CSc 461/561 CSc 461/561 Multimedia Systems Part B: 1. Lossless Compression

CSc 461/561 Summary (1) Information (2) Types of compression (3) Lossless compression algorithms (a)Shannon-Fano Algorithm (b)Huffman coding (c)Run-length coding (d)LZW compression (e)Arithmetic Coding (4) Example: Lossless image compression

CSc 461/ Information (1) Information is decided by three parts: The source The receiver The delivery channel We need a way to measure information: Entropy: a measure of uncertainty; min bits –alphabet set {s 1, s 2, …, s n } –probability {p 1, p 2, …, p n } –entropy: - p 1 log 2 p 1 - p 2 log 2 p 2 - … - p n log 2 p n

CSc 461/ Entropy examples (2) Alphabet set {0, 1} Probability: {p, 1-p} Entropy: H = - p log 2 p - (1-p) log 2 (1-p) –when p=0, H=0 –when p=1, H=0 –when p=1/2, H max =1 1 bit is enough!

CSc 461/ Types of compression (1) Lossless compression: no information loss Lossy compression: otherwise

CSc 461/ Compression Ratio (2) Compression ratio –B 0 : # of bits to represent before compression –B 1 : # of bits to represent after compression –compression ratio = B 0 /B 1

CSc 461/ Shannon-Fano algorithm (1) Fewer bits for symbols appear more often “divide-and-conquer” –also known as “top-down” approach –split alphabet set into subsets of (roughly) equal probabilities; do it recursively –similar to building a binary tree

CSc 461/ Shannon-Fano: examples (2)

CSc 461/ Shannon-Fano: results (3) Prefix-free code –no code is a prefix of other codes –easy to decode

CSc 461/ Shannon-Fano: more results (4) Encoding is not unique –roughly equal Encoding 2 Encoding 1

CSc 461/ Huffman coding (1) “Bottom-up” approach –also build a binary tree and know alphabet probability! –start with two symbols of the least probability s 1 : p 1 s 2 : p 2 s 1 or s 2 : p 1 +p 2 –do it recursively

CSc 461/ Huffman coding: examples (2) Encoding not unique; prefix-free code Optimality: H(S) <= L < H(S)+1 a2 (0.4) a1(0.2) a3(0.2) a4(0.1) a5(0.1) Sort 0.2 combineSort combine Sort combine Sort 1 combine Assign code

CSc 461/ Run-length coding Run: a string of the same symbol Example –input: AAABBCCCCCCCCCAA –output: A3B2C9A2 –compression ratio = 16/8 = 2 Good for some inputs (with long runs) –bad for others: ABCABC –how about to treat ABC as an alphabet?

CSc 461/ LZW compression (1) Lempel-Ziv-Welch (LZ77, W84) –Dictionary-based compression –no a priori knowledge on alphabet probability –build the dictionary on-the-fly –used widely: e.g., Unix compress LZW coding –if a word does not appear in the dictionary, add it –refer to the dictionary when the word appears again

CSc 461/ LZW examples (2) Input –ABABBABCABABBA Output –

CSc 461/ Arithmetic Coding (1) Arithmetic coding determines a model of the data -- basically a prediction of what patterns will be found in the symbols of the message. The more accurate this prediction is, the closer to optimality the output will be. Arithmetic coding treats the whole message as one unit.

CSc 461/ Arithmetic Coding (2)

CSc 461/ Arithmetic Coding (3)

CSc 461/ Arithmetic Coding (4)

CSc 461/ Arithmetic Coding (5)

CSc 461/ Lossless Image Compression (1)

CSc 461/ Lossless Image Compression (2)

CSc 461/ Lossless JPEG NNeighboring Pixels for Predictors in Lossless JPEG NeighPredictors for Lossless JPEG