07-06-2013, 03:31 PM
What is Curvelet?
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INTRODUCTION
Energized by the success of wavelets, the last two
decades saw the rapid development of a new field,
computational harmonic analysis, which aims to
develop new systems for effectively representing
phenomena of scientific interest. The curvelet transform
is a recent addition to the family of mathematical
tools this community enthusiastically builds
up. In short, this is a new multiscale transform with
strong directional character in which elements
are highly anisotropic at fine scales, with effective
support shaped according to the parabolic scaling
principle length2 ∼ width.
Curvelets for What?
It is well known that discontinuities destroy the
sparsity of a Fourier series. This is the Gibbs
phenomenon; we need many, many terms to reconstruct
a discontinuity to within good accuracy.
Wavelets, because they are localized and multiscale,
do much better in one dimension, but because
of their poor orientation selectivity, they do not represent
higher-dimensional singularities effectively.
What makes curvelets interesting and actually
motivated their development is that they provide
a mathematical architecture that is ideally adapted
for representing objects which display curvepunctuated
smoothness—smoothness except for
discontinuity along a general curve with bounded
curvature—such as images with edges, for example.
Beyond Scale-Space?
A beautiful thing about mathematical transforms
is that they may be applied to a wide variety of problems
as long as they have a useful architecture. The
Fourier transform, for example, is much more than
a convenient tool for studying the heat equation
(which motivated its development) and, by extension,
constant-coefficient partial differential equations.
The Fourier transform indeed suggests a
fundamentally new way of organizing information
as a superposition of frequency contributions, a
concept which is now part of our standard repertoire.
In a different direction, we mentioned before
that wavelets have flourished because of their
ability to describe transient features more accurately
than classical expansions. Underlying this
phenomenon is a significant mathematical architecture
that proposes to decompose an object
into a sum of contributions at different scales and
locations. This organization principle, sometimes
referred to as scale-space, has proved to be very
fruitful—at least as measured by the profound
influence it bears on contemporary science.
Multiscale Geometric Analysis?
Curvelets are new multiscale ideas for data representation,
analysis, and synthesis which, from a
broader viewpoint, suggest a new form of multiscale
analysis combining ideas of geometry and multiscale
analysis. Of course, curvelets are by no means
the only instances of this vision which perceives
those promising links between geometry and multiscale
thinking. There is an emerging community
of mathematicians and scientists committed to
the development of this field. In January 2003, for
example, the Institute for Pure and Applied Mathematics
at UCLA, newly funded by the National Science
Foundation, held the first international workshop
on this topic. The title of this conference: Multiscale
Geometric Analysis.