A New Detail-Preserving Regularization Scheme

Abstract

It is a challenging task to reconstruct images from their noisy, blurry, and/or incomplete measurements, especially those with important details and features such as medical magnetic resonance (MR) and CT images. We propose a novel regularization model that integrates two recently developed regularization tools: total generalized variation (TGV) by Bredies, Kunisch, and Pock; and shearlet transform by Labate, Lim, Kutyniok, and Weiss. The proposed model recovers both edges and fine details of images much better than the existing regularization models based on the total variation (TV) and wavelets. Specifically, while TV preserves sharp edges but suffers from oil painting artifacts, TGV "selectively regularizes" different image regions at different levels and thus largely avoids oil painting artifacts. Unlike the wavelet transform, which represents isotropic image features much more sparsely than anisotropic ones, the shearlet transform can efficiently represent anisotropic features such as edges, curves, and so on. The proposed model based on TGV and the shearlet transform has been tested in the compressive sensing context and produced high-quality images using fewer measurements than the state-of-the-art methods. The proposed model is solved by splitting variables and applying the alternating direction method of multiplier (ADMM). For certain sensing operators, including the partial Fourier transform, all the ADMM subproblems have closed-form solutions. Convergence of the algorithm is briefly mentioned. The numerical simulations presented in this paper use the incomplete Fourier, discrete cosine, and discrete wavelet measurements of MR images and natural images. The experimental results demonstrate that the proposed regularizer preserves various image features (including edges and textures), much better than the TV/wavelet based methods.

Image Reconstruction from Undersampled Fourier Data Using the Polynomial Annihilation Transform

“…2, the approximation is readily improved by employing the PA transform (16), for the l 1 regularization term in (2). Algorithm 2 was used in all cases with the number of iterations ranging from 30 − 40.…”

“…As mentioned in the introduction, in [28], the MATLAB CVX package was used to implement (2) with (16). While suitable for one-dimensional problems, it is not efficient for multiple dimensions.…”

“…While suitable for one-dimensional problems, it is not efficient for multiple dimensions. Below we describe how the split Bregman algorithm [12], can be adapted to incorporate the PA transform (16), into (2), and thus improve the overall efficiency.…”

“…We begin by defining J x := L m x and J y := L m y in (19), where L m x and L m y are the respective directional PA transform operators defined by in one dimension by (16). Bregman iteration of (19) is then given by the sequence of two unconstrained optimization problems of the form…”

Space-Time Regularization for Video Decompression

“…Many of the mathematical methods found in the literature for recovering compressed sensing data focus on the reconstruction of single image data-for example, those working on the preservation of edges [13,38,17,27,3] or texture [11,32,18]. However, the temporal aspect of videos introduces the need for different types of regularization which leverage the particular structure of the data and the compression.…”