Slide nén dữ liệu transform coding PJEG

Idea of Transform Coding 1) Transform the input pixels x0,x2, ,xN-1 into coefficients c0,c1, ,cN-1 (real values): CSE 592 Data Compression – The coefficients should have the property that most of them are near zero – Therefore most of the “energy” is compacted into a few coefficients Fall 2006 Lecture 14: Transform Coding JPEG 2) Scalar quantize the coefficient: – This is bit allocation! – Important coefficients should have more quantization levels (== represented with more accuracy) 3) Entropy encode the quantized values Decoding Block Diagram of Transform Coding 1) Entropy decode the quantized values Encoder input x 2) Compute approximate coefficients c’0,c’1, ,c’N-1 from the quantized values 3) Inverse transform c’0,c’1, ,c’N-1 to x’0,x’1, ,x’N-1 which is (we hope) a good approximation of the original x0,x1, ,xN-1 coefficients c transform symbols entropy decoding s symbols quantization bit allocation bit stream s entropy coding output coefficients decode symbols c’ b inverse transform x’ Decoder Mathematical Properties of Transforms Why Coefficients? A c=x T  a 00 L   M  a 0,N -1 L • Linear Transformation: – Defined by a real N x N matrix A = (aij)  a 00 L a 0,N-1   x   c    M   M  =  M   M aN-1,0 L aN-1,N-1   xN-1  c N-1  a N -1,0   c   x   M   M  =  M  a N -1, N -1  c N -1   x N -1   a N -1,0   a 00   x0        = + + c c M L M   N −1  M    a N -1,N -1   a 0,N -1   x N -1  • Orthonormality: A −1 = A T (transpose) basis vectors coefficients CuuDuongThanCong.com https://fb.com/tailieudientucntt Why Orthonomality? Example (with Compaction) 1    1 − 1 A = I ⇒ A −1 = A A= • The energy of the data equals the energy of the coefficients: N−1 ∑c i=0 i A T = A = A −1 = c T c = (Ax)T (Ax) N−1 = (x T A T )(Ax) = x T (A T A)x = x T x = ∑ x i orthonormal 1  b  2b     =  1 − 1 b   i =0 compaction Discrete Cosine Transform (1D)   aij =    i=0 N (2 j + 1)iπ cos N 2N Basis Vectors (1D) 1 0.5 0.5 0 i >0 -0.5 -0.5 -1 -1 row row N = 4: 5    65328 270598 - 270598 - 65328   A=   - - 5   .270598 - 65328 65328 - 270598 1 0.5 0.5 0 3 -1 -1 row row 10 Decomposition in Terms of Basis Vectors 11 Block Transform Image .5  653281     270598   x0  .5  270598  −.5    - 653281  c +  c +  c +  c =  x1  .5 − 270598 −.5  x2   653281            270598 − 653281          x3  DC coefficient -0.5 -0.5 Each 8x8 block is individually coded AC coefficients 12 13 CuuDuongThanCong.com https://fb.com/tailieudientucntt 2-Dimensional Block Transform 8x8 DCT Basis Vectors Transform Block of pixels X a00 a 10 A= a20  a30 x00 x01 x02 x03 x10 x11 x12 x13 x20 x21 x22 x23 x30 x31 x32 x33 a01 a02 a11 a12 a21 a22 a31 a32 a03  a13   a23   a33  N−1 Transform rows rij = ∑ a kj xik k =0 N −1 N−1 N−1 N −1 N −1 m= m =0 k =0 m= k = Transform columns c ij = ∑ aimrmj = ∑ a im ∑ akj x mk = ∑ Summary: C = AXA ∑a a x mk im kj T 14 15 Importance of Coefficients Quantization Tables • The DC coefficient is (in general) the most important • The AC coefficients (in general) become less important as they are farther from the DC coefficient • One example bit allocation (bits per coefficient): • For an nxn block we construct an nxn matrix Q: – Qij indicates how large the quantization interval is for coefficient cij 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • Encode cij with the label: compression 55 bits for 64 pixels = 86 bpp • Decode sij to: c'ij = sijQij 16 Example Quantization c' = ⋅ 