A. Leitao, J. Zubelli
We propose and investigate efficient numerical methods for inverse problems related to Magnetic Resonance Imaging (MRI). Our goal is to extend the recent convergence results for the Landweber-Kaczmarz method obtained in [Haltmeier, Leitao, Scherzer 2007], in order to derive a convergent iterative regularization method for an inverse problem in MRI.
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Spherical function regularization for parallel MRI reconstruction
本文提出基于球函数基的正则化方法，用于解决平行MRI中同时估计线圈灵敏度的优化难题，通过非线性ADMM求解模型并验证其有效性。
PET-MRI Joint Reconstruction by Joint Sparsity Based Tight Frame Regularization
本文提出基于紧框架的PET-MRI联合重建模型，利用紧框架系数的联合稀疏性，并采用非凸平衡方法考虑PET和MRI图像的不同正则性，通过近端交替最小化算法实现全局收敛，实验表明性能优于现有方法。
Recently, the finite-rate-of-innovation (FRI) based continuous domain regularization is emerging as an alternative to the conventional on-the-grid sparse regularization for the compressed sensing (CS) due to its ability to alleviate the basis mismatch between the true support of the shape in the continuous domain and the discrete grid. In this paper, we propose a new off-the-grid regularization for the CS-MRI reconstruction. Following the recent works on two dimensional FRI, we assume that the discontinuities/edges of the image are localized in the zero level set of a band-limited periodic fun…
GAN-based iterative motion estimation in HASTE MRI
针对磁共振成像因采集时间长易受运动破坏的问题，提出基于GAN预测的迭代运动估计方法，可实现数据一致重建，无需时间冗余，在脑与腹部HASTE数据上验证了有效性。
Iterative Refinement of A Modified Lavrentiev Regularization Method for De-convolution of the Discrete Helmholtz Type Differential Filter
本文提出并分析了改进的Lavrentiev正则化方法用于离散亥姆霍兹型微分滤波器的反卷积，通过利用亥姆霍兹滤波器的特性，改进了正则化方法，降低了近似解与原解之间的误差界，并推导了最优停止条件。
A randomized progressive iterative regularization method for data fitting problems
针对带随机噪声的数据拟合问题，提出一种适用于大规模矩阵计算且依期望收敛到最小二乘解的随机化渐进迭代正则化方法，并给出正则化参数最优估计以构造无需先验信息的自洽算法，数值实验验证了其在曲线与曲面拟合中的有效性。
Time Relaxation with Iterative Modified Lavrentiev Regularization
本文研究了一种新的时间松弛模型，结合迭代修正Lavrentiev正则化方法，通过合理选择时间松弛参数以加速未解析波动的指数衰减，从而在更少的计算步骤中获得更好的近似解。
Dynamic MRI Reconstruction from Undersampled Data with an Anatomical Prescan
本文提出了一种新的变分方法，结合时间平滑和空间总变分正则化，利用解剖MRI预扫描的结构先验信息来重建欠采样动态MRI数据，提升图像结构相似性并允许局部时间平滑变化。
Connected with the rise of interest in inverse problems is the development and analysis of regularization methods, which are a necessity due to the ill-posedness of inverse problems. Tikhonov-type regularization methods are very popular in this regard. However, its direct implementation for large-scale linear or non-linear problems is a non-trivial task. In such scenarios, iterative regularization methods usually serve as a better alternative. In this paper we propose a new iterative regularization method which uses descent directions, different from the usual gradient direction, that enable a…
On the convergence of an adaptive denoiser driven iterative regularization with early stopping
本文提出DDIR方法，通过自适应步长和后验停止规则，证明其在经典意义下稳定且收敛，用于图像去模糊和相位恢复CT的数值实验验证了其有效性。
IR Tools: A MATLAB Package of Iterative Regularization Methods and Large-Scale Test Problems
本文介绍了一个新的MATLAB工具包IR Tools，提供多种迭代求解器和大规模线性逆问题的测试问题，支持灵活的矩阵输入方式，并通过数值示例展示算法与测试问题。
Superiorized Regularization of Inverse Problems
本文提出将超正则化与迭代正则化结合，在保持算法简单性的同时利用额外先验信息，证明超正则化版本仍为正则化方法且考虑不同先验。
$L_{1/2}$ Regularization: Convergence of Iterative Half Thresholding Algorithm
本文研究了$L_{1/2}$正则化中迭代半阈值化算法的收敛性，证明其在特定条件下收敛到局部极小值，并具有线性收敛速度，为该算法在广泛应用中提供了理论保证。
In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a linear vanishing damping term, which can be viewed not only as an extension of the asymptotical regularization, but also as a continuous analog of the Nesterov's acceleration scheme. New iterative regularization methods are derived from this continuous model in combination with damped symplectic numerical schemes. The regularization property as well as convergenc…
Recently, inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications. After the discretization, many of inverse problems are reduced to linear systems. Due to the typical ill-posedness of inverse problems, the reduced linear systems are often ill-posed, especially when their scales are large. This brings great computational difficulty. Particularly, a small perturbation in the right side of an ill-posed linear system may cause a dramatical change in the solution. Therefore, regularization methods should be …
A general regularization strategy for singular Stokes problems and convergence analysis for corresponding discretization and iterative solution
针对奇异 Stokes 问题的最低阶弱 Galerkin 离散，提出秩一通用正则化策略，分析其最优收敛性及 MINRES/GMRES 块预条件迭代效果。
In this paper we the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type. We carry out a convergence analysis in the sense of regularization methods and discuss applicability to the problem of identifying the spatially varying diffusivity in an elliptic PDE from different sets of observations. Among these is a novel hybrid imaging techology known as impedance acoustic tomography, for which we provide numerical experiments.
