Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients

TL;DR

This paper introduces a novel multiplicative noise removal method combining wavelet thresholding, variational optimization with L1 fidelity, and exponential reconstruction, outperforming existing techniques.

Contribution

It proposes a new multi-stage denoising approach that integrates thresholding and variational methods with a specialized L1 fidelity criterion and a fast Douglas-Rachford splitting scheme.

Findings

Numerical results show superior denoising performance.

The method effectively preserves image mean.

Convergence of the minimization scheme is proven.

Abstract

We address the denoising of images contaminated with multiplicative noise, e.g. speckle noise. Classical ways to solve such problems are filtering, statistical (Bayesian) methods, variational methods, and methods that convert the multiplicative noise into additive noise (using a logarithmic function), shrinkage of the coefficients of the log-image data in a wavelet basis or in a frame, and transform back the result using an exponential function. We propose a method composed of several stages: we use the log-image data and apply a reasonable under-optimal hard-thresholding on its curvelet transform; then we apply a variational method where we minimize a specialized criterion composed of an $\ell^1$ data-fitting to the thresholded coefficients and a Total Variation regularization (TV) term in the image domain; the restored image is an exponential of the obtained minimizer, weighted in a way that the mean of the original image is preserved. Our restored images combine the advantages of shrinkage and variational methods and avoid their main drawbacks. For the minimization stage, we propose a properly adapted fast minimization scheme based on Douglas-Rachford splitting. The existence of a minimizer of our specialized criterion being proven, we demonstrate the convergence of the minimization scheme. The obtained numerical results outperform the main alternative methods.