Graphics Programs Reference
In-Depth Information
∂
n
I
∂
n
I
∂
r
∂
ϕ
cos
ϕ
r
sin
ϕ
[]
=
=
(2.6)
sin
ϕ
r
cos
ϕ
∂
n
Q
∂
n
Q
∂
r
∂
ϕ
The determinant of the matrix of derivatives is called the Jacobian, and in this
case it is equal to
J
=
r
()
(2.7)
Substituting Eqs. (2.4) and (2.7) into Eq. (2.5) and collecting terms yield
r
2
A
2
r
2πψ
2
rA
cos
ψ
2
ϕ
+
2ψ
2
-------------
-------------------
f r
ϕ
(
,
)
=
exp
-----------------
exp
(2.8)
The
pdf
for
r
alone is obtained by integrating Eq. (2.8) over
ϕ
2π
∫
2π
∫
r
2
A
2
r
ψ
2
1
2π
rA
cos
ψ
2
ϕ
+
2ψ
2
------
------
-------------------
f
()
=
fr
ϕ
(
,
) ϕ
d
=
exp
-----------------
exp
d
(2.9)
0
0
where the integral inside Eq. (2.9) is known as the modified Bessel function of
zero order,
2π
∫
1
2π
e
βθ
cos
------
I
0
()
=
d
(2.10)
0
Thus,
r
2
A
2
r
ψ
2
rA
ψ
2
+
2ψ
2
------
I
0
-----
f
()
=
exp
-----------------
(2.11)
A
ψ
2
which is the Rician probability density function. If (noise alone),
then Eq. (2.11) becomes the Rayleigh probability density function
⁄
=
0
r
2
2ψ
2
r
ψ
2
------
f
()
=
exp
---------
(2.12)
A
ψ
2
Also, when is very large, Eq. (2.11) becomes a Gaussian probability
density function of mean
(
⁄
)
ψ
2
A
and variance
:
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