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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B-YF. Istanbul 2004 
  
scale s vary defines the multiscale ridgelet dictionary 
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that is, a whole pyramid of local ridgelets at various lengths and 
locations. This is, of course, a massively overcomplete 
representation system and no formula like (9) is available for 
this multiscale ridgelet pyramid, because it is highly 
overcomplete. 
2.3 Discrete Ridgelet Transfrom(DRT) 
A basic strategy for calculating the continuous ridgelet 
transform is first to compute the Radon transform Rf (0,1) and 
second to apply a one-dimensional wavelet transform to the 
slices Af (0. -).. 
A fundamental fact about the Radon transform is the projection- 
slice formula(Deans, 1983) : 
f («cos 0, w sin 0) = [ron (10) 
This says that the Radon transform can be obtained by applying 
the one-dimensional inverse Fourier transform to the two- 
dimensional Fourier transform restricted to radial lines through 
the origin. 
This of course suggests that approximate Radon transforms for 
digital data can be based on discrete fast Fourier transforms. In 
outline, one simply does the following, 
1. 2D-FFT 
Compute the two-dimensional Fast Fourier Transform 
(FFT) of 1. 
2. Cartesian to polar conversion 
Using an interpolation scheme, substitute the sampled 
values of the Fourier transform obtained on the square 
lattice with sampled values of f on a polar lattice: that 
is, on a lattice where the points fall on lines through the 
origin. 
3. 1D-IFFT 
Compute the one-dimensional Inverse Fast Fourier 
Transform (IFFT) on each line; i.e., for each value of 
the angular parameter. 
The use of this strategy in connection with ridgelet transform 
has been discussed in the articles( Deans, 1983 ; Donoho, 1998). 
3. DIGITAL CURLET TRANSFORM 
The idea of curvelets (Candes er al, 1999;Starck et al, 
2002;Starck et al, 2003) is to represent a curve as a super- 
position of functions of various lengths and widths obeying the 
scaling law width = length” . This can be done by first 
decomposing the image into subbands, i.e., separating the 
object into a series of disjoint scales. Each scale is then 
analysed by means of a local ridgelet transform. 
61 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
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Radon Transform Ridgelet[Transform 
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Frequency 
Figure 1. Ridgelet transform flowgraph. Each of the 27 radial 
lines in the Fourier domain is processed separately. The 1-D 
inverse FFT is calculated along each radial line followed by a 
I-D nonorthogonal wavelet transform. In practice, the 1-D 
wavelet coefficients are directly calculated in the Fourier space. 
Curvelets are based on multiscale ridgelets combined with a 
spatial bandpass filtering operation to isolate different scales. 
This spatial bandpass filter nearly kills all multiscale ridgelets 
which are not in the frequency range of the filter. In other 
words, a curvelet is a multiscale ridgelet which lives in a 
prescribed frequency band. The bandpass is set so that the 
curvelet length and width at fine scales are related by a scaling 
law width = length” and so the anisotropy increases with 
decreasing scale like a power law. There is very special 
relationship between the depth of the multiscale pyramid and 
the index of the dyadic subbands; the side length of the 
localizing windows is doubled at every other dyadic subband, 
hence maintaining the fundamental property of the curvelet 
15 
transform which says that elements of length about 2^ serve 
for the analysis and synthesis of the / th subband [2,27]. 
While multiscale ridgelets have arbitrary dyadic length and 
arbitrary dyadic widths, curvelets have a scaling obeying 
width = length” . Loosely speaking, the curvelet dictionary is a 
subset of the multiscale ridgelet dictionary, but which allows 
reconstruction. 
The discrete curvelet transform of a continuum function 
f (31,x?) makes use of a dyadic sequence of scales, and a bank 
of filters (P f, A, f, A, f,L) with the property that the 
passband filter ^ is concentrated near the frequencies 
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