Dror Baron 
Compression of multidimensional functions with discontinuities
Discontinuities in data often provide vital information,
and representing these discontinuities sparsely is an important
goal for approximation and compression algorithms.
We considered an Mdimensional horizon class, in which
Mdimensional functions contain a smooth (M1)dimensional
singularity separating smooth regions. Such a function in M=2
dimensions appears above.
We characterized the metric entropy of these signals,
and provided multiscale representations that enable to achieve
the metric entropy. Our constructive solutions rely on a
hierarchical geometric tiling, where each atom of our tiling
dictionary contains polynomial surfaces  and is thus called
a surflet. An important insight for compression
is that the large number of higherorder coefficients need not
take too much coding length, because they can be quantized
coarsely. Additionally, coefficients are correlated between
scales, and prediction is used to reduce coding length.

V. Chandrasekaran,
M. B. Wakin,
D. Baron,
and R. G. Baraniuk,
"Representation and Compression of MultiDimensional Piecewise Functions
Using Surflets,"
IEEE Transactions on Information Theory,
vol. 55, No. 1, pp. 374400, January 2009
(pdf).

V. Chandrasekaran,
M. B. Wakin,
D. Baron,
and R. G. Baraniuk,
"Surflets: A Sparse Representation for Multidimensional Functions
Containing Smooth Discontinuities,"
2004 IEEE International Symposium on Information Theory
(ISIT2004), Chicago, IL, June 2004
(pdf).

V. Chandrasekaran,
M. B. Wakin,
D. Baron,
and R. G. Baraniuk,
"Compression of Higher Dimensional Functions Containing Smooth
Discontinuities,"
Proceedings of 38th Annual Conference on Information Sciences and
Systems (CISS2004), Princeton, NJ, March 2004
(pdf).
Slides from a talk I gave on this topic in June 2009 appear
here.
Last updated June 2009