Image Processing Reference
In-Depth Information
At high SNR, the asymptotic approximation of the modified Bessel function
is given by
e
z
Iz
()
∼
2
for
z
→∞.
(4.14)
ν
π
z
Then, given that
m
/→
A
1
for SNR
→
∞
, we find for the Rician PDF:
1
2
(
m
−
A
)
−
p
(
m
|
A
σ
)
=
e
ε
(
m
)
for
SNR
→ .
(4.15)
2
2
σ
m
2
πσ
2
Hence, for high SNR, the Rician PDF approaches a Gaussian PDF, with a
mean
. The transition between the two limits of the Rician
PDF can visually be appreciated in Figure 4.1.
A
and a variance
σ
2
4.2.2.2
Moments of the Rician PDF
The moments of the Rician PDF can be analytically expressed as a function of
the confluent hypergeometric function of the first kind
1
F
,
ν
ν
A
2
ν
ν
/
2
[]
(
=
Γ
1
+
F
−;;−
1
(4.16)
E
m
2
σ
2
)
11
2
2
2
σ
2
representing the Gamma function [20]. The even moments of the Rician
distribution (i.e., when
with
Γ
ν
is even) are simple polynomials. The expressions for the
SNR = 0
SNR = 1
SNR = 2
SNR = 3
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
m
4
5
6
FIGURE 4.1
The Rician PDF as a function of the magnitude variable M, drawn for various
values of the SNR where M
=
1. At SNR
=
0, the distribution equals a Rayleigh PDF,
whereas at high SNR (SNR
>
3), the Rician PDF approaches a Gaussian PDF.
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