第 四 章 VQ 加速運算與編碼表壓縮 4-

4.1 VQ Codeword Search 4-

Vector Quantization (VQ) 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-

4.2 PCA 4-

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-

Principal component analysis (PCA) D1 = [0.710 0.703] D2 = [-0.703 0.710] 4-

PCA 40 samples with 2 variables, X1 and X2 Covariance matrix λ1 =1160.139 λ2 =36.780 4-

Principal component analysis (PCA) λ1 =1160.139 λ2 =36.780 D1 = (0.710, 0.703) D2 = (-0.703, 0.710) 4-

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-

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-

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-

The Encoding algorithm using PCA Codebook The covariance matrix 4-

The Encoding algorithm using PCA From the covariance matrix, we compute D1: (0.5038, 0.4904, 0.4788, 0.5259), λ1=19552, D2: (-0.4915, -0.5126, 0.4293, 0.5580), λ2=151, D3: (-0.0294, -0.0292, 0.7658, -0.6418), λ3=86 and D4: (0.7098, -0.7042, -0.0108, -0.0134), λ4=6. D1: (0.5038, 0.4904, 0.4788, 0.5259) is a coordinate D1 reserves 98.77% information of the variance of the codewords. 4-

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, 0.4904, 0.4788, 0.5259) 4-

Encode an input vector v = (150, 145, 121, 130) Transform v to ρ= D1.v To search and find that 321.93 is the closest value to 272 For P5’ = 321.93, d(v, cw’5)=63.2 For P4’ = 162.60, d(v, cw’4)=122.3 For P6’ = 382.84, d(v, cw’6)=114.2 So, we choose cw’5 to replace the input vector v. 4-

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-

4-

4.3 快速歐幾里德距離演算法 4-

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

4.3.1 Truncated Look-up Table (TLUT) 4-

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

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

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-

Reduced Code LUT (RCLUT) 4-

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-

4-

Experimental Results 4-

4.4 VQ 編碼表的壓縮 4-

An example for indices of VQ 4-

4.4.1 Search-Order Coding (SOC) 4-

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

The block diagram of the proposed scheme 4-

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

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

Experimental results 4-

The comparison of two schemes in the unmatched index numbers 4-

The comparison of three schemes in compression results 4-

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

Discussions 4-

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-

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-