Summary: ENO related techniques in 2D
In the 80's, ENO (essentially non oscillatory) schemes have been de-
veloped by A. Harten. These schemes produce smooth reconstruction of
functions from discrete data, e.g. point values or cell averages of a function.
While similar linear schemes only allow good reconstructions in smooth re-
gions, sufficiently far away from singularities of the underlying function, the
nonlinear ENO procedure works well in a much larger region. an additional
technique, also proposed by Harten, is the so called subcell resolution tech-
nique, wich enlarges the region of good approximation for an ENO scheme to
an optimal extent (but, on the other hand, also brings up serious questions
of stability). This good resolution of singularities makes ENO and subcell
resolution techniques especially interesting for the use in numerical schemes
for conservation laws (mainly in connection with Ţnite volume schemes).
Very much in the spirit of wavelet theory, ENO schemes can also be
used to construct nonlinear multiresolution anlyses. Then, the interesting
features of ENO and subcell resolution promise an even smaller number of
relevant datail coefficients close to the singularities of a function. Equipped
with efficient coding strategies, these ENO multiresolution analyses seem to
be very interesting also for the compression of images, since the edges in an
image correspond to curves of singularities of a function in 2D.