# An introduction to Data Compression

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An introduction to Data Compression

General informations Requirements some programming skills (not so much...) knowledge of data structures ... some work! Office hours ... ... please write me an Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

What is compression? Intuitively compression is a method “to press something into a smaller space”. In our domains a better definition is “to make information shorter” Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Some basic questions What is information?
How can we measure the amount of information? Why compression is useful? How do we compress? How much we can compress? Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

What is information? - I Commonly the term information refers to the knowledge of some fact, circumstance or thought. For example we can think about reading a newspaper, news are the information. syntax letters, punctuation marks, white spaces, grammar rules ... semantics meaning of the words and of the sentences Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

What is information? - II
In our domain, information is merely the syntax, i.e. we are interested in the symbols of the alphabet used to express the information. In order to give a mathematical definition of information we need some principle of Information Theory Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

The fundamental concept
A key concept in Information Theory is that the information is conveyed by randomness Which information give us a biased coin, which outcome is always head? What about another biased coin, which outcome is head with 90% probability? We need a way to measure quantitatively the amount of information in some mathematical sense Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

The Uncertainty - I Suppose we have a discrete random variable and is a particular outcome with probability uncertainty The units are given by the base of the logarithms base 2  bits base 10  nats Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

The Uncertanty - II Suppose the random variable output
 each outcome has 1 bit of information  0 gives no information at all, while if the outcome is 1 the information is Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

The Entropy More useful is the entropy of a random variable with values in a space The entropy is a measure of the average uncertanty of the random variable Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

The entropy - examples Consider again a r.v. with only two possible outcomes, 0 and 1 In this case Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Compression and loss lossless lossy
decompressed message (file) is an exact copy of the original. Useful for text compression lossy some information is lost in the decompressed message (file). Useful for image and sound compression lgnore for a while lossy compression Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Definitions - I A source code from a r.v. is a mapping from to , the set of finite-length string from a D-ary alphabet. , codeword for , length of Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Definitions - II non-singular code (... trivial ...)
every element of is mapped in a different string of : extension of a code uniquely decodable code its extension is uniquely decodable Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Definitions - III prefix (better prefix-free) or istantaneous code a
no codeword is a prefix of any other codeword the advantage is that decoding has no need to look-ahead codewords a 11 b 110 ... 11? ... Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Examples Code 1 Code 2 Code 3 Code 4 1 01 10 2 110 010 00 3 11 4 111
10 2 110 010 00 3 11 4 111 singular not singular, but not uniquely decodable uniquely decodable, but not instantaneous instantaneous Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Kraft Inequality - I Theorem (Kraft Inequality)
For any instantaneous code over an alphabet of size D, the codeword lengths must satisfy Conversely, given a set of codeword lengths that satisfy this inequality there exists an istantaneous code with these word lengths Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Kraft Inequality - II Consider a complete D-ary tree
at level k, there are nodes a node at level has descendants that are nodes at level k level 0 level 1 level 2 level 3 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Kraft Inequality - III Proof
Consider a D-ary tree (not necessarily complete) representing the codewords, each path down the tree is a sequence of symbols, and each leaf (with its unique path) is a codeword. Let be the longest codeword. A codeword of length , being a leaf, imply that at level there are missing nodes Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Kraft Inequality - IV The total number of possible nodes at level is
Summing over all codewords Dividing by Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Kraft Inequality - V Proof
Suppose (without loss of generality) that codewords are ordered by length, i.e Consider a D-ary tree and start assigning each codeword to a node, starting from . For a generic codeword with length consider the set K of codewords with length , except i. Suppose there is no available node at level i. That is, Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Kraft Inequality - VI but this means that Then
that is absurd. Then the obtained tree represents an instantaneous code with desidered codeword lengths Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Models and coders model model compressed text text text encoder decoder The model supplies the probabilities of the symbols (or of the group of symbols, as we will see later) The coder encodes and decodes starting from these probabilities Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Good modeling is crucial
What happens if the true probability of the symbols to be coded are but we use ? Simply, compressed text will be longer, i.e. the average number of bits/symbol will be greater It is possible to calculate the difference in bit/symbol from the two mass probability p and q, known as relative entropy Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Finite-context models
in english text ... ... but A finite-context model of order m uses the previous m symbols to make the prediction Better modeling but we need to extimate much more probabilities Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Finite-state models a 0.5 b 0.5 b 0.01 1 2 a 0.99 Although potentially more powerful (e.g. they can model wheather an odd or even number of as have occurred consecutively), they are not so popular. Obviously the decoder uses the same model, so they are always in the same states Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Static models A models is static if we set up a reasonable probability distribution and use it for all the texts to be coded. Poor performance in case of different kind of sources (english text, financial data...) One solution is to have K different models and to send the index of the used model ... but cfr. the book Gadsby by E. V. Wright Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Adaptive models In order to solve the problems of static modeling, adaptive (or dynamic) models begin with a bland probability distribution, that is refined as more symbols of the text are known The encoder and the decoder have the same initial distribution, and the same rules to alter it There could be adaptive models of order m>0 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

The zero-frequency problem
The situation in which a symbol is predicted with probability zero should be avoided, as it cannot be coded One solution: the total number of symbols in the text is increased by 1. This 1/total probability is divided among all unseen symbols Another solution: to augment by 1 the count of every symbol Many more solutions... Which is the best? If text is sufficiently long the compression is similar Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a

Symbolwise and dictionary models
The set of all possible symbols of a source is called the alphabet Symbolwise models provide an extimated probability for each symbol in the alphabet Dictionary models instead replace substrings in a text with codewords that identify each substring in a collection, called dictionary or codebook Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a