Data CompressionData Compression

Systematisation of Techniques and Systematisation of Techniques and MethodsMethods

Dr.-Ing.habil. Tilo StrutzDr.-Ing.habil. Tilo Strutz

University of RostockUniversity of Rostock

based on: Strutz: „Datenkompression - Grundlagen, Verfahren und deren Anwendung in der Verarbeitung von Graustufen- und Farbbildern“, 2002, based on: Strutz: „Datenkompression - Grundlagen, Verfahren und deren Anwendung in der Verarbeitung von Graustufen- und Farbbildern“, 2002,

thesis for doctor-habilitatus degree, University of Rostock, also: ISBN 3-89825-575-1thesis for doctor-habilitatus degree, University of Rostock, also: ISBN 3-89825-575-1

Tilo Strutz, University of Rostock 2

BasicsBasicsObjective of data compression: Objective of data compression:

transfer of information of digital data into such a description transfer of information of digital data into such a description format (embodiment), which requires as less expense of format (embodiment), which requires as less expense of

storage/transmission as possiblestorage/transmission as possible

Application:Application:

storage or transmission

of digital data

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Basics (2)Basics (2)

• What are primal characteristics of digital data?

Digital Data

Origin Format Resolution

• digitisation of analogue data (photography, microphone recordings)

• digital generation (ASCII text, computer graphics, machine- code)

• one-dimensional (audio signals, ultrasonic echos, text)

• two-dimensional (image signals)

• three-dimensional (image sequences)• ...

• binary (e.g. black/ white)

• multivalued (e.g. grey values with 8 bit)

• multi-channel (e.g. colour images with 3 x 8 bit)

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SystematisationSystematisation• Motivation:Motivation:

– rapid development at the field of data compressionrapid development at the field of data compression

– urgent needs for systematised overviewurgent needs for systematised overview

• Goals of this study:Goals of this study:

– uniform classification of techniques and methods uniform classification of techniques and methods clear overview over this areaclear overview over this area

– support for understanding of compressionsupport for understanding of compression

– guidance to the design of new compression systemsguidance to the design of new compression systems(reasonable combination of function blocks)(reasonable combination of function blocks)

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Systematisation (2)Systematisation (2)

Data CompressionData Compression

Decorrelation Data ReductionCoding

– reversible– removal of redundancy

– reversible (with restrictions)– concentration of signal energy or signal information

– irreversible– removal of irrelevancy

Adaptation

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Data ReductionData Reduction

Sub-sampling Quantisation

Vector Quantisation Scalar Quantisation

- reduction of temporal resolution

Quantisation

- mapping of similar vectors to a common representative vector

- mapping of similar values to a common representative value

- resolution reduction of signal values

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QuantisationQuantisation

• goal: removal of irrelevant mattersgoal: removal of irrelevant matters

• partition into two steps:partition into two steps:

– QuantisationQuantisation:: (encoder)(encoder)

– ReconstructionReconstruction::(decoder)(decoder)

x .. signal value, q ... quantisation symbol, yq ... reconstruction value

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CodingCoding• reversible, removal of redundancyreversible, removal of redundancy

• information content of one symbol information content of one symbol ssii

• entropy (average information content)entropy (average information content)

– ! ! refers to independent symbolsrefers to independent symbols ! !

– What happens if symbols depend on each other?What happens if symbols depend on each other?

– theoretical limit of minimum bitratetheoretical limit of minimum bitrate

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– considers symbols as dependent of each other

– mapping of symbols to symbols of another alphabet or seeking after correlationen

– goal: (H x N)new < (H x N)old

– considers symbols as independent of each other

– mapping of symbols to code words

– goal: convergence of storage expense to signal entropy H

Entropy Coding Precoding

CodingCoding

H ... Entropy of alphabet, N ... Number of symbols to be transmitted

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– substitution of symbol srings by a bitstream

– algorithms: common arithmetic coding, binary arithmetic coding, range coder, ...

– substitution of symbols by bit strings

– algorithms: Shannon-Fano, Huffman, Golomb-Rice, ...

Code-word based Arithmetic

Entropy CodingEntropy Coding

• Optimum: transmitted bits per symbol = information content [bit]Optimum: transmitted bits per symbol = information content [bit]

prefix codes

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PrecodingPrecoding

Run Length Coding Miscellaneous

Block SortingPhrase Coding

- substitution of strings of arbitrary symbols by symbols of another alphabet (dictionary- based coding)

- increase of correlation between adjacent symbols by sorting

- substitution of strings of identical symbols by symbols of another alphabet

- techniques for provision of supplemental information improving the subsequent processing (e.g. quad-tree coding, bit- mapping, etc.)

