Abstract | ||
---|---|---|
Shearlet theory has become a central tool in analyzing and representing 2D
data with anisotropic features. Shearlet systems are systems of functions
generated by one single generator with parabolic scaling, shearing, and
translation operators applied to it, in much the same way wavelet systems are
dyadic scalings and translations of a single function, but including a precise
control of directionality. Of the many directional representation systems
proposed in the last decade, shearlets are among the most versatile and
successful systems. The reason for this being an extensive list of desirable
properties: shearlet systems can be generated by one function, they provide
precise resolution of wavefront sets, they allow compactly supported analyzing
elements, they are associated with fast decomposition algorithms, and they
provide a unified treatment of the continuum and the digital realm.
The aim of this paper is to introduce some key concepts in directional
representation systems and to shed some light on the success of shearlet
systems as directional representation systems. In particular, we will give an
overview of the different paths taken in shearlet theory with focus on
separable and compactly supported shearlets in 2D and 3D. We will present
constructions of compactly supported shearlet frames in those dimensions as
well as discuss recent results on the ability of compactly supported shearlet
frames satisfying weak decay, smoothness, and directional moment conditions to
provide optimally sparse approximations of cartoon-like images in 2D as well as
in 3D. Finally, we will show that these compactly supported shearlet systems
provide optimally sparse approximations of an even generalized model of
cartoon-like images comprising of $C^2$ functions that are smooth apart from
piecewise $C^2$ discontinuity edges. |
Year | Venue | Keywords |
---|---|---|
2011 | CoRR | functional analysis,satisfiability,sparse approximation |
DocType | Volume | Citations |
Journal | abs/1109.5993 | 11 |
PageRank | References | Authors |
0.80 | 28 | 3 |
Name | Order | Citations | PageRank |
---|---|---|---|
Gitta Kutyniok | 1 | 325 | 34.77 |
Jakob Lemvig | 2 | 24 | 3.89 |
Wang-Q Lim | 3 | 195 | 9.41 |