Computer Vision – Computer Vision – Compression(1)Compression(1)

Hanyang University

Jong-Il Park

            

Department of Computer Science and Engineering, Hanyang University

Image CompressionImage Compression

The problem of reducing the amount of data required to represent a digital image

Underlying basis Removal of redundant data

Mathematical viewpoint Transforming a 2-D pixel array into a statistically

uncorrelated data set

            

Department of Computer Science and Engineering, Hanyang University

Topics to be coveredTopics to be covered

Fundamentals Basic concepts of source coding theorem

Practical techniques Lossless coding Lossy coding

Optimum quantization Predictive coding Transform coding

Standards JPEG MPEG Recent issues

            

Department of Computer Science and Engineering, Hanyang University

History of image compressionHistory of image compression Theoretic foundation

C.E.Shannon’s works in 1940s

Analog compression Aiming at reducing video transmission bandwidth

Bandwidth compression Eg. Subsampling methods, subcarrier modulation…

Digital compression Owing to the development of ICs and computers Early 70s: Facsimile transmission – 2D binary image

coding Academic research in 70s to 80s Rapidly matured around 1990. standardization such

as JPEG, MPEG, H.263, …

            

Department of Computer Science and Engineering, Hanyang University

Data redundancyData redundancy

Data vs. information

Data redundancy Relative data redundancy

Three basic redundancies1. Coding redundancy

2. Interpixel redundancy

3. Psychovisual redundancy

ratio)n compressio(/ where,1

1 21 nnCC

R RR

D

            

Department of Computer Science and Engineering, Hanyang University

Coding redundancyCoding redundancy

Code: a system of symbols used to represent a body of information or set of events

Code word: a sequence of code symbols

Code length: the number of symbols in each code word

Average number of bits

)()(1

0kr

L

kkavg rprlL

            

Department of Computer Science and Engineering, Hanyang University

Eg. Coding redundancyEg. Coding redundancy

Reduction by variable length coding

            

Department of Computer Science and Engineering, Hanyang University

CorrelationCorrelation

Cross correlation

Autocorrelation

1

0

1

0

),(),(*1

),(),(M

m

N

n

nymxhnmfMN

yxhyxf

),(),(*),(),( vuHvuFyxhyxf

2|),(|),(),(*),(),( vuFvuFvuFyxfyxf

            

Department of Computer Science and Engineering, Hanyang University

Eg. CorrelationEg. Correlation

            

Department of Computer Science and Engineering, Hanyang University

Interpixel redundancyInterpixel redundancy

Spatial redundancy Geometric redundancy Interframe redundancy

            

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Eg. Interpixel redundancyEg. Interpixel redundancy

            

Department of Computer Science and Engineering, Hanyang University

Eg. Run-length codingEg. Run-length coding

            

Department of Computer Science and Engineering, Hanyang University

Psychovisual redundancyPsychovisual redundancy

+

            

Department of Computer Science and Engineering, Hanyang University

Image compression modelsImage compression models

Communication model

Source encoder and decoder

            

Department of Computer Science and Engineering, Hanyang University

Basic concepts in information theoryBasic concepts in information theory

Self-information: I(E)= - log P(E) Source alphabet A and symbols Probability of the events z Ensemble (A, z) Entropy(=uncertainty): Channel alphabet B Channel matrix Q

J

ijjj aPaPH )(log)()(z

Qzv

            

Department of Computer Science and Engineering, Hanyang University

Mutual information and capacityMutual information and capacity

Equivocation: Mutual information: Channel capacity C

Minimum possible I(z,v)=0 Maximum possible I over all possible choices of source

probabilities in z is the channel capacity

)()(),( v|zzvz HHI )( v|zH

)],([max vzzIC

            

Department of Computer Science and Engineering, Hanyang University

Eg. Binary Symmetric ChannelEg. Binary Symmetric Channel

Entropy

Mutualinformation

Channel capacity

0 0

1 1

1-pe

1-pe

pe

pe

pbs

1-pbs

BSC

            

Department of Computer Science and Engineering, Hanyang University

Noiseless coding theoremNoiseless coding theorem

Shannon’s first theorem for a zero-memory source

It is possible to make L’avg/n arbitrarily close to H(z) by coding infinitely long extensions of the source

Efficiency = entropy/ L’avg

Eg. Extension coding Extension coding better efficiency

nH

n

LH avg 1

)('

)( zz

            

Department of Computer Science and Engineering, Hanyang University

Extension codingExtension coding

A

B

A Efficiency =0.918/1.0=0.918

B Efficiency =0.918*2/1.89=0.97

Better efficiency

            

Department of Computer Science and Engineering, Hanyang University

Noisy coding theoremNoisy coding theorem

Shannon’s second theorem for a zero-memory channel:

For any R<C, there exists an integer r and code of block length r and rate R such that the probability of a block decoding error is arbitrarily small.

Rate-Distortion theory

The source output can be recovered at the decoder with an arbitrarily small probability of error provided that the channel has capacity C > R(D)+e.

x

x

x

x

NeverFeasible!

feasible

            

Department of Computer Science and Engineering, Hanyang University

Using mappings to reduce entropyUsing mappings to reduce entropy

1st order estimate of entropy

> 2nd order estimate of entropy

> 3rd order estimate of entropy

….

The (estimated) entropy of a properly mapped image (eg. “difference source”) is in most cases smaller than that of original image source.

How to implement ?

The topic of the next lecture!