Source Coding for Compression
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1 Source Coding for Compression Types of data compression: 1. Lossless -. Lossy removes redundancies (reversible) removes less important information (irreversible) Lec 16b.6-1 M1
2 Lossless Entropy Coding, e.g. Huffman Coding Example 4 possible messages, bits uniquely specifies each if p (A) = 0. 5 = 0 p (B) = 0. 5 = 1 1 forms comma-less p (C) = code = p (D) = = Example of comma-less message: codeword grouping (unique) Lec 16b.6- Then the average number of bits per message ( rate ) R = 0. 5 Define entropy 1 bit = H = i p i log Define coding efficiency (here H = 1.75) η c = H R 1 p i 1.75 bits/messa ge H of a message ensemble M
3 Simple Huffman Example Simple binary message with p{0} = p{1} = 0.5; each bit is independent, then entropy (H): (. 5 log 0. 5 ) = 1 bit/symbol R H = 0 = If there is no predictability to the message, then there is no opportunity for further lossless compression Lec 16b.6-3 M3
4 Lossless Coding Format Options MESSAGE CODED uniform blocks non-uniform non-uniform uniform non-uniform non-uniform Lec 16b.6-4 M4
5 Run-length Coding Example: N N 1111 N run-length coding Huffman code in this sequence n i ( 3, 7,, 4, 9, ) optional Note: opportunity for compression here comes from tendancy for long runs of 0 s or 1 s Simple: Better: { ; i = 1, } code p n i { n n ; i = 1,.. } code p i i 1 If n i correlated with n i 1 Lec 16b.6-5 M5
6 Other Popular Codes Arithmetic codes: (e.g. see Feb. 89, IEEE Trans. Comm., 37,, pp ) One of the best entropy codes for it adapts well to the message, but it involves some computation in real time. Lempel-Ziv-Welch (LZW) Codes: Deterministically compress digitial streams adaptively, reversibly, and efficiently Lec 16b.6-6 M6
7 Information-Lossy Source Codes Common approaches to lossy coding: 1) Quantization of analog signals ) Transform signal blocks; quantize the transform coefficients 3) Differential coding: code only derivatives or changes most of the time; periodically reset absolute value 4) In general, reduce redundancy and use predictability; communicate only unpredictable parts, assuming prior message was received correctly 5) Omit signal elements less visible or useful to recipient Lec 16b.6-7 N1
8 Discrete Fourier Transform (DFT): e.g X (n) [n = 0, 1,, N 1] n = 0 x (k) n = 1 Transform Codes - DFT = = N 1 k = 0 1 N N 1 n = 0 x(k)e n = X(n)e jn π jn π ( k N ) ( k N ) Inverse DFT IDFT Note: X(n) is complex N # s Lec 16b sharp edges of window ringing or sidelobes in the reconstructed decoded signal N
9 Example of DCT Image Coding Say 8 8 block: 8 8 real # s Can sequence coefficients, stopping when they are too small, e.g.: D.C. term may stop here Can classify blocks, and assign bits correspondingly Lec 16b.6-9 Image Types: A. Smooth image B. Horizontal striations C. Vertical striations D. Diagonals (utilize correlations) Contours of typical DCT coefficient magnitudes A B C D N3
10 Discrete Cosine and Sine Transforms (DCT and DST) Discrete Cosine Transform (DCT) 1 Discrete Sine Transform (DST) The DC basis function (n = 1) is an advantage, but the step functions at the ends produce artifacts at block boundaries of reconstructions unless n Lack of a DC term is a disadvantage, but zeros at end often overlap Lec 16b.6-10 N4
11 Lapped Transforms ~DC term Lapped Orthogonal Transform = (LOT) (1:1, invertible; orthogonal between blocks) block extended block Reconstructions ring less, but ring outside the quantized block Lec 16b.6-11 (a) Even Basis Functions ( b) Odd Basis Functions An optimal LOT for N =16, L = 3, and ρ = 0.95 N5
12 Lapped Transforms central block zeros, still lower sidelobes t MLT = Modulated Lapped Transform First basis function for MLT Ref: Henrique S. Malvar and D.H. Staelin, The LOT: Transform Coding Without Blocking Effects, IEEE Trans. on Acous., Speech, and Sign. Proc., 37(4), (1989). Lec 16b.6-1 N6
13 Karhounen-Loéve Transform (KLT) Maximizes energy compaction within blocks for jointly gaussian processes Example: Average pixel energy D.C. term Average energy DFT, DCT, DST 0 j transform LOT, MLT, KLT 0 n 0 j Note: The KLT for a first order Markov process is the DCT Lapped KLT (best) Lec 16b j N7
