A Fast search Algorithm for Vector
Quantization
授課老師：王立洋
老師
製作學生：M9535204
蔡鐘葳
Outline
▓ Full Search for VQ
▓ Principle
▓ Fast Search for VQ
▓ Experimental Results
▓ Reference
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Full Search for VQ
優點：
作法直接簡單
較少的失真以達到較佳的影像品質
 缺點：
 編解碼計算耗時
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Principle
Rejects those codewords that are impossible to be the
nearest codeword.
It produces the same output as the conventional full
search algorithm.
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Fast search Algorithm
ENNS:
Equal-average nearest neighbor search
IENNS:
Improved equal-average nearest neighbor search
EENNS:
Equal-average equal-variance nearest neighbor search
IEENNS:
Improved equal-average equal-variance nearest neighbor
search
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Preview (1/3)
Let X = (x1,x2,…,xk) be a k-dimensional vector.
X(h) = (x1,x2,…,xh) be a h-dimensional subvector of X.
Xf = (x1,x2,…,xk/2) be composed of the first half vector
components of X.
Xs = (xk/2+1,xk/2+2,…,xk) be composed of the remaining
vector components of X, where k is an even number.
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Preview (2/3)
The sum of the h vextor components for X(h) can be
expressed as:
h
SX ( h )   xi
i 1
where 1 ≦ h ≦ k and SX(h) is the sum of the hdimensional vector X(h) . If h = k, we denote the sum
of the vector X as SX .
SXf and SXs are the sum of the vector componenets of
Xf and Xs .
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Preview (3/3)
If the mean of the h-dimensional vector X(h) is
mX(h) = SX(h)/h, then the variance of the h-dimensional
vector X(h) can be expressed as:
VX ( h ) 
h
 (x  m
i
)
X (h)
2
i 1
Likewise, VXf and VXs are the sum of the vector
componenets of Xf and Xs .
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ENNS (1/3)
The ENNS algorithm takes advantage of the fact that
the nearest codeword is usually in the neighborhood
of the minimum squared sum distance.
Assuming the current minimum distortion is Dmin, the
main spirit of ENNS algorithm can be stated as
follows:
if
(SX－SCj)2 ≧ k · Dmin
then
D(X, Cj) ≧ Dmin
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ENNS (2/3)
Then the codewords Cj for which SX ≧ SCj＋ k  D min
or SX ≦ SCj－ k  D min are eliminated.
If the condition is satisfied
The search will be stopped in this direction.
Continued in another direction until the nearest codeword
is found.
The search shows in Fig. 1.
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ENNS (3/3)
Fig .1
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IENNS (1/3)
The basic inequality for IENNS algorithm is as
follows:
if
(SX(h)－SCj(h))2 ≧ v · Dmin
then
D(X, Cj) ≧ Dmin
where h ≦ v ≦ k. Let v = h = k/2 and if k is an even
number, two inequalities can be expressed as follows.
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IENNS (2/3)
a) For vector Xf and codeword Cjf
if
(SXf－SCjf)2 ≧ k/2 · Dmin
then
D(X, Cj) ≧ Dmin
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IENNS (3/3)
b) For vector Xs and codeword Cjs
if
(SXs－SCjs)2 ≧ k/2 · Dmin
then
D(X, Cj) ≧ Dmin
 Thus, besides the elimination criterion of ENNS,
inequalities (a) and (b) can be used to eliminate more
unlikely codewords.
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EENNS (1/2)
By ENNS, however, two vectors with the same mean
value may have a large distance.
Based on this condition, the EENNS algorithm
introduces the variance to reject more codewords.
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EENNS (2/2)
The EENNS algorithm uses the following elimination
criterion to eliminate unlikely codewords:
if
(VX－VCj)2 ≧ Dmin
then
D(X, Cj) ≧ Dmin
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IEENNS (1/7)
The IEENNS algorithm uses the variance and the sum
of a vector simultaneously, while the EENNS
algorithm uses them separately.
The following theorem:
Theorem 1:
k · D(X, Cj) ≧ (SX－SCj)2＋k · (VX－VCj)2
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IEENNS (2/7)
Based on this theorem, the following inequality can
be obtained:
if
(SX－SCj)2＋k · (VX－VCj)2 ≧ k · Dmin
then
D(X, Cj) ≧ Dmin
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IEENNS (3/7)
Theorem 2:
v · D(X, Cj) ≧ (SX(h)－SCj(h))2＋k · (VX(h)－VCj(h))2
where h ≦ v ≦ k
 Based on Theorem 2, three elimination criteria can be
stated as follows.
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IEENNS (4/7)
Based on this theorem, the following inequality can
be obtained:
if
(SX－SCj)2＋k · (VX－VCj)2 ≧ k · Dmin
then
D(X, Cj) ≧ Dmin
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IEENNS (5/7)
a) For vector X and codeword Cj, set c = h = k, the
following elimination criterion is obtained:
if
(SX－SCj)2＋k · (VX－VCj)2 ≧ k · Dmin
then
D(X, Cj) ≧ Dmin
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IEENNS (6/7)
b) For vector Xf and codeword Cjf, set v = h = k/2, the
following elimination criterion is obtained:
if
(SXf－SCjf)2＋k · (VXf－VCjf)2 ≧ k/2 · Dmin
then
D(X, Cj) ≧ Dmin
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IEENNS (7/7)
c) For vector Xs and codeword Cjs, set v = h = k/2, the
following elimination criterion is obtained:
if
(SXs－SCjs)2＋k · (VXs－VCjs)2 ≧ k/2 · Dmin
then
D(X, Cj) ≧ Dmin
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Experimental Results
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Unit: Second(s)
Reference
[1] Jeng-Shyang Pan, Zhe-Ming Lu, and Sheng-He Sun, “An
Efficient Encoding Algorithm for Vector Quantization Based
on Subvector Technique,” IEEE Signal Processing Lett.,
vol. 12, Mar, 2003.
[2] S. J. Baek, B. K. Jeon, and K. M. Sung, “A fast encoding
algorithm for vector quantization, “ IEEE Signal Processing
Lett., vol. 12, Mar, 2003.
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