It is now widely acknowledged that analyzing the intrinsic geometrical
features of a function/signal is essential in many applications.
In order to achieve this, several directional systems have been proposed
in the past.
The first breakthrough was achieved by Candes and Donoho who introduced
curvelets and showed that
curvelets provide an optimal approximation property for a special
class of 2D piecewise smooth functions, called cartoon-like images.
Since then, various directional systems have been proposed. However, only
band-limited directional systems providing an optimal approximation
have been constructed so far, except adaptive representation schemes.
In this talk, we will discuss a novel approach to construct
shearlets which not only have compact support in spatial domain but also
provide optimally sparse approximation of cartoon-like images in both 2D
We then show that shearlets can even provide optimal sparse approximation
for a large class of piecewise smooth functions.
Finally, we will discuss some applications of compactly supported shearlets
for imaging science and PDE.