Tomographic reconstruction methods for decomposing directional components

Kongskov, Rasmus Dalgas; Dong, Yiqiu

doi:10.3934/ipi.2018060

Cited by 3 publications

(2 citation statements)

References 27 publications

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“…A priori knowledge about the device (e.g. sparsity, high contrast, and direction [10,40]) can be used, given that the object is separated from the background.…”

Section: Example 1: Reconstruction Of a Geometric Abdomen From Unders...mentioning

confidence: 99%

Convergence of proximal gradient algorithm in the presence of adjoint mismatch ^*

Chouzenoux

Pesquet

Riddell³

et al. 2021

Inverse Problems

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We consider the proximal gradient algorithm for solving penalized least-squares minimization problems arising in data science. This first-order algorithm is attractive due to its flexibility and minimal memory requirements allowing to tackle large-scale minimization problems involving non-smooth penalties. However, for problems such as X-ray computed tomography, the applicability of the algorithm is dominated by the cost of applying the forward linear operator and its adjoint at each iteration. In practice, the adjoint operator is thus often replaced by an alternative operator with the aim to reduce the overall computation burden and potentially improve conditioning issues. In this paper, we propose to analyze the effect of such an adjoint mismatch on the convergence of the proximal gradient algorithm in an infinite-dimensional setting, thus generalizing the existing results on PGA. We derive conditions on the stepsize and on the gradient of the smooth part of the objective function under which convergence of the algorithm to a fixed point is guaranteed. We also derive bounds on the error between this point and the solution to the original minimization problem. We illustrate our theoretical findings with two image reconstruction tasks in computed tomography.

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“…A priori knowledge about the device (e.g. sparsity, high contrast, and direction [10,40]) can be used, given that the object is separated from the background.…”

Section: Example 1: Reconstruction Of a Geometric Abdomen From Unders...mentioning

confidence: 99%

Convergence of proximal gradient algorithm in the presence of adjoint mismatch ^*

Chouzenoux

Pesquet

Riddell³

et al. 2021

Inverse Problems

View full text Add to dashboard Cite

show abstract

“…, n to employ different anisotropic norms that favour one particular direction. This yields again an adaptive regularisation of image data via a decomposition into an isotropic and several anisotropic parts and can be employed, for instance, to recover certain line structures for denoising [109] or applications in CT imaging [125].…”

Section: Extensionsmentioning

confidence: 99%

Higher-order total variation approaches and generalisations

Bredies

Höller

2020

Inverse Problems

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Over the last decades, the total variation (TV) has evolved to be one of the most broadly-used regularisation functionals for inverse problems, in particular for imaging applications. When first introduced as a regulariser, higher-order generalisations of TV were soon proposed and studied with increasing interest, which led to a variety of different approaches being available today. We review several of these approaches, discussing aspects ranging from functional-analytic foundations to regularisation theory for linear inverse problems in Banach space, and provide a unified framework concerning well-posedness and convergence for vanishing noise level for respective Tikhonov regularisation. This includes general higher orders of TV, additive and infimal-convolution multi-order total variation, total generalised variation, and beyond. Further, numerical optimisation algorithms are developed and discussed that are suitable for solving the Tikhonov minimisation problem for all presented models. Focus is laid in particular on covering the whole pipeline starting at the discretisation of the problem and ending at concrete, implementable iterative procedures. A major part of this review is finally concerned with presenting examples and applications where higher-order TV approaches turned out to be beneficial. These applications range from classical inverse problems in imaging such as denoising, deconvolution, compressed sensing, optical-flow estimation and decompression, to image reconstruction in medical imaging and beyond, including magnetic resonance imaging, computed tomography, magnetic-resonance positron emission tomography, and electron tomography.

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Directional total generalized variation regularization

2019

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Tomographic reconstruction methods for decomposing directional components

Cited by 3 publications

References 27 publications

Convergence of proximal gradient algorithm in the presence of adjoint mismatch ^*

Convergence of proximal gradient algorithm in the presence of adjoint mismatch ^*

Higher-order total variation approaches and generalisations

Directional total generalized variation regularization

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Tomographic reconstruction methods for decomposing directional components

Cited by 3 publications

References 27 publications

Convergence of proximal gradient algorithm in the presence of adjoint mismatch *

Convergence of proximal gradient algorithm in the presence of adjoint mismatch *

Higher-order total variation approaches and generalisations

Directional total generalized variation regularization

Contact Info

Product

Resources

About

Convergence of proximal gradient algorithm in the presence of adjoint mismatch ^*

Convergence of proximal gradient algorithm in the presence of adjoint mismatch ^*