From Tensor-Driven Diffusion to Anisotropic Wavelet Shrinkage

Diffusion processes driven by anisotropic diffusion tensors are known to be well-suited for structure-preserving denoising. However, numerical implementations based on finite differences introduce unwanted blurring artifacts that deteriorate these favourable filtering properties. In this paper we introduce a novel discretisation of a fairly general class of anisotropic diffusion processes on a 2-D grid. It leads to a locally semi-analytic scheme (LSAS) that is absolutely stable, simple to implement and offers an outstanding sharpness of filtered images. By showing that this scheme can be translated into a 2-D Haar wavelet shrinkage procedure, we establish a connection between tensor-driven diffusion and anisotropic wavelet shrinkage for the first time. This result leads to coupled shrinkage rules that allow to perform highly anisotropic filtering even with the simplest wavelets.