International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B-YF. Istanbul 2004 
  
  
2. CONTINUOUS RIDGELET TRANSFORM 
The two-dimensional continuous ridgelet transform in R° can 
be defined as follows(Candes, 1999; Starck et al, 2002). We 
pick a smooth univariate function y:R—R with sufficient 
decay and satisfying the admissibility condition 
car (|l ae « o (1) 
which holds if, say, v has a vanishing mean [y (r)dt=0 - We 
will suppose that vw is normalized so that n 
0 
each b e R and each 0 e[0,2z) , we 
2.52.92 4 
lv ca^ ^aést: 
For each a» 0, 
= Baris : 2 2 
define the bivariate ridgelet V bo R' —R by 
{x)= aux, cos 0 * x, sinQ — b)/a) (2) 
Y a.b,0 
A ridgelet is constant along lines x, cos 0 + x, sin 0 = const . 
Transverse to these ridges it is a wavelet. 
3552 12924 =. 
Given f e L (R^), we define its ridgelet coefficients by 
W y (a,b. 8) = [vio 00rends (3) 
If fe LARRY, then we have the exact reconstruction 
formula 
T dt 
r= | | [rr@bowapom? d 10 1 
0 — 0 
and a Parseval relation 
2x o oc 
do 
J fo” dx = ] | fe (a,b ey “a e (5) 
—o0 0 
Hence, much like the MA or Fourier transforms, the identity 
(4) expresses the fact that one can represent any arbitrary 
function as a continuous superposition of ridgelets. Discrete 
analogs of (4) and (5) exists; see (Candes, 1999) or (Donoho, 
2000) for a slightly different approach. 
2.1 Radon Transform 
A basic tool for calculating ridgelet coefficients is to view 
ridgelet analysis as a form of wavelet analysis in the Radon 
domain. 
The Radon transform A: (RY) — D ([0, 22], L (R)) is 
defined by 
Rf (0,1) = [rman cos 0+x, sin O-t) dx, dx) (6) 
60 
where ó is the Dirac distribution. The ridgelet coefficients 
9$ 
/ 
transform via 
(a,b, 0) of an object / are given by analysis of the Radon 
9t (a,b, 0) - [nrto. a ^v - 01 axi (7) 
Hence, the ridgelet transform is precisely the application of a 1- 
dimensional wavelet transform to the slices of the Radon 
transform where the angular variable © is constant and f is 
varying. 
2.2 Ridgelet Pyramids 
Let Q denote a dyadic square Q - [4 j28 un - 0/2 )x 
[£, 12 Ah + 1)/2°) and let Q be the collection of all such 
dyadic squares. We write Q, for the collection of all dyadic 
squares of scale s . Associated to the squares QeQg we 
construct a partition of energy as follows. With & a nice 
smooth window obeying S dt e (x x5 -k2 )»1 , we dilate 
and transport & to all squares Q at scale s , producing a 
collection of windows (c, ) such that the ER QeQs , make 
up a partition of unity. We also let 7, donote the transport 
operator acting on functions g via 
S sS 5 
(709)x,x,)= 2 g@ x, mA ne aka) 
With these notations, it is not hard to see that 
da dO 
fo, = IE f, 0, TU 7 TV nb. EN. (8) 
rai 
3. 
e 4 
and, therefore, summing the above equalities across squares at a 
given scale gives 
2 m 
fe Y fay = 2 Io fo Q TV > T Vaso 3 
QeQ, 
         
The identity (9) expresses the fact that one can represent any 
function as a superposition of elements of the form 
QUT Wb o 5 that is, of ridgelet elements localized near the 
squares Q For the function T M as bo is the ridgelet 
with parameters obeying 
V qo 59.00 p yıng 
s 
ag=2 ‘a, bo=b+k; 275 cos0+k32 * sin0, 09-0 
and, thus, œ, TV ano is a windowed ridgelet, supported near 
a,b, 
the square Q, hence the name local ridgelet transform. 
The previous paragraph discussed the construction of local 
ridgelets of fixed length, roughly 275 ( s fixed). Letting the 
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