Selection of optimal stopping time for nonlinear diffusion filtering

Pavel Mrazek
Center for Machine Perception
Czech Technical University
http://cmp.felk.cvut.cz/

Introduction

Many image restoration techniques start from a noisy image and create a sequence of possible solutions. To obtain a good filtering result, we have to choose one image from this sequence. In the context of nonlinear diffusion, the level of filtering is influenced by the time of the diffusion process: at early stages, the diffused image is close to the input data, later the results become more simplified and dominated by the (piecewise constant) model inherent in the NL diffusion equations. So, what stopping time T to choose, and when to stop the filtering procedure?

Main idea

Let f stand for the ideal, noise-free data, n for noise; the filtering starts from the input image u(0)=f+n and creates a series u(t).

In [1] we exploit the following idea: if the noise n in the input image is uncorrelated with the data f, and if the filtering procedure removes the noise first before influencing important data features, then we want to stop when the 'filtering noise' u(0)-u(T) is uncorrelated with the 'filtered signal' u(T).

In reality, the filtering procedure removes noise and blurs the data simultaneously. However, we found that minimizing the correlation of the 'noise' u(0)-u(T) and the 'signal' u(T) with respect to T gives often a very good estimate of the optimal stopping time for which the filtered result u(T) is closest to the ideal data f.

Examples

	Triangle and rectangle artificial data, non-Gaussian noise, shows that the stopping time selection can be combined with various filtering methods
	Cymbidium data - combination of the autonomous stopping time selection with anisotropic nonlinear diffusion filter, applied to images with varied amount of additive Gaussian noise
	Baboon - natural texture, several NL diffusion filters, with and without additional noise
	Inclined surface - artificial noisy data, time selection used with classical, and monotonicity-enhancing anisotropic NL diffusion filter

Further information