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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