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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, August 15, 2000}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Jean-Pierre Antoine}
\centerline{\sl Inst. Theoretical Physics}
\centerline{\sl Cath. Univ. Louvain}
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\centerline{\bf Two-dimensional directional wavelets as a symmetry scanner}
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It is well-known that the  2-D continuous wavelet transform (CWT) is a powerful
tool for detecting various features in a picture or a pattern. If the
relevant features  have a preferred direction, the  analysis tool necessary
for detecting them is the full 2-D CWT, including the  rotation parameter,
in addition to the usual translations and dilations, and  one must choose a
wavelet with some directional selectivity, such as a Morlet or a conical
wavelet. This is standard, for instance, in directional filtering or edge
detection, two
classical problems in image processing. In this talk, we will describe
another application
of  such directional wavelets, namely the determination of the symmetries
of a pattern.

Given a 2-D signal (an object, a pattern,$\ldots$),  its scale-angle measure
(or wavelet spectrum) is defined as the integral over positions of the
squared modulus of its wavelet transform. This quantity may also be viewed
as a partial energy density in the scale and angle variables, that is, in
spatial frequency space.   When  the object is invariant (even
approximately, or statistically), under discrete rotations,   discrete
dilations, or a combination of both, its scale-angle  measure is a periodic
function of the scale and angle variables. Thus, the symmetries in question
can be read off the scale-angle measure. In addition, a voting algorithm
allws one to list exaustively all such symmetries.

This is illustrated on several types of pictures: a ``twisted snowflake'';
a Penrose tiling; various  dot patterns (octagonal, decagonal, $\dots$),
closely linked to particular quasi-periodic lattices; a pattern generated
by the Faraday instability in fluid dynamics;  quasi-periodic patterns
derived from the root diagram of certain infinite-dimensional Lie algebras.
The technique could also be  used for uncovering hidden symmetries of
physical objects, such as quasicrystals or nanotubes, through their X-ray
diffraction patterns.

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