Non-local regularization of inverse problems

Abstract

This article proposes a new framework to regularize linear inverse problems using the total variation on non-local graphs. This nonlocal graph allows to adapt the penalization to the geometry of the underlying function to recover. A fast algorithm computes iteratively both the solution of the regularization process and the non-local graph adapted to this solution. We show numerical applications of this method to the resolution of image processing inverse problems such as inpainting, super-resolution and compressive sampling.

Sensitivity encoding reconstruction with nonlocal total variation regularization

“…It suggests that the proposed method is rather insensitive to the choice of d and m within a range of values. We empirically This choice is consistent with those used in other nonlocal TV literatures (18)(19)(20)(21)(22)(23)(24) with similar image sizes. Figure 8 shows the compressed sensing reconstruction results using NLTV and TV regularizations with a net reduction factor of 4.…”

“…The parameter d > 0 controls the nonlocality of the method and also allows speeding up computation. The parameter r corresponds to the noise level in general and is usually set to be the standard deviation of the noise (20,23,24). f(p þ) and f(q þ) are vectors representing the neighborhoods of pixels p and q of image f, respectively.…”

“…In this study, we investigate NLTV for SENSE regularization to address the issue of blocky effect with TV-regularized SENSE. NLTV has demonstrated its superior performance to TV (21)(22)(23)(24) and also nonlocal H 1 (18,20) in other regularization applications. In NLTV, the gradient for the regularization term is calculated with pixels belonging to the whole image, instead of only the nearest neighboring pixels as used in TV regularization.…”

“…A drawback of TV and MRF-based regularization methods is that they both use local information only, which may cause blocky effects with a loss of fine structures while preserving edges in reconstruction. Nonlocal regularization methods (18)(19)(20)(21)(22)(23)(24) such as nonlocal TV (NLTV) and nonlocal H 1 have recently been studied in image denoising to address the issue of blocky effect by employing nonlocal pixels for calculating the gradients in the regularization term. Nonlocal H 1 is regarded as a variational equivalent of the nonlocal mean filter and has been used in SENSE reconstruction for improving the signal-to-noise ratio (25).…”

Image denoising for real‐time MRI

“…However, the regularization of global image properties such as total variation entails spurious block‐shaped artifacts or loss of details when applied to in vivo data. New methods based on nonlocal regularization combine patch‐based denoising with iterative reconstruction and outperform global methods. The 4D collaborative filtering presents another notable approach in the same context.…”