Lecture 4: Data Compression Techniques TSBK01 Image Coding and Data Compression Jörgen Ahlberg Div. of Sensor Technology Swedish Defence Research Agency (FOI)

Outline  Huffman coding  Arithmetic coding  Application: JBIG  Universal coding  LZ-coding  LZ77, LZ78, LZW  Applications: GIF and PNG

Repetition  Coding: Assigning binary codewords to (blocks of) source symbols.  Variable-length codes (VLC) and fixed- length codes.  Instantaneous codes ½ Uniqely decodable codes ½ Non-singular codes ½ All codes  Tree codes are instantaneous.  Tree code, Kraft’s Inequality.

Creating a Code: The Data Compression Problem  Assume a source with an alphabet A and known symbol probabilities {p i }.  Goal: Chose the codeword lengths as to minimize the bitrate, i.e., the average number of bits per symbol  l i ¢ p i.  Trivial solution: l i = 0 8 i.  Restriction: We want an instantaneous code, so  2 -l i · 1 (KI) must be valid.  Solution (at least in theory): l i = – log p i

In practice…  Use some nice algorithm to find the code tree –Huffman coding –Tunnstall coding

Huffman Coding  Two-step algorithm: 1.Iterate: –Merge the least probable symbols. –Sort. 2.Assign bits. a d b c Merge Sort Assign Get code

Coding of the BMS  Trick: Code blocks of symbols (extended source).  Example: p 1 = ¼, p 2 = ¾.  Applying the Huffman algorithm directly: 1 bit/symbol. Block P (block) Code 009/160) 013/1610 approx /16110bits/symbol 111/16111

Huffman Coding: Pros and Cons +Fast implementations. + Error resilient: resynchronizes in ~ l 2 steps. -The code tree grows exponentially when the source is extended. -The symbol probabilities are built-in in the code. Hard to use Huffman coding for extended sources / large alphabets or when the symbol probabilities are varying by time.

Arithmetic Coding  Shannon-Fano-Elias  Basic idea: Split the interval [0,1] according to the symbol probabilities.  Example: A = {a,b,c,d}, P = {½, ¼, 1/8, 1/8}.

b c a Start in b. Code the sequence c c a. cb a 0.9 Code the sequence c c a. ) Code the interval [0.9, 0.96] Bit IntervalDecoder 1c c a bc cba

An Image Coding Application  Consider the image content in a local environment of a pixel as a state in a Markov model.  Example (binary image):  Such an environment is called a context.  A probability distribution for X can be estimated for each state. Then arithmetic coding is used.  This is the basic idea behind the JBIG algorithm for binary images and data. X

Flushing the Coder  The coding process is ended (restarted) and the coder flushed –after a given number of symbols (FIVO) or –When the interval is too small for a fixed number of output bits (VIFO).

Universal Coding  A universal coder doesn’t need to know the statistics in advance. Instead, estimate from data.  Forward estimation: Estimate statistics in a first pass and transmit to the decoder.  Backward estimation: Estimate from already transmitted (received) symbols.

Universal Coding: Examples 1.An adaptive arithmetic coder 2.An adaptive dictionary technique –The LZ coders [Sayood 5] 3.An adaptive Huffman coder [Sayood 3.4] Arithmetic coder Statistics estimation

Ziv-Lempel Coding (ZL or LZ)  Named after J. Ziv and A. Lempel (1977).  Adaptive dictionary technique. –Store previously coded symbols in a buffer. –Search for the current sequence of symbols to code. –If found, transmit buffer offset and length.

LZ77 abcabd acab d eee f c Search bufferLook-ahead buffer Output triplet d0e2f 2 Transmitted to decoder: If the size of the search buffer is N and the size of the alphabet is M we need bits to code a triplet. Variation: Variation: Use a VLC to code the triplets! PKZip, Zip, Lharc, PNG, gzip, ARJ

Drawback with LZ77  Repetetive patterns with a period longer than the search buffer size are not found.  If the search buffer size is 4, the sequence a b c d e a b c d e a b c d e a b c d e … will be expanded, not compressed.

LZ78  Store patterns in a dictionary  Transmit a tuple  Transmit a tuple

LZ78 Output tuple Dictionary: 1 a 2 b 3 c 4 a b 5 a b c abcaba bc 0a Transmitted to decoder: 0b0c1b4c Decoded:a b c a a b b c Strategy needed for limiting dictionary size!

LZW  Modification to LZ78 by Terry Welch,  Applications: GIF, v42bis  Patented by UniSys Corp.  Transmit only the dictionary index.  The alphabet is stored in the dictionary in advance.

LZW Output: dictionary index Encoder dictionary: 1 a 2 b 3 c 4 d 5 a b abcaba bc 1 Transmitted: 2355 Decoded: abca b 6 bc 7 ca 8 aba 9 abc Decoder dictionary: 1 a 2 b 3 c 4 d 5 a b 6 bc 7 ca 8 aba Input sequence:

And now for some applications: GIF & PNG

GIF  CompuServe Graphics Interchange Format (1987, 89).  Features: –Designed for up/downloading images to/from BBSes via PSTN. –1-, 4-, or 8-bit colour palettes. –Interlace for progressive decoding (four passes, starts with every 8th row). –Transparent colour for non-rectangular images. –Supports multiple images in one file (”animated GIFs”).

GIF: Method  Compression by LZW.  Dictionary size 2 b+1 8-bit symbols –b is the number of bits in the palette.  Dictionary size doubled if filled (max 4096).  Works well on computer generated images.

GIF: Problems  Unsuitable for natural images (photos): –Maximum 256 colors ( ) bad quality). –Repetetive patterns uncommon ( ) bad compression).  LZW patented by UniSys Corp.  Alternative: PNG

PNG: Portable Network Graphics  Designed to replace GIF.  Some features: –Indexed or true-colour images ( · 16 bits per plane). –Alpha channel. –Gamma information. –Error detection.  No support for multiple images in one file. –Use MNG for that.  Method: –Compression by LZ77 using a 32KB search buffer. –The LZ77 triplets are Huffman coded.  More information:

Summary  Huffman coding –Simple, easy, fast –Complexity grows exponentially with the block length –Statistics built-in in the code  Arithmetic coding –Complexity grows linearly with the block size –Easily adapted to variable statistics ) used for coding of Markov sources  Universal coding –Adaptive Huffman or arithmetic coder –LZ77: Buffer with previously sent sequences –LZ77: Buffer with previously sent sequences –LZ78: Dictionary instead of buffer –LZ78: Dictionary instead of buffer –LZW: Modification to LZ78 –LZW: Modification to LZ78

Summary, cont  Where are the algorithms used? –Huffman coding: JPEG, MPEG, PNG, … –Arithmetic coding: JPEG, JBIG, MPEG-4, … –LZ77: PNG, PKZip, Zip, gzip, … –LZW: compress, GIF, v42bis, …

Finally  These methods work best if the source alphabet is small and the distribution skewed. –Text –Graphics  Analog sources (images, sound) require other methods –complex dependencies –accepted distortion