In this work we consider the image reconstruction problem of sparsely sampled dynamic contrast-enhanced (DCE) magnetic resonance imaging (MRI). DCE-MRI is a technique for acquiring a series of MR images before, during and after intravenous contrast agent administration, and it is used to study microvascular structure and perfusion. To overcome the ill-posedness of the related spatio-temporal inverse problem, we use regularization. In regularization one of the main problems is how to determine the regularization parameter which controls the balance between data fitting term and regularization t…
Bi-level regularization via iterative mesh refinement for aeroacoustics
本研究揭示了有限元离散PDE的自适应网格细化与双层正则化算法的联系，通过基于数据噪声的网格细化实现正则化与收敛，并在航空声学Helmholtz方程中证明其优于经典Landweber算法。
We investigate the Cauchy problem for linear elliptic operators with $C^\infty$-coefficients at a regular set $Ω\subset R^2$, which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold $Γ\subset \partialΩ$ and our goal is to reconstruct the trace of the $H^1(Ω)$ solution of an elliptic equation at $\partial Ω/ Γ$. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solv…
A non-iterative method for electrical property tomography based on a simple formula
本文提出了一种非迭代的重建方法，利用MRI系统数据和拉莫尔频率下的谐波电磁场数据，通过精确的分析公式求解对比源函数，采用几何方法获取全感生场，从而实现电性质断层成像。
Parameter Selection by GCV and a $χ^2$ test within Iterative Methods for $\ell_1$-regularized Inverse Problems
针对ℓ₁正则化反问题的Split Bregman、Majorization-Minimization迭代方法，将GCV和χ²检验等方法拓展用于迭代内广义Tikhonov正则参数选择，图像去模糊实验验证了该自动选参策略的有效性。
Stopping criterion for iterative regularization of large-scale ill-posed problems using the Picard parameter
本文提出了一种新的停止准则，用于大规模病态逆问题的Krylov子空间迭代正则化。通过扩展Picard参数，有效分离噪声主导的傅里叶系数与信号主导的系数，并通过向量化方案将二维问题降维处理。
A preconditioned Krylov subspace method for linear inverse problems with general-form Tikhonov regularization
本文提出基于预条件Golub-Kahan双对角化的Krylov子空间迭代方法，用于求解一般形式Tikhonov正则化线性反问题，并进一步设计两种混合正则化算法以克服半收敛不稳定性，数值实验验证了其优越性能。
Scalable iterative data-adaptive RKHS regularization
本文提出可扩展的迭代数据自适应 RKHS 正则化方法 iDARR，通过广义 Golub-Kahan 双对角化过程构造 Krylov 子空间正交基，高效求解不适定线性反问题，并在图像去模糊等实验中展现出优于传统范数的准确性与稳定性。
We propose a set of iterative regularization algorithms for the TV-Stokes model to restore images from noisy images with Gaussian noise. These are some extensions of the iterative regularization algorithm proposed for the classical Rudin-Osher-Fatemi (ROF) model for image reconstruction, a single step model involving a scalar field smoothing, to the TV-Stokes model for image reconstruction, a two steps model involving a vector field smoothing in the first and a scalar field smoothing in the second. The iterative regularization algorithms proposed here are Richardson's iteration like. We have e…
We analyze sparse frame based regularization of inverse problems by means of a diagonal frame decomposition (DFD) for the forward operator, which generalizes the SVD. The DFD allows to define a non-iterative (direct) operator-adapted frame thresholding approach which we show to provide a convergent regularization method with linear convergence rates. These results will be compared to the well-known analysis and synthesis variants of sparse $\ell^1$-regularization which are usually implemented thorough iterative schemes. If the frame is a basis (non-redundant case), the three versions of sparse…