14 Value Vector Quntization ( VQ ) Example: consider pairs of samples y = a,b as vectors. [ ] a b b y 1 y VQ assigns each cell an integer number, unique b p(a,b) Can Huffman code cell numbers more general Lec 16b.6-14 a a VQ is better because more probable cells are smaller and well packed. VQ is n-dimensional (n = 4 to 16 is typical). There is a tradeoff between performance and computation cost P1
15 Reconstruction Errors When such block transforms are truncated (high frequency terms omitted) or quantized, their reconstructions tend to ring The reconstruction error is the superposition of the truncated (omitted or imperfectly quantized) sinusoids. t t window functions (blocks) [original f(t)] Reconstructed signal from truncated coefficients and quantization errors Ringing and block-edge errors can be reduced by using orthogonal overlapping tapered transforms (e.g., LOT, ELT, MLT, etc.) Lec 16b.6-15 P
16 Smoothing with Pseudo-Random Noise (PRN) Problem: Coarsely quantized images are visually unacceptable Solution: Add spatially white PRN to image before quantization, and subtract identical PRN from quantized reconstruction; result shows no quantization contours (zero!). PRN must be uniformly distributed, zero mean, with range equal to quantization interval. s(x,y) Q channel Q 1 - s(x, ˆ y) PRN(x,y) PRN(x,y) Lec 16b.6-16 P3
17 Smoothing with Pseudo-Random Noise (PRN) s(x,y) Q 1 Q channel s(x, ˆ y) - PRN(x,y) PRN(x,y) s(x) sprn(x) A p{prn} A 0 Q 1 [ Q ( s(x) )] x -A/ 0 A/ PRN 0 x s(x) A 0 Lec 16b.6-17 d d A x 0 s(x ˆ ) A s(x ˆ ) = Q s(x) PRN(x) PRN(x) x [ ] 0 fil tered sˆ (x) = ŝ(x) h(x) x P4
18 Example of Predictive Coding s(t) t δ - code δ s δ (t ) δ (t ) channel ~s(t ) s(t ˆ ) decode δ s decode δ s δ (t ) s(t ˆ ) predictor (3 ) = computation time predictor (3 ) t - t - delay s(t ˆ ) The predictor can simply predict using derivatives, or can be very sophisticated, e.g. full image motion compensation. Lec 16b.6-18 P5
19 Joint Source and Channel Coding Source coding Channel coding high priority data high degree of protection channel medium priority medium protection lowest priority lowest degree of protection For example: lowest priority data may be highest spatial (or time) frequency components. Lec 16b.6-19 P6
20 Prefiltering and Postfiltering Problem s(t) (sound, image, etc.) truth channel sampler f(t) g(t) h(t) n 1 (t) prefilter postfilter i(t) n (t) - O(t) O(t) out observer response function ( ) MSE (minimize) Given s(t), f(t), i(t), O(t), n 1 (t), and n (t) [channel plus receiver plus quantization noise], choose g(t), h(t) to minimize MSE. Lec 16b.6-0 P7
21 Typical solution: Lec 16b.6-1 some sharpening Interpretation: Solution: s(f Prefiltering and Postfiltering ) G(f ) 0 Given f o g(t) Mexican-hat function f By boosting the weaker signals relative to the stronger ones prior to adding aliasing and n (t), better weak-signal (high-frequency) performance follows. Prefilters and postfilters first boost and then attenuate weak signal frequencies. t S(f ), N(f ) S(f ) g(t) (Ref: H. Malvar, MIT EECS PhD thesis, 1987) f o N (f ) f 0, aliasing f o h(t) s(f) G(f) H(f) t Net N (f ) aliasing f P8
22 Analog Communications Double Sideband Synchronous Carrier DSBSC : Received signal = A s(t) cos ω t n(t) c c s(t) cos ω c (t) S(f) W kt kt = N R n (t) = N o 4W 0 f c f Lec 16b.6- R1
23 DSBSC Receiver [ ] A s(t) n (t) cos ω t n (t) sin ω t c c c s c sig(f) LPF y(t) y LPF (t) W kt SNR out =? cos ω c t Let n(t) = n (t) cos ω t n (t)sin ω t c c s c slowly varying slowly varying 0 f c f So: n (t) = n c (t) n s (t) = n c = ns = N o W Lec 16b.6-3 c c c s c c [ A s(t n (t ] cos n (t ( )( cos ) y(t) = ) ) ω t ) sin ω t ω t n s (t) = sin ω c t (filtered out by low-pass filter) R
24 So: n DSBSC Carrier (t) = n c (t) n s (t) = n c = ns = o [ ] c s(t) c (t) cos ωc n s (t) s c t cos c y(t) = A n t in ω ω t n s (t) = sin ω c t (filtered out by low-pass filter) cos ω c t = 1( 1 cos ω c t) Therefore y 1 LPF (t) = [ A c s(t) n c ( t) ] (low-pass filtered ) out out = c t) n c = [ c N ow ] s let max = 1 4N ow where carrier power P c = A c S N A s ( (t) P (t) ( ) N W Lec 16b.6-4 ( ) = "CNR" = "Carrier-to-Noise Ratio" for s = 1 DSBSC R3
25 Single-sideband SSB Systems (Synchronous carrier) W S N = out out Ps c (t) N W o -fc 0 P s fc N watts f Note: Both signal and noise are halved, so S N = S N out out SSBSC out out DSBSC Lec 16b.6-5 R4
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