Efficient determination of multiple regularization parameters in a generalized L-curve framework

Belge, M.; Kilmer, Misha E.; Miller, Eric L.

doi:10.1088/0266-5611/18/4/314

Cited by 208 publications

(162 citation statements)

References 21 publications

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“…A linear combination of operators (each with its own regularization parameter i ) can be used as a side constraint (e.g., the Sobolev norm (14): ¥ i i 2 ʈL i rʈ 2 ). In such cases, a multidimensional extension to the L-curve method is required to optimize all the parameters simultaneously (the L-hypersurface was recently introduced to such effect (22)). …”

Section: Discussionmentioning

confidence: 99%

Quantification of bolus‐tracking MRI: Improved characterization of the tissue residue function using Tikhonov regularization

Calamante

Gadian

Connelly

2003

Magnetic Resonance in Med

123

129

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Quantification of cerebral blood flow (CBF) and the tissue residue function (R) using bolus-tracking MRI requires deconvolution of the arterial input function (AIF). Currently, the most commonly used deconvolution method is singular value decomposition (SVD), which has been shown to produce accurate estimations of CBF. However, this method introduces unwanted oscillations in the time course of R, and there are situations in which the actual shape is of interest (e.g., in calculating flow heterogeneity and assessing bolus dispersion). In such cases, the conventional SVD method may no longer be suitable, and an alternative approach may be required. This work describes the implementation of Tikhonov regularization with the L-curve criterion to quantify CBF and obtain a better characterization of R. The methodology is tested on simulated and patient data, and the results are compared to those found using the conventional SVD approach. Although both methods produce similar CBF values, the deconvolved R shape obtained using SVD is dominated by oscillations and fails to characterize the shape in the presence of dispersion. On the other hand, the use of the proposed regularization method improves the char- Key words: perfusion; dynamic susceptibility contrast MRI; deconvolution; singular value decomposition; arterial input function Bolus-tracking MRI, also known as dynamic susceptibility contrast MRI, is becoming a well-established technique for measuring cerebral blood flow (CBF) (1). The most important clinical application by far is in the investigation of cerebral ischemia. Although some limitations have been reported (2,3), this technique is playing an increasingly important role in the diagnosis, assessment, and management of acute stroke patients (4 -7).The fundamental equation for CBF quantification is (8):where C(t) is the tissue concentration time curve, C a (t) is the arterial input function (AIF; i.e., the concentration of contrast entering the tissue of interest at time t), R(t) is the tissue residue function (i.e., the fraction of contrast agent concentration at time t for the case of an ideal instantaneous bolus injected at t ϭ 0), and the symbol V indicates the convolution operation (1). Bolus-tracking MRI relies on the deconvolution of the AIF to obtain the product function CBF ⅐ R(t), and the calculation of CBF from the initial (or maximum) value of this function (8). Currently, the most common deconvolution method is singular value decomposition (SVD), which has been shown to produce fairly accurate estimations of CBF (8). However, this method introduces unwanted oscillations 1 in the shape of R(t) (9,10), and there are situations in which the actual shape of R(t)-not just its initial value-is of interest. For example, the determination of flow heterogeneity and oxygen delivery relies on characterization of the whole residue function (11). In addition, the presence of bolus dispersion has been shown to be a significant source of error in CBF quantification (11,12), and accurate determination of the sha...

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Section: Discussionmentioning

confidence: 99%

Quantification of bolus‐tracking MRI: Improved characterization of the tissue residue function using Tikhonov regularization

Calamante

Gadian

Connelly

2003

Magnetic Resonance in Med

123

129

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“…The truncation is expected to keep important components, since the directions removed from X k+ +1 are perpendicular to the current best approximation x k+1 , and also to the previous best approximations x k , x k−1 , …, x 1 . If the rotation and truncation are combined in one step, then the computational cost of the method is O(( + 1)(n + m + p 1 + · · · + p )), which quickly becomes smaller than the (re)orthogonalization cost as k grows.…”

Section: Subspace Expansion For Multiparameter Tikhonovmentioning

confidence: 99%

“…See for example [1,2,6,14,16,20,20]. In particular, there is no obvious multiparameter extension of the discrepancy principle.…”

Section: A Multiparameter Selection Strategymentioning

confidence: 99%

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Multidirectional Subspace Expansion for One-Parameter and Multiparameter Tikhonov Regularization

Zwaan

Hochstenbach

2016

J Sci Comput

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Tikhonov regularization is a popular method to approximate solutions of linear discrete ill-posed problems when the observed or measured data is contaminated by noise. Multiparameter Tikhonov regularization may improve the quality of the computed approximate solutions. We propose a new iterative method for large-scale multiparameter Tikhonov regularization with general regularization operators based on a multidirectional subspace expansion. The multidirectional subspace expansion may be combined with subspace truncation to avoid excessive growth of the search space. Furthermore, we introduce a simple and effective parameter selection strategy based on the discrepancy principle and related to perturbation results.This work is part of the research program "Innovative methods for large matrix problems", which is (partly) financed by the Netherlands Organization for Scientific Research (NWO).

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“…Methods for determining suitable regularization parameters for this minimization problem are discussed in [2,3,9,18].…”

Section: Multi-parameter Tikhonov Regularizationmentioning

confidence: 99%

A Golub–Kahan-Type Reduction Method for Matrix Pairs

Hochstenbach

Reichel

2015

J Sci Comput

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Efficient determination of multiple regularization parameters in a generalized L-curve framework

Cited by 208 publications

References 21 publications

Quantification of bolus‐tracking MRI: Improved characterization of the tissue residue function using Tikhonov regularization

Quantification of bolus‐tracking MRI: Improved characterization of the tissue residue function using Tikhonov regularization

Multidirectional Subspace Expansion for One-Parameter and Multiparameter Tikhonov Regularization

A Golub–Kahan-Type Reduction Method for Matrix Pairs

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