Directional TGV-Based Image Restoration under Poisson Noise

Abstract

We are interested in the restoration of noisy and blurry images where the texture mainly follows a single direction (i.e., directional images). Problems of this type arise, for example, in microscopy or computed tomography for carbon or glass fibres. In order to deal with these problems, the Directional Total Generalized Variation (DTGV) was developed by Kongskov et al. in 2017 and 2019, in the case of impulse and Gaussian noise. In this article we focus on images corrupted by Poisson noise, extending the DTGV regularization to image restoration models where the data fitting term is the generalized Kullback–Leibler divergence. We also propose a technique for the identification of the main texture direction, which improves upon the techniques used in the aforementioned work about DTGV. We solve the problem by an ADMM algorithm with proven convergence and subproblems that can be solved exactly at a low computational cost. Numerical results on both phantom and real images demonstrate the effectiveness of our approach.

Automatic parameter selection for the TGV regularizer in image restoration under Poisson noise

“…The ADMM with the adaptive adjustment of the regularization parameter λ has been implemented starting from the implementation of the method used in [16,17], available as a MATLAB code at https://github.com/diserafi/respond. The outer iterations of the alternating scheme as well as the inner ADMM iterations are stopped as soon as…”

“…By following the computations carried out in [16], briefly reported here for the sake of completeness, and after introducing the variables x := (u, w) and z := (z 1 , z 2 , z 3 , z 4 ) and the matrices…”

“…Under the adoption of periodic boundary conditions, the matrices A, D h and D v are Block Circulant with Circulant Blocks (BCCB), and they can be thus diagonalized by means of the 2D Fast Fourier Transform (FFT). By exploiting this property, which is block-wise inherited by the matrix H T H, in [16] the authors have shown that the update of x, that is the update of u and w = (w 1 , w 2 ), can be obtained in a very efficient manner by three applications of the forward 2D FFT and three applications of the inverse 2D FFT. We refer the reader to [16] for the extended computations.…”

“…By exploiting this property, which is block-wise inherited by the matrix H T H, in [16] the authors have shown that the update of x, that is the update of u and w = (w 1 , w 2 ), can be obtained in a very efficient manner by three applications of the forward 2D FFT and three applications of the inverse 2D FFT. We refer the reader to [16] for the extended computations.…”

upU-Net Approaches for Background Emission Removal in Fluorescence Microscopy

“…The first term is the fit–to–data or data fidelity , which measures the discrepancy between the registered data and the recovered image. The choice for this functionality depends on the noise affecting the data: in presence of Gaussian noise the most suitable one is the Least Square functional [ 3 ], whilst in presence of Poisson noise the (generalized) KullBack–Leibler function [ 4 , 5 ] is widely employed. The second term occurring in the minimization is the so-called regularization term, which has the role of considering some characteristics of the desired image and controlling the influence of the noise in the reconstruction.…”