Curvelets are a non-adaptive technique for multi-scale object representation. Being an extension of the wavelet concept, they are becoming popular in similar fields, namely in image processing and scientific computing.
Wavelets generalize the Fourier transform by using a basis that represents both location and spatial frequency. For 2D or 3D signals, directional wavelet transforms go further, by using basis functions that are also localized in orientation. A curvelet transform differs from other directional wavelet transforms in that the degree of localisation in orientation varies with scale. In particular, fine-scale basis functions are long ridges; the shape of the basis functions at scale j is
2-j
2-j/2
Curvelets are an appropriate basis for representing images (or other functions) which are smooth apart from singularities along smooth curves, where the curves have bounded curvature, i.e. where objects in the image have a minimum length scale. This property holds for cartoons, geometrical diagrams, and text. As one zooms in on such images, the edges they contain appear increasingly straight. Curvelets take advantage of this property, by defining the higher resolution curvelets to be more elongated than the lower resolution curvelets. However, natural images (photographs) do not have this property; they have detail at every scale. Therefore, for natural images, it is preferable to use some sort of directional wavelet transform whose wavelets have the same aspect ratio at every scale.
When the image is of the right type, curvelets provide a representation that is considerably sparser than other wavelet transforms. This can be quantified by considering the best approximation of a geometrical test image that can be represented using only
n
n
O(1/\sqrt{n})
O(1/n)
O({(logn)}3/{n2})
Efficient numerical algorithms exist for computing the curvelet transform of discrete data. The computational cost of the discrete curvelet transforms proposed by Candès et al. (Discrete curvelet transform based on unequally-spaced fast Fourier transforms and based on the wrapping of specially selected Fourier samples) is approximately 6–10 times that of an FFT, and has the same dependence of
O(n2logn)
n x n
To construct a basic curvelet
\phi
The number of wedges is
Nj=4 ⋅
| ||||||
2 |
2-j
Let
\boldsymbol{\xi}=\left(\xi1,\xi2\right)T
2}, | |
r=\sqrt{\xi | |
2 |
\omega=\arctan
\xi1 | |
\xi2 |
We use the ansatz for the dilated basic curvelets in polar coordinates:
\hat{\phi}j,0,0
| ||||
:=2 |
W(2-j
r)\tilde{V} | |
Nj |
(\omega),r\ge0,\omega\in[0,2\pi),j\inN0
To construct a basic curvelet with compact support near a ″basic wedge″, the two windows
W
\tilde{V} | |
Nj |
W(r)
(0,infty)
\tilde{V} | |
Nj |
\tilde{V} | |
Nj |
Then the admissibility yields
infty | |
\sum | |
j=-infty |
\left|W(2-jr)\right|2=1,r\in(0,infty).
N
N
2\pi
\tilde{V}N
\left[
-2\pi | |
N |
,
2\pi | |
N |
\right]
N-1 | |
\sum | |
l=0 |
2 | ||
\tilde{V} | \left(\omega- | |
N |
2\pil | |
N |
\right)=1
\omega\in\left[0,2\pi\right)
\tilde{V}N
2\pi
V\left(
N\omega | |
2\pi |
\right)
Nj-1 | |
\sum | |
l=0 |
\left|
| ||||
2 |
\hat{\phi}j,0,0\left(r,\omega-
2\pil | |
Nj |
\right)\right|2=\left|W(2-jr)
Nj-1 | |
\right| | |
l=0 |
2 | ||
\tilde{V} | \left(\omega- | |
Nj |
2\pil | |
N |
\right)=\left|W(2-jr)\right|2
For a complete covering of the frequency plane including the region around zero, we need to define a low pass element
\hat{\phi}-1:=W0(\left|\xi\right|)
2(r) | |
W | |
0 |
infty | |
j=0 |
W(2-jr)2