Image Processing Reference
In-Depth Information
To construct a magnitude image from the complex data, the magnitude is
computed on a pixel-by-pixel basis
m
=
ωω
r
2
+
2
,
(4.9)
i
where
is the magnitude variable corresponding to the magnitude observation As
root extraction is a nonlinear transformation, the PDF of the magnitude data is
no longer expected to be Gaussian [18,19].
The PDF of the magnitude data is found by transforming the joint PDF of
the real and imaginary data, given in Equation 4.1, into polar coordinates:
m
m
.
2
)
2
1
( os os)
A
ϕφ
σ
−
( in
A
ϕϕφ
σ
−
sin
−
−
p
=
e
e
m
,
(4.10)
2
2
2
2
(,
mA
φϕσ
1
,,)
2
πσ
2
where
φ
denotes the phase variable corresponding to the phase observation
φ
.
The last
factor,
m
, is the Jacobian of the transformation (see
Appendix
). Integration of Equation
4.10 over a full cycle of
cos
φ
leads us to the PDF that characterizes magnitude data:
22
2
I
A
mA
+
−
pA
(
| , =
σ
)
e
ε
(
m
)
,
2
(4.11)
σ
m
0
σ
σ
2
2
with
denoting the zeroth-order modified Bessel function of the first kind. The
unit step Heaviside function
I
0
ε
(.) is used to indicate that the expression for the
PDF of
m
is valid for nonnegative values of
m
only. The preceding distribution
is called the
, after S. O. Rice, who derived it in the context
of communication theory in 1944 [18]. Note that the shape of the Rician PDF
depends on the signal-to-noise ratio (SNR), which is here defined as the ratio
Rician distribution
A
/
σ
.
Figure 4.1
shows the Rician PDF as a function of the magnitude variable
for various values of the SNR.
4.2.2.1
Asymptotic Approximation of the Rician
Distribution
This subsection describes the behavior of the Rician distribution for very low and
very high SNR:
•
For low SNR, the modified Bessel function is given by
ν
z
Iz
()
∼
2
Γ
(
ν
+
1
)
for
z
→ .
0
(4.12)
ν
Hence, the Rician PDF then leads to a Rayleigh distribution:
2
m
m
−
p
m
(
m
|
σ
)
=
e
ε
(
m
)
for
SNR
→.
0
(4.13)
2
2
σ
σ
2
The Rayleigh PDF characterizes the random intensity distribution of nonsig-
nal background areas such as air.
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