Image Processing Reference
In-Depth Information
decision levels are determined by minimizing the average distortion (MSE) given by
the equation
ð
d
i
X
L
2
p
(
x
)d
x
D
¼
(
x
r
i
)
(
2
:
48
)
i
¼
1
d
i
1
To minimize the average distortion D, we set the derivatives of D with respect to r
k
and d
k
equal to zero. This yields
@
D
@
d
k
¼
2
(
d
k
r
k
)
p
(
d
k
)
2
(
d
k
r
k
þ
1
)
p
(
d
k
) ¼
0
d
k
ð
(
2
:
49
)
@
D
@
r
k
¼
(
x
r
k
)
p
(
x
)d
x
¼
2
0
d
k
1
Solving these equations results in
r
i
þ
r
i
þ
1
2
d
i
¼
i
¼
1, 2,
...
, L
1
with
d
0
¼
x
min
d
L
¼
x
max
(
2
:
50
)
and
Ð
d
i
d
i
1
xp
(
x
)d
x
Ð
d
i
d
i
1
p
(
x
)d
x
r
i
¼
i
¼
1, 2,
...
, L
(
2
:
51
)
The above MMSE solution, also known as the Lloyd
Max quantizer [6], has
the following properties: Decision levels are halfway between two adjacent recon-
struction levels and reconstruction levels are given by the centroids of the signal
probability density that are enclosed between two adjacent decision levels. The
Lloyd
-
Max quantizer results in the minimum-mean-square quantization error for a
given number of reconstruction levels. The MMSE quantizer decision and recon-
struction levels are solutions to a set of integral equations. In rare occasions (such
as the Laplacian distribution), a closed form solution exists. However, in most
cases, a numerical solution needs to be determined. The following popular iterative
technique due to Lloyd can be used to design an MMSE quantizer with L recon-
struction levels:
-
Step 1:
Divide the range of the signal values into L uniform reconstruction levels.
Step 2:
For the current set of reconstruction levels, compute the optimum decision
levels using Equation 2.50.
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