- application also multidimensional or hierarchical

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DecorrelationDecorrelation

• goal: concentration of signal energy / informationgoal: concentration of signal energy / information

DecorrelationDecorrelation

Prediction

– forecast of signal values

Filterbanks Transformation

– decomposition of signals in (overlapping) frequency domains

– decomposition of signals in basis functions

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PredictionPrediction

Sender Receiver

Recursive !Recursive !

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Prediction (2)Prediction (2)

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• common discrete signal transformationcommon discrete signal transformation

• inverse transformationinverse transformation

• matrix notationmatrix notation

TransformationTransformation

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Transformation (2)Transformation (2)• matrix for inverse transformation: matrix for inverse transformation: BB

• columns are basis vectorscolumns are basis vectors

B =

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Transformation (3)Transformation (3)

• columns are basis vectorscolumns are basis vectors

• signal reconstruction by sum of weighted basis signal reconstruction by sum of weighted basis functionsfunctions

– transform coefficients transform coefficients XX[[kk]] are the weights are the weights

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FilterbanksFilterbanks

• Two-Channel FilterbankTwo-Channel Filterbank

low-passlow-pass

high-passhigh-pass

upsamplingupsampling

subband signals/subband signals/

transformation coefficientstransformation coefficients

subsamplingsubsampling

g1[n]2 2

+

2 2 g0[n]h0[n]

h1[n]x[n] x‘[n]

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Filterbanks (2)Filterbanks (2)• cascade of 2-channel filterbankscascade of 2-channel filterbanks

• frequency decomposition:frequency decomposition:

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Filterbanks (3)Filterbanks (3)• Wavelet Filterbank Wavelet Filterbank Wavelet Transformation Wavelet Transformation

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Filterbanks (3)Filterbanks (3)• Wavelet Filterbank Wavelet Filterbank Wavelet Transformation Wavelet Transformation

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SystematisationSystematisation

Data CompressionData Compression

Decorrelation Data ReductionCoding

– reversible– removal of redundancy

– reversible (with restrictions)– concentration of signal energy or signal information

– irreversible– removal of irrelevancy

Adaptation

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• e.g. chrominances of Yxx-colour space

• feedback at prediction with quantisation

• prediction, transformation, filterbank

• maybe combination of different methods

• prefixcode, arithmetic code

• scalar quantisation or vector quantisation

• It depends on the application, whether function blocks are used or not

Flow ChartFlow Chart

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Characteristic PropertiesCharacteristic Properties

• DecorrelationDecorrelation– signal information will be concentrated to few values / signal information will be concentrated to few values /

symbolssymbols

– number of symbols / samples remains the samenumber of symbols / samples remains the same

– is reversible dependent on accuracy of calculationsis reversible dependent on accuracy of calculations

– result of decorrelation is tolerant against quantisation result of decorrelation is tolerant against quantisation (concerning signal reconstruction)(concerning signal reconstruction)

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Characteristic Properties (2)Characteristic Properties (2)• Entropy CodingEntropy Coding

– mapping of symbols to a bitstreammapping of symbols to a bitstream

– presumption: symbols are independent on each otherpresumption: symbols are independent on each other

– operation is invertibleoperation is invertible

• PrecodingPrecoding– mapping of one symbol alphabet to anothermapping of one symbol alphabet to another

– uses statistical relations between symbolsuses statistical relations between symbols

– number of symbols / samples can changenumber of symbols / samples can change

– operation is invertibleoperation is invertible

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Characteristic Properties (3)Characteristic Properties (3)

• Data ReductionData Reduction– decrease of irrelevancydecrease of irrelevancy

– precondition: relevant and irrelevant precondition: relevant and irrelevant constituents areconstituents are separated separated to the greatest possible extent to the greatest possible extent

– quantisation: number of samples remains the same quantisation: number of samples remains the same

– subsampling: number of samples decreasessubsampling: number of samples decreases

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Characteristic Properties (4)Characteristic Properties (4)

• AdaptationAdaptation– fixed signal modelfixed signal model

• updating of model parameterupdating of model parameter

• switching between parameter setsswitching between parameter sets

– different signal modelsdifferent signal models

– context-based context-based finite-state-machine finite-state-machine