Effective new methods for automated parameter selection in regularized inverse problems

Sanders, Toby; Platte, Rodrigo B.; Skeel, Robert D.

doi:10.1016/j.apnum.2020.01.015

Cited by 15 publications

(12 citation statements)

References 37 publications

(69 reference statements)

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“…Here, T is the regularization operator often set as a first or second order finite difference operator [13,14]. For example in 1D, the first order finite difference operators is given as…”

Section: Regularization Operatorsmentioning

confidence: 99%

“…The extension of these operators to 2D images is done naturally by taking differences in both the vertical and horizontal dimensions, and mathematically this can be handled by taking appropriate Kronecker products (see [14] for details). For deconvolution problems, there is a major computational advantage to writing the operators appearing in (11) as circulant, making these operators convolutional operators.…”

Section: Regularization Operatorsmentioning

confidence: 99%

“…For deconvolution problems, there is a major computational advantage to writing the operators appearing in (11) as circulant, making these operators convolutional operators. Hence, when T is a rth order finite difference operator, then it is also diagonalized by the Fourier transform given by (see [13,14] for details)…”

Section: Regularization Operatorsmentioning

confidence: 99%

See 2 more Smart Citations

Fixed Point Iterations for SURE-based PSF Estimation for Image Deconvolution

Sanders¹

2022

Preprint

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Stein's unbiased risk estimator (SURE) has been shown to be an effective metric for determining optimal parameters for many applications. The topic of this article is focused on the use of SURE for determining parameters for blind deconvolution. The parameters include those that define the shape of the point spread function (PSF), as well as regularization parameters in the deconvolution formulas. Within this context, the optimal parameters are typically determined via a brute for search over the feasible parameter space. When multiple parameters are involved, this parameter search is prohibitively costly due to the curse of dimensionality. In this work, novel fixed point iterations are proposed for optimizing these parameters, which allows for rapid estimation of a relatively large number of parameters. We demonstrate that with some mild tuning of the optimization parameters, these fixed point methods typically converge to the ideal PSF parameters in relatively few iterations, e.g. 50-100, with each iteration requiring very low computational cost.

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“…Here, T is the regularization operator often set as a first or second order finite difference operator [13,14]. For example in 1D, the first order finite difference operators is given as…”

Section: Regularization Operatorsmentioning

confidence: 99%

Section: Regularization Operatorsmentioning

confidence: 99%

See 1 more Smart Citation

Fixed Point Iterations for SURE-based PSF Estimation for Image Deconvolution

Sanders¹

2022

Preprint

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“…To this end we note that typically µ is chosen to reflect both the SNR given in (4.1) and the regularity in the signal. For instance, µ might be chosen to approximate σ ξ , where σ is the the standard deviation of the complex noise η in (2.5) and ξ is the standard deviation of data in the transformed domain under L is used in [49]. Since σ and ξ are explicitly known in our test problems, to obtain the "best" case scenario we simply pick samples from N σ ξ , σ .…”

Section: Individual Image Recoverymentioning

confidence: 99%

Sequential image recovery from noisy and under-sampled Fourier data

Xiao,

Glaubitz,

Gelb

et al. 2022

Preprint

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A new algorithm is developed to jointly recover a temporal sequence of images from noisy and under-sampled Fourier data. Specifically we consider the case where each data set is missing vital information that prevents its (individual) accurate recovery. Our new method is designed to restore the missing information in each individual image by "borrowing" it from the other images in the sequence. As a result, all of the individual reconstructions yield improved accuracy. The use of high resolution Fourier edge detection methods is essential to our algorithm. In particular, edge information is obtained directly from the Fourier data which leads to an accurate coupling term between data sets. Moreover, data loss is largely avoided as coarse reconstructions are not required to process inter-and intra-image information. Numerical examples are provided to demonstrate the accuracy, efficiency and robustness of our new method.

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“…However, an often encountered difficulty for the ℓ 1 -regularized inverse problem (1.2) is the selection of an appropriate regularization parameter λ. This parameter can critically influence the quality of the regularized reconstruction x λ [31,21,35,46,41]. In part for this reason, many statistical approaches have been proposed for regularized inverse problems [38,12,47].…”

Section: Introduction Many Applications Seek To Solve the Linear Inve...mentioning

confidence: 99%

Generalized sparse Bayesian learning and application to image reconstruction

Glaubitz¹,

Gelb²,

Song³

2022

Preprint

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Image reconstruction based on indirect, noisy, or incomplete data remains an important yet challenging task. While methods such as compressive sensing have demonstrated high-resolution image recovery in various settings, there remain issues of robustness due to parameter tuning. Moreover, since the recovery is limited to a point estimate, it is impossible to quantify the uncertainty, which is often desirable. Due to these inherent limitations, a sparse Bayesian learning approach is sometimes adopted to recover a posterior distribution of the unknown. Sparse Bayesian learning assumes that some linear transformation of the unknown is sparse. However, most of the methods developed are tailored to specific problems, with particular forward models and priors. Here, we present a generalized approach to sparse Bayesian learning. It has the advantage that it can be used for various types of data acquisitions and prior information. Some preliminary results on image reconstruction/recovery indicate its potential use for denoising, deblurring, and magnetic resonance imaging.

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Effective new methods for automated parameter selection in regularized inverse problems

Cited by 15 publications

References 37 publications

Fixed Point Iterations for SURE-based PSF Estimation for Image Deconvolution

Fixed Point Iterations for SURE-based PSF Estimation for Image Deconvolution

Sequential image recovery from noisy and under-sampled Fourier data

Generalized sparse Bayesian learning and application to image reconstruction

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