Almost every lossy compression system contains a lossless compression system

Transcription

1 Lossless compression in lossy compression systems Almost every lossy compression system contains a lossless compression system Lossy compression system Transform Quantizer Lossless Encoder Lossless Decoder Lossless compression system Dequantizer Inverse Transform We discuss the basics of lossless compression first, then move on to lossy compression Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no.

2 Topics in lossless compression Binary decision trees and variable length coding Entropy and bit-rate Prefix codes, Huffman codes, Golomb codes Joint entropy, conditional entropy, sources with memory Fax compression standards Arithmetic coding Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 2

3 Example: 2 Questions Alice thinks of an outcome (from a finite set), but does not disclose her selection. Bob asks a series of yes/no questions to uniquely determine the outcome chosen. The goal of the game is to ask as few questions as possible on average. Our goal: Design the best strategy for Bob. Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 3

4 Example: 2 Questions (cont.) Which strategy is better? (=no) (=yes) A B C D E F A B C D E F Observation: The collection of questions and answers yield a binary code for each outcome. Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 4

5 Fixed length codes A B C D E F G H Average description length for K outcomes Optimum for equally likely outcomes Verify by modifying tree l av log 2 K Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 5

6 Variable length codes If outcomes are NOT equally probable: Use shorter descriptions for likely outcomes Use longer descriptions for less likely outcomes Intuition: Optimum balanced code trees, i.e., with equally likely outcomes, can be pruned to yield unbalanced trees with unequal probabilities. The unbalanced code trees such obtained are also optimum. Hence, an outcome of probability p should require about log 2 p bits Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 6

7 Consider a discrete, finite-alphabet random variable Information associated with the event =x Entropy of is the expected value of that information Unit: bits Entropy of a random variable Alphabet {,, 2,..., K } PMF f x P x for each x H E h f x log 2 f x x h x log 2 f x Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 7

8 Information and entropy: properties Information h x Information h (x) strictly increases with decreasing probability f (x) Boundedness of entropy Equality if only one outcome can occur H( ) log 2 Equality if all outcomes are equally likely Very likely and very unlikely events do not substantially change entropy plog p for p or p 2 Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 8

9 Example: Binary random variable H plog p ( p)log ( p) 2 2 H Equally likely deterministic p Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 9

10 Consider IID random process (or source ) where each sample (or symbol ) possesses identical entropy H() n H() is called entropy rate of the random process. Noiseless Source Coding Theorem [Shannon, 948] The entropy H() is a lower bound for the average word length R of a decodable variable-length code for the symbols. Conversely, the average word length R can approach H(), if sufficiently large blocks of symbols are encoded jointly. Redundancy of a code: Entropy and bit-rate n R H Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no.

11 Given IID random process with alphabet and PMF f x Task: assign a distinct code word, c x, to each element, x, where cx is a string of c bits, such that each x symbol x n can be determined, even if the codewords are directly concatenated in a bitstream Codes with the above property are said to be uniquely decodable. Prefix codes Variable length codes n No code word is a prefix of any other codeword Uniquely decodable, symbol by symbol, in natural order,, 2,..., n,... c xn Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no.

12 Example of non-decodable code Encode sequence of source symbols,,,, 2 3 Resulting bit-stream Encode sequence of source symbols,, 3,, Resulting bit-stream Same bit-stream for different sequences of source symbols: ambiguous, not uniquely decodable BTW: Not a prefix code. Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 2

13 Unique decodability: McMillan and Kraft conditions Necessary condition for unique decodability [McMillan] x c x 2 Given a set of code word lengths c x satisfying McMillan condition, a corresponding prefix code always exists [Kraft] Hence, McMillan inequality is both necessary and sufficient. Also known as Kraft inequality or Kraft-McMillan inequality. No loss by only considering prefix codes. Prefix code is not unique. Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 3

14 Prefix Decoder Shift register to hold longest code word Code word LUT... Input buffer Advance c x bits Code word length LUT... Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 4

15 Binary trees and prefix codes Any binary tree can be converted into a prefix code by traversing the tree from root to leaves. Any prefix code corresponding to a binary tree meets McMillan condition with equality x c x 2 Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 5

16 Binary trees and prefix codes (cont.) Augmenting binary tree by two new nodes does not change McMillan sum. Pruning binary tree does not change McMillan sum. 2 l l l l McMillan sum for simplest binary tree 2 2 Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 6

