Edge detection and noise removal by use of a partial differential equation with automatic selection of parameters

Abstract

This work deals with noise removal by the use of an edge preserving method whose parameters are automatically estimated, for any application, by simply providing information about the standard deviation noise level we wish to eliminate. The desired noiseless image u(x), in a Partial Differential Equation based model, can be viewed as the solution of an evolutionary differential equation ut (x) = F(uxx, ux, u, x, t) which means that the true solution will be reached when t → ∞. In practical applications we should stop the time ‘‘t’’ at some moment during this evolutionary process. This work presents a sufficient condition, related to time t and to the standard deviation σ of the noise we desire to remove, which gives a constant T such that u(x, T) is a good approximation of u(x). The approach here focused on edge preservation during the noise elimination process as its main characteristic. The balance between edge points and interior points is carried out by a function g which depends on the initial noisy image u(x, t0), the standard deviation of the noise we want to eliminate and a constant k. The k parameter estimation is also presented in this work therefore making, the proposed model automatic. The model’s feasibility and the choice of the optimal time scale is evident through out the various experimental results.