A nonlinear fourth-order PDE for image denoising in Sobolev spaces with variable exponents and its numerical algorithm

Image restoration is a very challenging task in image analysis and plays important roles in various fields such as medical imaging. In particular, in ultrasound imaging the obtained images are usually highly corrupted with multiplicative noise which makes important features hard to detect and to preserve. In this work, we use a mathematical model based on a minimization problem. To preserve the important features of the image, we consider a variable exponent function p(x) chosen adaptively based on the map provided by edge-detectors which are constructed form high-order derivatives. The Euler-Lagrange equation of the minimization problem gives rise to a nonlinear p(x)-biharmonic PDE. We then propose a numerical scheme based on the convexity splitting (CS) method for the ultrasound image denoising and we prove its stability and convergence results. Finally, some numerical results are presented to illustrate the effectiveness of our approach.