17 Instantaneous variable length encoding without redundancy A code without redundancy, i.e. R H( ) Example requires all individual code word lengths l log 2 k k f All probabilities would have to be binary fractions: f k l ( ) 2 k.75 bits H R.75 bits Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 7

18 Huffman Code Design algorithm for variable length codes proposed by Huffman (952) always finds a code with minimum redundancy. Obtain code tree as follows: Pick the two symbols with lowest probabilities and merge them into a new auxiliary symbol. 2 Calculate the probability of the auxiliary symbol. 3 If more than one symbol remains, repeat steps and 2 for the new auxiliary alphabet. 4 Convert the code tree into a prefix code. Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 8

19 Huffman Code - Example Fixed length coding: Huffman code: Entropy Redundancy of the Huffman code: R fixed 4 bits/symbol R Huffman 2.77 bits/symbol H( ) 2.69 bits/symbol.8 bits/symbol Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 9

20 Redundancy of prefix code for general distribution Huffman code redundancy Theorem: For any distribution f, a prefix code can be found, whose rate R satisfies Proof Left hand inequality: Shannon s noiseless coding theorem Right hand inequality: H R H Choose code word lengths bit/symbol Resulting rate cx log2 f x 2 R f x log2 f x x x H f x log f x Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 2

21 Vector Huffman coding Huffman coding very inefficient for H() << bit/symbol Remedy: Combine m successive symbols to a new block-symbol Huffman code for block-symbols Redundancy H R H Can also be used to exploit statistical dependencies between successive symbols Disadvantage: exponentially growing alphabet size m m Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 2

22 Truncated Huffman Coding Idea: reduce size of Huffman code table and maximum Huffman code word length by Huffman-coding only the most probable symbols. Combine J least probable symbols of an alphabet of size K into an auxillary symbol ESC Use Huffman code for alphabet consisting of remaining K-J most probable symbols and the symbol ESC log J If ESC symbol is encoded, append 2 bits to specify exact symbol from the full alphabet Results in increased average code word length trade off complexity and efficiency by choosing J Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 22

23 Adaptive Huffman Coding Use, if source statistics are not known ahead of time Forward adaptation Measure source statistics at encoder by analyzing entire data Transmit Huffman code table ahead of compressed bit-stream JPEG uses this concept (even though often default tables are transmitted) Backward adaptation Measure source statistics both at encoder and decoder, using the same previously decoded data Regularly generate identical Huffman code tables at transmitter and receiver Saves overhead of forward adaptation, but usually poorer code tables, since based on past observations Generally avoided due to computational burden at decoder Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 23

24 Unary coding Geometric source x Alphabet,,... PMF f x 2, x Optimal prefix code with redundancy is unary code ( comma code ) c "" c "" c "" c "" 2 3 Consider geometric source with faster decay x PMF f x, with ; x 2 Unary code is still optimum prefix code (i.e., Huffman code), but not redundancy-free Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 24

25 Golomb coding For geometric source with slower decay x PMF f x, with ; x 2 Idea: Express each x as x mx q x r with x q x and x m r x mod m Distribution of new random variables m m mxq f x f mx i f i q q q i i xr f x for r r x m r m q and r statistically independent. Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 25

26 Golomb coding (cont.) Golomb coding m Choose integer divisor 2 Encode x q optimally by unary code Encode x r by a modified binary code, using code word lengths k a log 2 m k b log 2 m Concatenate bits for x q and x r In practice, m=2 k is often used, so x r can be encoded by constant code word length log m 2 Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 26

27 Golomb code examples Unary Code... Unary Code... Constant length code m=2 Unary Code... Constant length code m=4 Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 27

28 Golomb parameter estimation Expected value for geometric distribution x E x E E x Approximation for E m E m m E E m m k 2 k E 2 2 max, log2 E 2 Rule for optimum performance of Golomb code Reasonable setting, even if E does not hold Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 28

29 Adaptive Golomb coder (JPEG-LS) Initialize A ˆ and N For each n,,2, A Set k max, log2 2N Code symbol x using Golomb code with parameter If N x N max n Initial estimate of mean Set A A/ 2 and N N / 2 Update A A x and N N n k Pick the best Golomb code Avoid overflow and slowly forget the past Bernd Girod: EE398A Image and Video Compression Entropy and Lossless Coding no. 29