Edge Preserving Smoothing with Euclidean Shortening flow

Edge preserving smoothing is a locally adaptive process, where the size of the blurkernel, applied to suppress the noise, is a local function of the edge strength [2, 3]. Blurring is described by a partial differential equation, the diffusion equation. At strong edges, the 'conductivity' of the diffusion is reduced.This was first introduced into the realm of computer vision by Perona and Malik [4] in 1991. It was a huge success, as it also enhanced the remaining edges. But at strong edges, the noise remained, and there was a parameter k that had to be set. Alvarez [5] came up with an elegant solution for both issues, by proposing a new nonlinear image evolution scheme, where the local edge direction was taken into account. This solution is known as Euclidean Shortening Flow.We first give the theory and implementation of Perona & Malik nonlinear diffusion, then we focus on Euclidean shortening flow. The last section discusses an implementation on a noise ultrasound image. This paper is based on Chapter 21 of the book "Front-End Vision & Multi-Scale Image Analysis" (ter Haar Romeny, 2003)