25-08-2012, 10:28 AM
Introduction of Curvelet Transform
Curvelets.ppt (Size: 1.61 MB / Downloads: 251)
Introduction
Curvelet Transform is a new multi-scale representation most suitable for objects with curves.
Developed by Candès and Donoho (1999).
Still not fully matured.
Seems promising, however.
Point and Curve Discontinuities
A discontinuity point affects all the Fourier coefficients in the domain.
Hence the FT doesn’t handle points discontinuities well.
Using wavelets, it affects only a limited number of coefficients.
Hence the WT handles point discontinuities well.
Discontinuities across a simple curve affect all the wavelets coefficients on the curve.
Hence the WT doesn’t handle curves discontinuities well.
Curvelets are designed to handle curves using only a small number of coefficients.
Hence the CvT handles curve discontinuities well.
Curvelet Transform
The Curvelet Transform includes four stages:
Sub-band decomposition
Smooth partitioning
Renormalization
Ridgelet analysis
Sub-band Decomposition
The sub-band decomposition can be approximated using the well known wavelet transform:
Using wavelet transform, f is decomposed into S0, D1, D2, D3, etc.
P0 f is partially constructed from S0 and D1, and may include also D2 and D3.
s f is constructed from D2s and D2s+1.
Ridgelet Analysis
Each normalized square is analyzed in the ridgelet system:
The ridge fragment has an aspect ratio of 2-2s2-s.
After the renormalization, it has localized frequency in band ||[2s, 2s+1].
A ridge fragment needs only a very few ridgelet coefficients to represent it.
Digital Ridgelet Transform (DRT)
Unfortunately, the (current) DRT is not truly orthonormal.
An array of nn elements cannot be fully reconstructed from nn coefficients.
The DRT uses n2n coefficients for almost perfect reconstruction
Still a lot of research need to be done…