A review of P.D.E. models in image processing and image analysis

The years 1985-2000 have seen the emergence of several nonlinear P.D.E. models in image restoration and image analysis. Before that date, the heat equation and the reverse heat equation had been considered as relevant, one as a model of image smoothing compatible with Shannon conditions, and one as a restoration model proposed by Gabor. We try in this review to organize the P.D.E. models according to their genealogy from the initial heat equation and according to their very diverse use: some are useful for image denoising, some for image deblurring, some for invariant smoothing in view of shape recognition. Some permit to define easily active contours (snakes), some may be used for a nonlinear interpolation of sparse images. We show many experiments illustrating these different applicative aspects.