24 = 48 • c= 54.2, Q = 17 Example Quantization Table (JPEG) • c = 54.2, Q = 24 s =  54.2 + 0.5 =   24 • c = 54.2, Q = 12   c ij sij =  + 0.5   Qij   54.2 s= + 0.5 =   12 c' = ⋅12 = 60   54.2 s= + 0.5 =   c' = ⋅ = 54 16 12 11 12 10 14 16 19 24 26 40 58 51 60 61 55 14 14 18 24 49 72 13 17 33 35 64 92 16 22 37 55 78 95 24 29 56 64 87 98 40 51 68 81 103 112 57 87 109 104 121 100 69 80 103 113 120 103 56 62 77 92 101 99 Increase the bit rate = halve the size of quantization intervals Decrease the bit rate = double the size of quantization intervals 18 19 CuuDuongThanCong.com https://fb.com/tailieudientucntt JPEG (1987) Block Transmission • Let P = [pij], < i,j < N be an image with < pij < 256 • Center the pixels around zero: • DC coefficient: – DC coefficients don’t change much from block to neighboring block Hence, their labels change even less – Predictive coding with differences is used to code the DC label – xij = pij - 128 • Code 8x8 blocks of X using the DCT • Choose a quantization table: – The table depends on the desired bit rate and is built into JPEG • AC coefficients: • Quantize the coefficients according to the quantization table: – Do a zig-zag coding – The quantization symbols can be positive or negative • Transmit the labels (in a coded way) for each block 20 Zig-Zag Coding 21 Example Block of Labels • AC labels are coded in zig-zag order: – Usually use a special entropy coding to take advantage of the ordering of the bit allocation (quantization) -8 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Coding order of AC labels –8 0 0 1 0 0 22 Coding Labels Coding AC Label Sequence • Categories of labels – – – – • A symbol has three parts (Z,C,B) {0} {-1, 1} {-3,-2,2,3} {-7,-6,-5,-4,4,5 7} – Z for number of zeros preceding a label < Z < 15 – C for the category of the of the label – B for a C-1 bit number for the actual label • End of Block symbol (EOB) means the rest of the block is zeros EOB = (0,0,-) • Example: –8 0 0 1 0 0 • Label is indicated by two numbers Cat,B • Examples label C,B -4 23 3, 4, (0,3,2)(0,5,7)(0,3,3)(4,2,1)(0,2,1)(2,2,1)(0,0,-) 24 25 CuuDuongThanCong.com https://fb.com/tailieudientucntt Coding AC Label Sequence • Z,C have a prefix code • B is a C-1 bit number 0 Z 1010 Notes on Transform Coding • Video Coding: Partial prefix code table C – MPEG uses DCT – H.263 and H.261 use DCT 00 01 100 1100 11011 11110001 1110 11111001 1111110111 111010 111110111 111111110101 • Audio Coding: – MP3 = MPEG Layer uses DCT • Alternative Transforms: – Lapped transforms remove some of the blocking artifacts – Wavelet transforms not need to use blocks at all (0,3,2) (0,5,7) (0,3,3) (4,2,1) (0,2,1) (2,2,1) (0,0,-) 100 10 11010 0111 100 11 1111111000 01 11111001 1010 46 bits representing 64 pixels = 72 bpp 26 27 CuuDuongThanCong.com https://fb.com/tailieudientucntt ... CuuDuongThanCong.com https://fb.com/tailieudientucntt Coding AC Label Sequence • Z,C have a prefix code • B is a C-1 bit number 0 Z 1010 Notes on Transform Coding • Video Coding: Partial prefix code table C –... 111110111 111111110101 • Audio Coding: – MP3 = MPEG Layer uses DCT • Alternative Transforms: – Lapped transforms remove some of the blocking artifacts – Wavelet transforms not need to use blocks... entropy coding to take advantage of the ordering of the bit allocation (quantization) -8 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Coding order of AC labels –8 0 0 1 0 0 22 Coding