1. 第 四 章
VQ 加速運算與編碼表壓縮 4-

2. 4.1 VQ Codeword Search 4-

3. Vector Quantization (VQ)4-

4. Euclidean Distance The dimensionality of vector = k (= w*h)
An input vector x = (x1, x2, …, xk)
A codeword yi = (yi1, yi2, …, yik)
The Euclidean distance between x and yi 4-

5. 4.2 PCA 4-

6. Principal component analysis (PCA)
Given a set of points Y1, Y2, …, and YM where every Yi is characterized by a set of variables X1, X2, …, and XN. We want to find a direction D = (d1, d2, …, dN), where such that the variance of points projected onto D is maximized. 4-

7. Principal component analysis (PCA)
D1 = [ ]
D2 = [ ] 4-

8. PCA 40 samples with 2 variables, X1 and X2 Covariance matrix
λ1 =
λ2 =36.780 4-

9. Principal component analysis (PCA)
λ1 =
λ2 =36.780
D1 = (0.710, 0.703)
D2 = (-0.703, 0.710) 4-

10. Principal component analysis (PCA)
Algorithm of PCA
Start by coding the variables Y = (Y1, Y2, …YN) to have zero means and unit variances.
Calculate the covariance matrix C of the samples.
Find the eigenvalues λ1, λ2, …, λN, for C,
where λi λi+1, i = 1, 2, …, N-1. Let D1, D2, … DN denote the corresponding eigenvectors.
D1 is the first principal component direction, D2 is the second principal component direction, … , DN is the Nth principal component direction . 4-

11. The Encoding Algorithm Using PCA Technique
(A) the codebook processing:
For each codeword cwi in codebook matrix B, evaluate projected value pi by pi=D1.cwi. Then, all the codewords in the matrix B are transformed into 256 real numbers. The set of these real numbers can be expressed as P = {p1, p2, …, p256}.
Sort the values in P. We obtain the new ordered set P’ = {p’1, p’2, …, p’256} and their corresponding vector cw’1, cw’2 ,… cw’256 form an ordered codebook B’={cw’1, cw’2 ,… cw’256}. 4-

12. The Encoding Algorithm Using PCA Technique
(B) The Encoding Algorithm]
For the input vector x, evaluate the projected value px by px = D1.x.
Search the set P’ for r values P’i’s which are the r closest ones to Px.
Compute their distances from the set {cw’k, cw’k+1 ,… cw’k+r-1} to find a vector cw’j such that (cw’j, x) has a minimum distortion. Here, let the searching range be r and assume that the corresponding codeword of P’j is cw’j.
Store or transmit the index j of cw’j 4-

13. The Encoding algorithm using PCA
Codebook
The covariance matrix 4-

14. The Encoding algorithm using PCA
From the covariance matrix, we compute
D1: (0.5038, , , ), λ1=19552,
D2: ( , , , ), λ2=151,
D3: ( , , , ), λ3=86 and
D4: (0.7098, , , ), λ4=6.
D1: (0.5038, , , ) is a coordinate
D1 reserves 98.77% information of the variance of the codewords. 4-

15. The Encoding algorithm using PCA
The new sorted codebook and the corresponding projected value of codewords
Codebook
The sorted codewords
The projected values
D1: (0.5038, , , ) 4-

16. Encode an input vector v = (150, 145, 121, 130) Transform v to ρ= D1.v
To search and find that is the closest value to 272
For P5’ = , d(v, cw’5)=63.2
For P4’ = , d(v, cw’4)=122.3
For P6’ = , d(v, cw’6)=114.2
So, we choose cw’5 to replace the input vector v. 4-

17. Experimental results
The quality of the encoded image is evaluated by the peak signal-to noise ration PSNR, which is defined as
For an m*m image, the mean-square error (MSE) is defined as
Where xij and denote the original and quantized gray levels, respectively. 4-

18. 4-

19. 4.3 快速歐幾里德距離演算法 4-

20. Look-up Table (LUT)
The content of each pixel belongs to [0, m-1] 4-

21. 4.3.1 Truncated Look-up Table (TLUT)4-

22. Truncated Look-up Table (TLUT)1 2 … 4-

23. Truncated Look-up Table (TLUT)
For example,
r = 256/32 = 8
(17-34)2= 289
Error = |289 – 576|
= 287 4-

24. 4.3.2 Reduced Code LUT (RCLUT)
Suppose m = 2t.
Express xj (or yij) by
Define low nibble L(xj) = the lowest bits of xj=
Define high nibble H(xj) = the lowest bits of xj= 4-

25. Reduced Code LUT (RCLUT)4-

26. Reduced Code LUT (RCLUT)
H(xj), H(yij), L(xj), L(yij) are within the range [0.21/2-1]
Look-up Table (LUT) 4-

27. 4-

28. Experimental Results 4-

29. 4.4 VQ 編碼表的壓縮 4-

30. An example for indices of VQ4-

31. 4.4.1 Search-Order Coding (SOC)4-

32. 4.4.2. Locally Adaptive scheme
An example of segmentation with region size 4*4
for indices of VQ 4-

33. The block diagram of the proposed scheme4-

34. The compressing steps of our example10 10 01 31
207
211 8 7 35 10 11 11
100
011
011 4-

35. The decoding steps of our example10 31
207
211 8 7 35 10 01 10 11 11
100
011
011 4-

36. Experimental results 4-

37. The comparison of two schemes in the unmatched index numbers4-

38. The comparison of three schemes in compression results4-

39. The comparison of proposed method and SOC scheme in bit rate (bit/pixel)4-

40. Discussions 4-

41. Discussion of the VQ scheme
Advantage:
The bit rate of the VQ scheme is low.
bit rate = bpp.
Example:
When the codebook is composed of 256 codewords. Each block used is of 4*4 pixels.
bit rate = 0.5 bpp
VQ has a simple decoding structure. 4-

42. Discussion of the VQ scheme
Disadvantage:
The reconstructed image quality highly depends on the codebook performance.
Therefore, how to design a representative codebook is an important issue of VQ scheme.
The VQ scheme requires a lot of computation cost in codebook generation process and VQ encoding process. 4-