Digital Signal Processing Reference
In-Depth Information
More precisely, the monogenic concept is basically made of a Radon transform
joined with a 1D phase analysis, so we can say that the Radon transform is respon-
sible for
isotropy
. This crucial point has always been a deep issue in the discrete
world, while at the core of monogenic analysis.
In the past, we have proposed [
7
] a discrete Radon transform with exact recon-
struction is designed with the help of discrete analytical geometry. This last part
studies the use of this well established discrete representation to perform monogenic
analysis.
We now present the algorithm implementing the discrete Radon transform.
15.5 The Radon Domain for Numerical Monogenic Transform
It is shown in [
5
] that the Riesz transform is equivalent to an independent Hilbert
transform (1D) on each Radon projection, combined with a sine-like weighting de-
pending on
θ
:
(t)e
j
θ
{
R
s
}
θ
(t)
={
H
s
θ
}
(15.29)
where
H
s
θ
is the Hilbert transform of
s
θ
defined by
s
θ
(
(t).
1
(π
{
H
s
θ
}
(t)
=
·
)
∗
(15.30)
·
)
The Hilbert transform is well known and already integrated, for example, in some
analytic
wavelet transforms. Performing some monogenic analysis (based on the
Riesz transform) then reduces to a more classical problem in the Radon domain.
The Radon domain represents 2D functions by a set of 1D projections at several
orientations. It forms a fundamental link with 1D and 2D Fourier transforms and so
handles isotropic filtering well.
15.5.1 The Radon Transform
Given a 2D function
s(x,y)
, its projection into the Radon domain along direction
θ
is defined by
s
θ
(t)
=
s(τ
sin
θ
+
t
cos
θ,
−
τ
cos
θ
+
t
sin
θ)dτ.
(15.31)
R
The Radon transform can be obtained by applying the 1D inverse Fourier trans-
form to the 2D Fourier transform restricted to radial lines going through the origin
(this is exactly what we are going to do in the discrete Fourier domain with the help
of discrete analytical lines):
e
−
jωt
Rs(θ,t)dt
s(ω
cos
θ,ω
sin
θ)
=
R
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