bul 2004
Ticients
> Radon
(7)
1ofa |-
Radon
ind / is
12° ie
all such
| dyadic
=Qg we
a nice
ve dilate
lucing a
;, make
ransport
(8)
jares at a
dO
4
(9)
h
sent any
1e form
near the
ridgelet
0=0
rted near
of local
tting the
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
D
(ye ois2 sS QeQ,. a» 0, be RO c[0,27)) by
Q^
Yabo T OI WV, po
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
a 1 wo
Radon Transform Ridgelet[Transform
2
e
<
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
2s+2
]. €.2.
2s
[2.2