Chapter 8-b Lossy Compression Algorithms

8/13/2019 Chapter 8-b Lossy Compression Algorithms

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Chapter 5: Lossy Compression

Algorithms

Multimedia Systems (ITEC 305)

Dr. Abir Akram EL ABED


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Introduction Lossless compression algorithms do not deliver compression ratios that are

high enough

Most multimedia compression algorithms are lossy What is lossycompression?

Compressed data is not the same as the original data, but a close

approximation of it

Yields a much higher compression ratio than that of lossless compression

2

A general

compression

system

model


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Techniques

3

Source coding is based on changing the content of the original signal

Compression rates may be high but at a price of loss of information Good compression rates may be achieved with source encoding with

(occasionally) lossless or (mostly) little perceivable loss of information

There are three broad methods that exist:

1) Transform Coding Transformation from one domain: time (e.g. 1D audio,video:2D imagery

over time) or Spatial (e.g. 2D imagery) domain to the frequency domain via

Discrete Cosine Transform (DCT) Heart of JPEG and MPEG Video,

MPEG Audio

Fourier Transform (FT)

MPEG Audio

2) Differential Encoding

3) Vector Quantisation


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Distortion Measures

4

The three most commonly used distortion measures in image compression

are:

1) Mean Square Error(MSE) 2

wherexn,yn, andNare the input data sequence, reconstructed datasequence, and length of the data sequence respectively

2) Signal to noise ratio (SNR), in decibel units (dB),

where is the average square value of the original data sequence

and is the MSE

3) Peak signal to noise ratio (PSNR),

2 2

1

1 ( )N

n n

n

x yN

2

x2

d

2

10 210log x

d

SNR

2

10 210log

peak

d

x

PSNR


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Quantization

5

Reduce the number of distinct output values to a much

smaller set

Main source of the lossin lossy compression

Three different forms of quantization:

1) Uniform: midrise and midtread quantizers

2) Nonuniform: companded quantizer

3) Vector Quantization


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Quantization

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A uniform scalar quantizer partitions the domain of

input values into equally spaced intervals, except

possibly at the two outer intervals

Output or reconstruction value to each interval =

midpoint of the interval

Length of each interval = step size ()

Two types of uniform scalar quantizers:

1) Midrise quantizers have even numberof output

levels

= 1 => = .

2) Midtread quantizers have odd number of output

levels, including zero as one of them

= 1 => = + .

Midrise quantizers

Midtread quantizers


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1) Uniform Scalar Quantization

7

A uniform scalar quantizer partitions the domain of

input values into equally spaced intervals, except

possibly at the two outer intervals

Output or reconstruction value to each interval =

midpoint of the interval

Length of each interval = step size ()

Two types of uniform scalar quantizers:

1) Midrise quantizers have even numberof output

levels

= 1 => = .

2) Midtread quantizers have odd number of output

levels, including zero as one of them

= 1 => = + .

Midrise quantizers

Midtread quantizers


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2) Companded quantizer

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Companded quantization is nonlinear

A compander consists of a compressor function G, a

uniform quantizer, and an expander function G1

The two commonly used companders are the -law and A-

law companders

Companded


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3) Vector Quantization (VQ)

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According to Shannons original work on information theory, any compression

system performs better if it operates on vectors or groups of samples rather than

individual symbols or samples Form vectors of input samples by simply concatenating a number of consecutive

samples into a single vector

Instead of single reconstruction values as in scalar quantization, in VQ code

vectorswith ncomponents are used

A collection of these code vectors form the codebook

Basic vector

quantization

procedure


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ec or uan sa onEncoding/Decoding

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Search Engine:

Group (Cluster) data into vectors

Find closest code vectors


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Example

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Consider Vectors of 2x2 blocks, and only allow 8 codes in

table

9 vector blocks present in above

9 vector blocks, so only one has to be vector quantisedhere

Resulting code book for above image

A small block of

images and intensity

values


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Transform Coding

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The rationale behind transform coding:

If Yis the result of a linear transform Tof the input vector Xin

such a way that the components of Y are much less

correlated, then Ycan be coded more efficiently than X.

If most information is accurately described by the first few

components of a transformed vector, then the remaining

components can be coarsely quantized, or even set tozero, with little signal distortion.

Discrete Cosine Transform (DCT) will be studied first


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Spatial Frequency and DCT

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Spatial frequency indicates how many times pixel values change across an

image block

DCT formalizes this notion with a measure of how much the image contents

change in correspondence to the number of cycles of a cosine wave per block

Role of DCT is to decompose the original signal into its DC and AC

components

Role of IDCT is toreconstruct(re-compose) the signal

Definition of DCT:

Given an input function f(i, j) over two integer variables i and j (a piece of an

image)

2D DCT transforms it into a new function F(u, v), with integer uand vrunning

over the same range as iandj. The general definition of the transform is:

i, u= 0, 1, . . . ,M1; j, v= 0, 1, . . . ,N1; and constants C(u) and C(v) are

determined by

1 1

0 0

2 ( ) ( ) (2 1) (2 1)( , ) cos cos ( , )

2 2

M N

i j

C u C v i u j vF u v f i j

M NMN

2 0,( ) 2

1 .

ifC

otherwise


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2D Discrete Cosine Transform (2D DCT)

where i,j, u, v= 0, 1, . . . , 7

2D Inverse Discrete Cosine Transform (2DIDCT)

The inverse function is almost the same, with the roles of f(i, j)andF(u, v) reversed, except that now C(u)C(v) must stand inside

the sums:

wherei,j, u, v= 0, 1, . . . , 7.14

7 7

0 0

( ) ( ) (2 1) (2 1)( , ) cos cos ( , )

4 16 16i j

C u C v i u j vF u v f i j

7 7

0 0

( ) ( ) (2 1) (2 1)( , ) cos cos ( , )4 16 16

u v

C u C v i u j vf i j F u v


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1D Discrete Cosine Transform (1D DCT):

where i= 0, 1, . . . , 7, u= 0, 1, . . . , 7.

1D Inverse Discrete Cosine Transform (1DIDCT):

where i= 0, 1, . . . , 7, u= 0, 1, . . . , 7.15

7

0

( ) (2 1)( ) cos ( )2 16

i

C u i uF u f i

7

0

( ) (2 1)( ) cos ( )2 16u

C u i uf i F u


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1DD

CTbasis

functions

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Graphical Illustration of

8 8 2D DCT basis

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Exampleso

f1DDiscreteC

osineTransfor

m:(a)ADC

signalf

1(i),

(b)AnAC

sign

alf

2(i),(c)f3(

i)

=f1

(i)+f2

(i),

and(d)ana

rbitrarysignal

f(i)..

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(a)

(b)

(c)

(d)

Chapter 8-b Lossy Compression Algorithms

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