Image processing


The goal of image restoration is to filter out noise or patterns lacking coherence (like inhomogeneous tissues in medical imaging) while preserving as much as possible the edges of the coherent patterns. This requires the derivation  of nonlinear diffusion models.
Most available models in this field have a scale-time interpretation: as time goes on, more and more small scales get removed from the image. A stopping time needs therefore to be given,  on the basis in principle of the minimal scales to be retained. In practice this time parameter depends on the noise level, and large scale edges have to be traced back on the processed image, making these models  inadequate for highly degraded images.
Our approach between 92 and 98 has been to  consider the image as a perturbed version of a unique true image that we wish to obtain on the asymptotics of  our diffusion models.  This is done by coupling a diffusion equation with an evolution equation for the diffusion tensor which trains the filter to recognize the significant gradients of the image

Main features of the algorithm

  • built-in anisotropy through a diffusion tensor
  • the attractor of the system depend on a contrast parameter
  • integral approximation of the system leads to an integro-differential system that can be interpreted as a neural network with Hebbian learning rules
  • reinitialization of the filter with the processed gradients and the original image allows contrast enhancement


We present here the results of the algorithm on synthetic and real images. In the second example, the filter is used in a pre-processing stage to enable active contour techniques to detect the edges of the 2 ventricles in an ultrasound image of the heart.

Triangle-rectangle test case. 
A noisy image : 70% of the pixels have random grey values    The processed image after thresholding 

Ultrasound image processing. 
Top-left: original image; top-right: processed image; bottom-left: initialization of the snakes; bottom-right: the snakes, running on the processed images, have successfully converged to the edges of the ventricles  


G.-H. Cottet and L. Germain, Image Processing through Reaction Combined with Nonlinear Diffusion, Mathematics of Computation, 61, 659-673, 1993. pdf

G.-H. Cottet, Neural networks: continuous approach and applications to image processing, J. Biological Systems, 3, 1995.
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G.-H. Cottet and M. El Ayyadi,  A Volterra type model for image processing,   IEEE Transactions on image processing,  7, 1998.
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