Sparsity promoting regularization for effective noise suppression in SPECT image reconstruction

Zheng, Wei; Li, Si; Król, Andrzej; Schmidtlein, C. Ross; Zeng, Xueying; Xu, Yuesheng

doi:10.1088/1361-6420/ab23da

Cited by 11 publications

(9 citation statements)

References 63 publications

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“…Form (4.2) relates to regularization by the envelope of the ℓ 0 norm and form (4.3) relates to regularization by the capped ℓ 1 norm [11]. For specific examples of f , see [31] for inverting incomplete Fourier transform, [32,33] for image/signal processing, [34] for medical image reconstruction and machine learning [16,17,21,29]. Employing the partition A j , j ∈ Z d+1 , of R d and the definition of • 0 , we have an alternative representation of function f :…”

Section: Sparse Regularization In the Spacial Domainmentioning

confidence: 99%

“…The aim of this work is to understand a global minimizer of regularization problems whose objective functions have the form of a fidelity term plus a regularization term involving the ℓ 0 norm. Regularization problems of this type appear frequently in recent studies of machine learning [16,17,21,29], computer graphics [8,27], signal processing [6,15,33], image processing [24,25,32], medical imaging [34] and statistics [9,35]. Many published results have demonstrated that the use of the ℓ 0 norm in regularization models promotes sparsity for the regularized solutions or the transformed regularized solutions.…”

Section: Introductionmentioning

confidence: 99%

“…where g : R d × R d ′ → R is a continuous function, not necessarily convex. Typical examples of function f in the form (1.4) include the objective functions in wavelet inpainting with the ℓ 0 sparse regularization [24], inverting an incomplete Fourier transform [31] and medical image reconstruction [34]. In these applications, the function g is a sum of two or three convex functions which measure the data fidelity and define other convex constraints.…”

Section: Introductionmentioning

confidence: 99%

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Sparse Regularization with the $\ell_0$ Norm

Xu¹

2021

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We consider a minimization problem whose objective function is the sum of a fidelity term, not necessarily convex, and a regularization term defined by a positive regularization parameter λ multiple of the ℓ 0 norm composed with a linear transform. This problem has wide applications in compressed sensing, sparse machine learning and image reconstruction. The goal of this paper is to understand what choices of the regularization parameter can dictate the level of sparsity under the transform for a global minimizer of the resulting regularized objective function. This is a critical issue but it has been left unaddressed. We address it from a geometric viewpoint with which the sparsity partition of the image space of the transform is introduced. Choices of the regularization parameter are specified to ensure that a global minimizer of the corresponding regularized objective function achieves a prescribed level of sparsity under the transform. Results are obtained for the spacial sparsity case in which the transform is the identity map, a case that covers several applications of practical importance, including machine learning, image/signal processing and medical image reconstruction.

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Section: Sparse Regularization In the Spacial Domainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sparse Regularization with the $\ell_0$ Norm

Xu¹

2021

Preprint

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“…It is desirable to develop solution representations of the minimum norm interpolation problem and the regularization problem convenient for algorithmic development. Inspired by the suc-cess of the fixed-point approach used in solving several types of finite dimensional problems such as machine learning [2,44,45,46,56], image processing [11,41,42,47,51], medical imaging [40,43,85] and solutions of semi-discrete inverse problems [27,37], we develop solution representations for the minimum norm interpolation problem and the regularization problem by using a fixed-point formulation via the proximity operator of the functions appearing in the objective function or constraints. This formulation has great potential for convenience of designing iterative algorithms for solving these problems.…”

Section: Introductionmentioning

confidence: 99%

Representer Theorems in Banach Spaces: Minimum Norm Interpolation, Regularized Learning and Semi-Discrete Inverse Problems

Wang,

2020

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Constructing or learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation problem, regularized learning problem or, in general, a semidiscrete inverse problem, in certain functional spaces. The choice of an appropriate space is crucial for solutions of these problems. When they are considered in an infinite dimensional Hilbert space, which gives a linear method for their solutions, the remarkable representer theorem reduces their solutions to finding a finite number of coefficients of elements in the space. Motivated by sparse representations of the reconstructed functions such as compressed sensing and sparse learning, much of the recent research interest has been directed to considering these problems in certain Banach spaces in order to obtain their sparse solutions, which is a feasible approach to overcome challenges coming from the big data nature of most practical applications. It is the goal of this paper to provide a systematic study of the representer theorems for these problems in Banach spaces. There are a few existing results for these problems in a Banach space, with all of them regarding implicit representer theorems. We aim at obtaining explicit representer theorems based on which convenient solution methods will then be developed. For the minimum norm interpolation, the explicit representer theorems enable us to express the infimum in terms of the norm of the linear combination of the interpolation functionals. For the purpose of developing efficient computational algorithms, we establish the fixed-point equation formulation of solutions of these problems. We reveal that unlike in a Hilbert space, in general, solutions of these problems in a Banach space may not be able to be reduced to truly finite dimensional problems (with certain infinite dimensional components hidden). We demonstrate how this obstacle can be removed, reducing the original problem to a truly finite dimensional one, in the special case when the Banach space is ℓ 1 (N).

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“…We consider in this paper the convergence rate analysis of fixed-point algorithms. Fixed-point type algorithms have been popular in solving nondifferentiable convex or nonconvex optimization problems such as image processing [16,25,30,32,33,41], medical imaging [24,29,38,47], machine learning [14,27,28,36], and compressed sensing [21,48]. Existing fixed-point type algorithms for optimization including the gradient descent algorithm [8,39], the proximal point algorithm [37], the proximal gradient algorithm [7,35], the forward-backward splitting algorithm [15,45] and the fixed-point proximity algorithm [25,29,32,33].…”

mentioning

confidence: 99%

Convergence Rate Analysis for Fixed-Point Iterations of Generalized Averaged Nonexpansive Operators

Lin¹,

Xu²

2021

Preprint

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We estimate convergence rates for fixed-point iterations of a class of nonlinear operators which are partially motivated from solving convex optimization problems. We introduce the notion of the generalized averaged nonexpansive (GAN) operator with a positive exponent, and provide a convergence rate analysis of the fixed-point iteration of the GAN operator. The proposed generalized averaged nonexpansiveness is weaker than the averaged nonexpansiveness while stronger than nonexpansiveness. We show that the fixed-point iteration of a GAN operator with a positive exponent converges to its fixed-point and estimate the local convergence rate (the convergence rate in terms of the distance between consecutive iterates) according to the range of the exponent. We prove that the fixed-point iteration of a GAN operator with a positive exponent strictly smaller than 1 can achieve an exponential global convergence rate (the convergence rate in terms of the distance between an iterate and the solution). Furthermore, we establish the global convergence rate of the fixed-point iteration of a GAN operator, depending on both the exponent of generalized averaged nonexpansiveness and the exponent of the Hölder regularity, if the GAN operator is also Hölder regular. We then apply the established theory to three types of convex optimization problems that appear often in data science to design fixed-point iterative algorithms for solving these optimization problems and to analyze their convergence properties.

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Sparsity promoting regularization for effective noise suppression in SPECT image reconstruction

Cited by 11 publications

References 63 publications

Sparse Regularization with the $\ell_0$ Norm

Sparse Regularization with the $\ell_0$ Norm

Representer Theorems in Banach Spaces: Minimum Norm Interpolation, Regularized Learning and Semi-Discrete Inverse Problems

Convergence Rate Analysis for Fixed-Point Iterations of Generalized Averaged Nonexpansive Operators

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