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{\Large \bf Image Analysis and P.D.E.'s}
%ou bien : Image analysis and P.D.E.'s, ou bien Image local analysis and
%P.D.E.'s ou bien Image scale space and P.D.E's
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{\bf Fr\'{e}d\'{e}ric Guichard and Jean-Michel Morel }


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{\it {\bf Abstract.}  \newline

It is well known that a conveniently rescaled iterated convolution of a linear 
positive kernel converges to a gaussian.  As a consequence, all iterative linear
smoothing methods of a signal or an image boil down to the application to the signal
of the heat equation.  This book explains how a similar analysis can be performed for
image iterative smoothing by a wide class of nonlinear operators, the contrast invariant
operators.  These monotone operators have a property which makes them suitable  for 
image analysis: they commute with contrast changes of the images. Among them, the
median operator which computes a  local ``mean value" independent of constrast
changes.  We prove that all monotone and contrast invariant
operators, are asymptotically equivalent (when they become more and more
local) to a motion of the image by its curvature.  The iteration of these
filters is  equivalent to the application to the image of a
nonlinear heat equation.  We give in parallel a classification of all image
multiscale  smoothing methods (the so called ``scale space" methods in Computer
Vision).  It is shown that both approaches (classification of iterative
filtering methods or of ``scale spaces") yield the same, recently
discovered, partial differential equations. Experiments are presented with
both classical and new, contrast invariant and monotone numerical schemes.
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