Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients

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), apply a variational method on the log data or shrink their coefficients in a frame (e.g. a wavelet basis), 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 hybrid criterion composed of an l data-fidelity to the thresholded coefficients and a Total Variation regularization (TV) term in the log-image domain; the restored image is an exponential of the obtained minimizer, weighted in a such 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. Theoretical results on our hybrid criterion are presented. For the minimization stage, we propose a properly adapted fast 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 clearly outperform the main alternative methods especially for images containing tricky geometrical structures.