Hybrid and iteratively reweighted regularization by unbiased predictive risk and weighted GCV for projected systems
本文研究了大规模病态问题的投影解的Tikhonov正则化，通过无偏预测风险估计、广义交叉验证和不一致原理确定正则化参数，并展示了无偏预测风险估计在中等至严重病态问题中的有效性，同时解释了无加权广义交叉验证的局限性。
A CCBM-based generalized GKB iterative regularization algorithm for inverse Cauchy problems
本文针对反柯西问题，提出基于耦合复边界法的CCBM-GKB迭代正则化算法，结合GKB双对角化与Givens旋转求解算子方程，并证明其正则化性质与收敛率，数值实验表明其远快于Landweber方法。
On the filtering effect of iterative regularization algorithms for linear least-squares problems
本文分析了线性最小二乘问题中经典和最新梯度方法对奇异值的过滤方式，探讨了缩放矩阵和非负约束对解恢复的影响。
New Pair of Primal Dual Algorithms for Bregman Iterated Variational Regularization
本文提出了一种新的对偶原问题算法，用于解决一般形式的Tikhonov型泛函的最小化问题，通过迭代变分正则化程序近似求解，并在变分源条件假设下分析了迭代和连续意义下的总误差估计，给出了基于Bregman距离惩罚的非光滑总变分（TV）泛函的惩罚项选择，并通过GPS-断层成像问题验证了算法的稳定性。
Many inverse problems are concerned with the estimation of non-negative parameter functions. In this paper, in order to obtain non-negative stable approximate solutions to ill-posed linear operator equations in a Hilbert space setting, we develop two novel non-negativity preserving iterative regularization methods. They are based on fixed point iterations in combination with preconditioning ideas. In contrast to the projected Landweber iteration, for which only weak convergence can be shown for the regularized solution when the noise level tends to zero, the introduced regularization methods e…
These lecture notes for a graduate class present the regularization theory for linear and nonlinear ill-posed operator equations in Hilbert spaces. Covered are the general framework of regularization methods and their analysis via spectral filters as well as the concrete examples of Tikhonov regularization, Landweber iteration, regularization by discretization for linear inverse problems. In the nonlinear setting, Tikhonov regularization and iterative regularization (Landweber, Levenberg-Marquardt, and iteratively regularized Gauß-Newton methods) are discussed. The necessary background from fu…
Iterative regularization for ensemble data assimilation in reservoir models
本文提出利用迭代正则化方法开发集合方法解决贝叶斯反问题，设计了IR-enLM和IR-ES方法，通过正则化Levenberg-Marquardt方案提升后验估计的鲁棒性。
On the convergence of iterative regularization method assisted by the graph Laplacian with early stopping
提出一种数据辅助迭代正则化方法IRMGL+Ψ，结合图拉普拉斯正则化与经典迭代技术，并证明差异原则作为早停准则的可靠性，实验表明伴随算子初始化效果最优。
Bi-level iterative regularization for inverse problems in nonlinear PDEs
针对非线性演化偏微分方程中未知空间依赖参数的不适定逆问题，提出结合约化与全耦合设置的双层Landweber格式，推导迭代停止准则与收敛性，可用于磁性粒子成像中相关方程的参数识别。
Inverse Problem Regularization for 3D Multi-Species Tumor Growth Models
本文提出一种三维多物种PDE肿瘤生长模型及两阶段反演算法，通过压缩感知和加权正则化解决单时间点MRI数据校准的病态问题，实现了对不可观测肿瘤物种的稳定估计与浸润性细胞量化。
Regularization of ill-posed problems with non-negative solutions
本文综述了用于重构无限维空间中非负解的变分与迭代方法，重点讨论了基于熵最小化或约束的变分方法及涉及投影或非负性保持乘法更新的迭代方法，并指出一些开放性问题。
Iterative Methods for an Inverse Eddy Current Problem with Total Variation Regularization
本文研究逆涡流问题中的电导率重构，将其表述为带全变差正则化的约束优化问题，采用有限元离散并提出交替方向乘子法求解，证明了收敛性并通过数值实验验证了方法的有效性。
Data-driven Smooth-to-hard Spectral Regularization for General Linear Inverse Problems
该研究提出用于一般线性逆问题的数据驱动平滑到硬谱正则化方法，基于逻辑斯蒂平滑到硬收缩器的弱凸性保证，以MRI为例验证参数选择控制，为线性逆问题求解提供新的正则化方案。
Regularization of constrained maximum likelihood iterative algorithms by means of statistical stopping rule
提出一种新的统计停止准则，用于约束最大似然迭代算法求解不适定逆问题，并在高斯和泊松情形下证明其正则化性质，通过RHESSI图像重建验证其优于经典准则。
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