1. Basics of Compression Goals: to understand how image/audio/video signals are compressed to save storage and increase transmission efficiency to understand in detail common formats like GIF, JPEG and MPEG

2. What does compression do Reduces signal size by taking advantage of correlation Spatial Temporal Spectral

3. Compression Issues Lossless compression Coding Efficiency Compression ratio Coder Complexity Memory needs Power needs Operations per second Coding Delay

4. Compression Issues Additional Issues for Lossy Compression Signal quality Bit error probability Signal/Noise ratio Mean opinion score

5. Compression Method Selection Constrained output signal quality? – TV Retransmission allowed? – interactive sessions Degradation type at decoder acceptable? Multiple decoding levels? – browsing and retrieving Encoding and decoding in tandem? – video editing Single-sample-based or block-based? Fixed coding delay, variable efficiency? Fixed output data size, variable rate? Fixed coding efficiency, variable signal quality? CBR Fixed quality, variable efficiency? VBR

6. Fundamental Ideas Run-length encoding Average Information Entropy For source S generating symbols S 1 through S N Self entropy: I(s i ) = Entropy of source S: Average number of bits to encode  H(S) - Shannon Differential Encoding To improve the probability distribution of symbols

7. Huffman Encoding Let an alphabet have N symbols S 1 … S N Let p i be the probability of occurrence of S i Order the symbols by their probabilities p 1  p 2  p 3  …  p N Replace symbols S N-1 and S N by a new symbol H N- 1 such that it has the probability p N-1 + p N Repeat until there is only one symbol This generates a binary tree

8. Huffman Encoding Example Symbols picked up as K+W {K,W}+? {K,W,?}+U {R,L} {K,W,?,U}+E {{K,W,?,U,E},{R,L}} Codewords are generated in a tree- traversal pattern SymbolProbabilityCodeword K0.0510101 L0.201 U0.1100 W0.0510100 E0.311 R0.200 ?0.11011

9. Properties of Huffman Codes Fixed-length inputs become variable-length outputs Average codeword length: Upper bound of average length: H(S) + p max Code construction complexity: O(N log N) Prefix-condition code: no codeword is a prefix for another Makes the code uniquely decodable

10. Huffman Decoding Look-up table-based decoding Create a look-up table Let L be the longest codeword Let c i be the codeword corresponding to symbol s i If c i has l i bits, make an L-bit address such that the first l i bits are c i and the rest can be any combination of 0s and 1s. Make 2^(L-l i ) such addresses for each symbol At each entry, record, (s i, l i ) pairs Decoding Read L bits in a buffer Get symbol s k, that has a length of l k Discard l k bits and fill up the L-bit buffer

11. Constraining Code Length The problem: the code length should be at most L bits Algorithm Let threshold T=2 -L Partition S into subsets S 1 and S 2 S 1 = {s i |p i > T} … t-1 symbols S 2 = {s i |p i  T} … N-t+1 symbols Create special symbol Q whose frequency q = Create code word c q for symbol Q and normal Huffman code for every other symbol in S 1 For any message string in S 1, create regular code word For any message string in S 2, first emit c q and then a regular code for the number of symbols in S 2

12. Constraining Code Length Another algorithm Set threshold t like before If for any symbol s i, p i  T, set p i = T Design the codebook If the resulting codebook violates the ordering of code length according to symbol frequency, reorder the codes How does this differ from the previous algorithm? What is its complexity?

13. Golomb-Rice Compression Take a source having symbols occurring with a geometric probability distribution P(n)=(1-p 0 ) p n 0 n  0, 0<p 0 <1 Here is P(n) the probability of run-length of n for any symbol Take a positive integer m and determine q, r where n=mq+r. G m code for n: 0…01 + bin(r) Optimal if m = q has log 2 m bits if r >sup(log 2 m)

14. Golomb-Rice Compression Now if m=2 k Get q by k-bit right shift and r by last r bits of n Example: If m=4 and n=22 the code is 00000110 Decoding G-R code: Count the number of 0s before the first 1. This gives q. Skip the 1. The next k bits gives r

15. The Lossless JPEG standard Try to predict the value of pixel X as prediction residual r=y-X y can be one of 0, a, b, c, a+b-c, a+(b- c)/2,b+(a-c)/2 The scan header records the choice of y Residual r is computed modulo 2 16 The number of bits needed for r is called its category, the binary value of r is its magnitude The category is Huffman encoded cb aX