solve linear and nonlinear inverse problems, specifically, sparse-view computed tomography image reconstruction and phase retrieval. Our experiments demonstrate that the proposed method outperforms existing unsupervised or self-supervised solutions in terms of reconstruction quality.

SIAM影像科学杂志，第16卷，第4期，2247-2284页，2023年12月。摘要。深度学习已被证明是解决成像逆问题的有力工具，大部分相关工作都是基于监督学习的。在许多应用中，收集真值图像是一项具有挑战性和昂贵的任务，并且具有真值图像训练数据集的先决条件限制了其适用性。本文提出一种不需要任何训练样本的自监督深度学习方法来求解逆成像问题。该方法基于卷积神经网络对潜在图像进行重新参数化，并利用基于朗格万动力学的蒙特卡罗(MC)方法逼近潜在图像的最小均方误差估计，从而实现重建。为了在图像重建的背景下有效地采样网络权重，我们提出了一种称为Adam- ld的Langevin MC方案，该方案的灵感来自于深度学习中著名的优化器Adam。该方法适用于求解线性和非线性逆问题，特别是稀疏视图计算机断层扫描图像重建和相位检索。我们的实验表明，该方法在重建质量方面优于现有的无监督或自监督解决方案。

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