Structured Vector Quantization Using Linear Transforms Author Cagri

Structured Vector Quantization Using Linear Transforms Author : Cagri O Etemoglu and Vladimir Cuperman Source : IEEE Transaction on Signal Processing, VOL. 51, NO. 6, JUNE 2003 Speaker : 曾皇傑 Date : 05/4/2004 1

Outline l l l 2 VQ (vector quantization) MSVQ (multi stage vector quantization) ATVQ (affine transformation vector quantization) SRMVQ (scaled rotation matrices vector quantization) Performance

MSVQ Input vector is quantized sequentially by a group of quantizers --assume L-stage MSVQ l Best set of indices are used as encoding relation --input vector X encoded into L indices (i 0, i 1, …. , i. L-1) l 4

MSVQ 5

MSVQ 6

MSVQ l Example for two stage MSVQ Input X=(10, 20, 15, 30) C 1, 5=(6, 10, 11, 20) C 2, 20=(5, 8, 2, 11) X-C 1, 5=(4, 10, 4, 10) X X’=(5, 20) =(11, 18, 13, 31) codebook 1 7 codebook 2

ATVQ l Linear transform matrix Vector 1 …… …… Vector N Matrix N Codebook i 8 Matrix 1 Transform matrix codebook i

ATVQ l Input X=(10, 20, 15, 30) T 1, 5*c 2, 9=(4, 10, 4, 10) C 1, 5=(6, 10, 11, 20) T X codebook 1 9 1, i * codebook 2 New codebook 2 X’=(5, 15)

ATVQ l MSVQ的結果可以寫成以下數學表示式 L X’=∑Ci ci: 第i個編碼簿 L: stage的個數 i=1 例: x’=(5, 20)=C 1, 5+C 2, 20=(6, 10, 11, 20)+(5, 8, 2, 11)=(11, 18, 13, 31) l ATVQ的結果則是 L X’=∑Ti-1 Ci Tj: 第J個linear transformation matrix code book i=1 = 10 20 15 30 1234 0201 1001 0121 + 1001 10 6 10 11 20 x’=(5, 9)=C 1, 5+T 1, 5*C 2, 9=

ATVQ-codebook training l T 1, i的第j個row, 為rij. T的最小norm rij. T=zij. TYij+ j=1…M M: vector的維度 zij. T=[x 1, j…. . . x||Ui||, j] - avg(Xk, j) Xk, j є Ui Yij=[C 2, 1……C 2, ||Ui||] - avg(C 2, k) N 2 註: Ui=∪X , Xi, j為input vector用到C 1第i個codeword且用 J=1 i, j 到C 2第j個codeword 11

ATVQ-codebook training l l 求C 1, i=avg(X - T 1 C 2) 求C 2, j C 2, , j=min_norm(A+ b) T 1, ||Vj|| X 1 - C 1, 1 B= …. . T 1, 1 …. . A= 12 X є Ui X||Vj|| - C 1, ||Vj|| N 1 註: Vj=∪Xi, j i=1

ATVQ-codebook training Input set 13 Initial C 1 & T 1 Initial C 2 New C 1 & T 1 New C 2 Quantized set

SRMVQ l Scalar codebook & Rotation matrix codebook Vector 1 …… Vector N scalar N RMatrix 1 …… …… 14 scalar 1 RMatrix N

SRMVQ S 1, 5*R 1, 5*c 2, 9=(4, 10, 4, 10) C 1, 5=(6, 10, 11, 20) S 1, i *R 1, i X codebook 1 15 X’=(5, 15) codebook 2 New codebook 2

SRMVQ l 範例: 設一vector維度為 3 αi, 1=30 αi, 2=60 RM 1, i= cos 30 -sin 30 0 sin 30 cos 30 0 1 cos 60 0 sin 60 0 1 0 -sin 60 0 cos 60 在儲存RM時只要存一個連續sinα, cosα的 資料而不用存整各M*M的矩陣 17

ATVQ與SRMVQ之儲存空間比較 l l l 18 Transform matrix codebook需要N 1*(M*M) Scalar codebook需要N 1*1 Rotation matrix codebook需要N 1*2*(m-1) N 1為codebook 1之大小 ATVQ要: N 1(M 2+M)+N 2 M SRMVQ要: N 1(3 M-2)+N 2 M

Performance Rate bits Test 2 SRMVQ MSVQ 10=5+5 15. 99 15. 73 10=5+5 10. 75 10. 59 12=6+6 16. 97 16. 55 12=6+6 11. 21 11. 03 14=7+7 17. 97 17. 40 14=7+7 11. 66 11. 57 WSNR= 19 Test 1 ATVQ MSVQ 1 N N-1 ∑ K=0 ||SK||2 WK ||SK-SK’||2 WK

Performance Complexity MSVQ N 1 M+N 2 M multiplication N 1 M+N 2 M+M addition ATVQ N 1 M+N 2(M 2+M) multiplication N 1 M+N 2(M 2+M-1)+M addition SRMVQ N 1 M+N 2(M+1)+4(M-1) multiplication N 1 M+N 2 M+3 M-2 addition l 20

Performance Complexity Vector dim (M) 21 Memory (word) ATVQ MSVQ M=48 79872 m 78320 a 6144 m 6128 a 76800 3072 M=10 4480 m 4106 a 1280 m 1226 a 3840 640