Linearized Krylov subspace Bregman iteration with nonnegativity constraint

Buccini, Alessandro; Pasha, Mirjeta; Reichel, Lothar

doi:10.1007/s11075-020-01004-6

Cited by 10 publications

(11 citation statements)

References 36 publications

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“…It is a well known result, and it is illustrated in [57] that in the context of Tikhonov regularization with the regularization parameter determined by the discrepancy principle that the quality of the computed solution may be increased by carrying out a few more iterations of the Arnoldi process. A similar observation was concluded in [17] as well that adding few more vectors in the subspace after the discrepancy principle is satisfied enhances the quality of the computed solution. In this work, we stop the iterations using the discrepancy principle.…”

Section: Solving the Problem In The Subspace Generated By Fgksupporting

confidence: 79%

“…where among different choices of L are discretization of the first or the second derivative operator, framelet or wavelet transforms [15,17], or a weighting matrix aiming to reconstruct solutions with certain properties. For instance, if is chosen to be a transformation operator to the framelet domain, then by minimizing (3.13) with 0 < p ≤ 1 we seek to reconstruct solutions with sparse representation in the transformed framelet domain.…”

Section: The Minimization Problemmentioning

confidence: 99%

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A Krylov subspace type method for Electrical Impedance Tomography

et al. 2021

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Electrical Impedance Tomography (EIT) is a well-known imaging technique for detecting the electrical properties of an object in order to detect anomalies, such as conductive or resistive targets. More specifically, EIT has many applications in medical imaging for the detection and location of bodily tumors since it is an affordable and non-invasive method, which aims to recover the internal conductivity of a body using voltage measurements resulting from applying low frequency current at electrodes placed at its surface. Mathematically, the reconstruction of the internal conductivity is a severely ill-posed inverse problem and yields a poor quality image reconstruction. To remedy this difficulty, at least in part, we regularize and solve the nonlinear minimization problem by the aid of a Krylov subspace-type method for the linear sub problem during each iteration. In EIT, a tumor or general anomaly can be modeled as a piecewise constant perturbation of a smooth background, hence, we solve the regularized problem on a subspace of relatively small dimension by the Flexible Golub-Kahan process that provides solutions that have sparse representation. For comparison, we use a well-known modified Gauss-Newton algorithm as a benchmark. Using simulations, we demonstrate the effectiveness of the proposed method. The obtained reconstructions indicate that the Krylov subspace method is better adapted to solve the ill-posed EIT problem and results in higher resolution images and faster convergence compared to reconstructions using the modified Gauss-Newton algorithm.

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Section: Solving the Problem In The Subspace Generated By Fgksupporting

confidence: 79%

Section: The Minimization Problemmentioning

confidence: 99%

A Krylov subspace type method for Electrical Impedance Tomography

et al. 2021

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“…We have shown that the proposed method outperforms the classical TV approach. Matter of future research include the application of the proposed method to more general inverse problems as well as the integration of the considered method with the ℓ p − ℓ q minimization proposed in [12,[14][15][16]32,36] or with iterative regularization methods like, e.g., Linearized Bregman splitting [11,13,[18][19][20] and Iterated Tikhonov with general penalty terms [3,4,7,10]. Another line of future research is the construction of more sophisticated graphs ω which can better exploit the structure of the given image itself.…”

Section: Discussionmentioning

confidence: 99%

Graph Laplacian for image deblurring

Bianchi¹,

Buccini²,

Donatelli³

et al. 2021

Preprint

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Image deblurring is relevant in many fields of science and engineering. To solve this problem, many different approaches have been proposed and among the various methods, variational ones are extremely popular. These approaches are characterized by substituting the original problem with a minimization one where the functional is composed of two terms, a data fidelity term and a regularization term. In this paper we propose, in the classical ℓ 2 − ℓ 1 minimization with the non-negativity constraint of the solution, the use of the graph Laplacian as regularization operator. Firstly, we describe how to construct the graph Laplacian from the observed noisy and blurred image. Once the graph Laplacian has been built, we solve efficiently the proposed minimization problem splitting the convolution operator and the graph Laplacian by the alternating direction method of multipliers (ADMM). Some selected numerical examples show the good performances of the proposed algorithm.

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“…Nonnegativity constraint. In many application, such as medical imaging and astronomical imaging, the pixels of the desired solution are nonnegative [7,8,28], that is, the exact solution of (1.3) is known to live in the closed and convex set Ω 0 = {u ∈ R n : u ≥ 0, = 1, 2, . .…”

Section: Iterative Methods and Parameter Selection Techniquesmentioning

confidence: 99%

Efficient edge-preserving methods for dynamic inverse problems

Pasha¹,

Saibaba²,

Gazzola³

et al. 2021

Preprint

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We consider efficient methods for computing solutions to dynamic inverse problems, where both the quantities of interest and the forward operator (measurement process) may change at different time instances but we want to solve for all the images simultaneously. We are interested in large-scale ill-posed problems that are made more challenging by their dynamic nature and, possibly, by the limited amount of available data per measurement step. To remedy these difficulties, we apply regularization methods that enforce simultaneous regularization in space and time (such as edge enhancement at each time instant and proximity at consecutive time instants) and achieve this with low computational cost and enhanced accuracy. More precisely, we develop iterative methods based on a majorization-minimization (MM) strategy with quadratic tangent majorant, which allows the resulting least squares problem to be solved with a generalized Krylov subspace (GKS) method; the regularization parameter can be defined automatically and efficiently at each iteration.Numerical examples from a wide range of applications, such as limited-angle computerized tomography (CT), space-time image deblurring, and photoacoustic tomography (PAT), illustrate the effectiveness of the described approaches.

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Linearized Krylov subspace Bregman iteration with nonnegativity constraint

Cited by 10 publications

References 36 publications

A Krylov subspace type method for Electrical Impedance Tomography

A Krylov subspace type method for Electrical Impedance Tomography

Graph Laplacian for image deblurring

Efficient edge-preserving methods for dynamic inverse problems

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