April 22, 2012

Hyperbolets (on WITS: Where is the Starlet)

A new conference is born: UCCV 2013, The 1st IEEE Workshop on User-Centred Computer Vision, due in Florida, Tampa. on January 2013. It has been added to SIVA Conferences.

While the shearlets are enjoying some spread (cf. Shearlets from MIA 2012 or this paper), some of their contributors are involved in hyperbolets, or hyperbolic wavelets, closely related cousins. Here they are (as on WITS: where is the Starlet):

Hyperbolets

In short: An example of multi-composite wavelets with hyperbolic scaling law
Etymology: From the hyperbola (wiki entry), with a potential reference (article no available on 2011/05/26) to the parabolic scaling law of the shearlets
Origin: Glenn R. Easley, Demetrio Labate, Vishal M. Patel: Multi-composite wavelet estimation, Proceedings of SPIE Volume 8138, Wavelets and Sparsity XIV, Aug. 2011 (local copy)
Abstract: In this work, we present a new approach to image denoising by using a general representation known as wavelets with composite dilations. These representations allow for waveforms to be defined not only at various scales and locations but also at various orientations. For this talk, we present many new representations such as hyperbolets and propose combining multiple estimates from various representations to form a unique denoised image. In particular, we can take advantage of different representations to sparsely represent important features such as edges and texture independently and then use these estimates to derive an improved estimate.
The hyperbolet construction is further refined in:
G. R. Easley, D. Labate and V. M. Patel, Hyperbolic shearlets, IEEE International Conference on Image Processing (ICIP), Orlando, FL, 2012, submitted (local copy)
G. R. Easley, D. Labate, and V. M. Patel, Directional multiscale processing of images using wavelets with composite dilations, submitted 2011 (local copy)
Contributors: Glenn R. Easley (no personal page), Demetrio Labate, Vishal M. Patel
Some properties:
hyperbolet frequency plane

Tiling of the frequency domain associated with an hyperbolic system of wavelets with composite dilations.
Closely related to shearlets
Anecdote:
Usage:
See also: The above work might be related to Glenn R. Easley, Demetrio Labate: Critically Sampled Wavelets with Composite Dilations (local copy), preprint, 2011, which develops interesting critically sampled directional wavelet schemes (DWTShear, CShear, QDWTShear)
Comments:

More on the topic:
2D wavelets: A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity