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Lossy Compression Algorithms-Multimedia Applications-Lecture Slides, Slides of Multimedia Applications

Biju Patnaik University of Technology Multimedia Applications

This lecture was delivered by Dr. Paresh Sapan at Biju Patnaik University of Technology, Rourkela. This lecture is part of lecture series on Multimedia Applications course. It includes: Lossy, Compression, Algorithms, Distortion, Measures, Rate, Distortion, Theory, Quantization, Transform, Coding

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Lecture 18-19 Lossy Compression

Algorithms (Chapter 8 )

• 8.1 Introduction

• 8.2 Distortion Measures

• 8.3 The Rate-Distortion Theory

• 8.4 Quantization

• 8.5 Transform Coding

• Sections 8.5.2 to 8.9 reading-optional!

• 8.10 Further Exploration

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Lecture 18-19 Lossy Compression

Algorithms (Chapter 8 )

- 8.1 Introduction • 8.2 Distortion Measures • 8.3 The Rate-Distortion Theory • 8.4 Quantization • 8.5 Transform Coding • Sections 8.5.2 to 8.9 reading-optional! • 8.10 Further Exploration

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8.1 Introduction

Lossless compression algorithms do not deliver

compression

ratios

that are high enough.

Hence, most multimedia com-

pression algorithms are

lossy

What is

lossy compression

but a close approximation of it.The compressed data is not the same as the original data,

less compression.Yields a much higher compression ratio than that of loss-

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8.4 Quantization

Reduce

the

number

of

distinct

output

values

to

a

much

smaller set.

Main source of the “loss” in lossy compression.

Three different forms of quantization.

Uniform: midrise and midtread quantizers.

Nonuniform: companded quantizer.

Vector Quantization.

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Uniform Scalar Quantization

two outer intervals.values into equally spaced intervals, except possibly at theA uniform scalar quantizer partitions the domain of input

taken to be the midpoint of the interval.The output or reconstruction value corresponding to each interval is

The length of each interval is referred to as the

step size

, denoted by

the symbol ∆.

Two types of uniform scalar quantizers:

Midrise quantizers have even number of output levels.

as one of them (see Fig. 8.2).Midtread quantizers have odd number of output levels, including zero

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−

− 4 − 3 − 2 − 1 4

0.5 1.5 2.5 3.

−

Q(X)

x ∆^ /

∆/

−

(^) −

3.0 4.

−

(a)

(b)

Fig. 8.2:

Uniform Scalar Quantizers: (a) Midrise, (b) Midtread.

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Companded quantization

G

(^) − 1

Uniform quantizer

X

X ^

G

Fig. 8.4:

Companded quantization.

Companded quantization

is

nonlinear

As shown above, a

compander

consists of a

compressor func-

tion

G

, a uniform quantizer, and an

expander function

G

The two commonly used companders are the

-law and

A

-law

companders.

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code vectorFind closest

Table Lookup

Index

Encoder

Decoder

...

(^10987654321)

...

(^10987654321)

Fig. 8.5:

Basic vector quantization procedure.

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8.5 Transform Coding

The rationale behind transform coding

If

is the result of a linear transform

of the input vector

in such a way that the components of

are much less

correlated, then

can be coded more efficiently than

componentsIf most information is accurately described by the first few

of

a

transformed

vector,

then

the

remaining

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

Discrete

Cosine

Transform

(DCT)

will

be

studied

first.

Omitted:

In addition, we will examine the Karhunen-Lo`

eve Transform

(KLT) which

optimally

decorrelates the components of the input

X

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Given an input functionDefinition of DCT:

f

i, j

) over two integer variables

i

and

j

(a piece of an image), the 2D DCT transforms it into a new

function

F

u, v

with integer

u

and

v

running over the same

range as

i

and

j

. The general definition of the transform is:

F

u, v

C

u

C

v

M N

M (^) − 1

i ∑

N (^) − 1

j ∑

cos

i

uπ

M

cos

j

vπ

N

f

i, j

where

i, u

,... , M

j, v

,... , N

1; and the con-

stants

C

u

) and

C

v

) are determined by

C

√ 2

if

otherwise.

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Typical use2D Discrete Cosine Transform (2D DCT):

M

N

F

u, v

C

u

C

v

i ∑

j ∑

cos

i

uπ

cos

j

vπ

f

i, j

where

i, j, u, v

7, and the constants

C

u

) and

C

v

) are

The inverse function is almost the same, with the roles of 2D Inverse Discrete Cosine Transform (2D IDCT):determined by Eq. (8.5.16).

f

i, j

and

F

u, v

) reversed, except that now

C ( u ) C ( v

) must stand in-

side the sums:

f˜

i, j

u ∑

v ∑

C

u

C

v

cos

i

uπ

cos

j

vπ

F

u, v

where

i, j, u, v

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The 0th basis function (

u (^) = 0)

The 1st basis function (

u (^) = 1)

The 2nd basis function (

u (^) = 2)

The 3rd basis function (

u (^) = 3)

Fig. 8.6:

The 1D DCT basis functions.

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The 4th basis function (

u (^) = 4)

The 5th basis function (

u (^) = 5)

The 6th basis function (

u (^) = 6)

The 7th basis function (

u (^) = 7)

Fig. 8.6 (cont’d):

The 1D DCT basis functions.

Depiction of Continuous → Discrete D CT Basis.

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Signal

(^) f 3 ( i ) =

(^) f 1 ( i ) +

(^) f 2 ( i )

DCT output

F

3 ( u )

(c)

An arbitrary signal

(^) f ( i )

DCT output

F

u )

(d)

Fig.

8.7 (cont’d):

Examples of 1D Discrete Cosine Transform:

(c)

f

i

f

i

f

i

), and (d) an arbitrary signal

f

i

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After 0th iteration (DC)

After 1st iteration (DC + AC1)

After 2nd iteration (DC + AC1 + AC2)

− 100 − 50 0

100

0 1 2 3 4 5 6 7

After 3rd iteration (DC + AC1 + AC2 + AC3)

Fig. 8.

An example of 1D IDCT.

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Lossy Compression Algorithms-Multimedia Applications-Lecture Slides, Slides of Multimedia Applications

Related documents

Partial preview of the text

Download Lossy Compression Algorithms-Multimedia Applications-Lecture Slides and more Slides Multimedia Applications in PDF only on Docsity!

Lecture 18-19 Lossy Compression

Algorithms (Chapter 8 )

8.1 Introduction

Lossless compression algorithms do not deliver

compression

ratios

that are high enough.

Hence, most multimedia com-

pression algorithms are

lossy

What is

lossy compression

but a close approximation of it.The compressed data is not the same as the original data,

less compression.Yields a much higher compression ratio than that of loss-

8.4 Quantization

Reduce

the

number

of

distinct

output

values

to

a

much

smaller set.

Main source of the “loss” in lossy compression.

Three different forms of quantization.

Uniform: midrise and midtread quantizers.

Nonuniform: companded quantizer.

Vector Quantization.

Uniform Scalar Quantization

two outer intervals.values into equally spaced intervals, except possibly at theA uniform scalar quantizer partitions the domain of input

taken to be the midpoint of the interval.The output or reconstruction value corresponding to each interval is

The length of each interval is referred to as the

step size

, denoted by

the symbol ∆.

Two types of uniform scalar quantizers:

Midrise quantizers have even number of output levels.

as one of them (see Fig. 8.2).Midtread quantizers have odd number of output levels, including zero

Q(X)

Q(X)

(a)

(b)

Fig. 8.2:

Uniform Scalar Quantizers: (a) Midrise, (b) Midtread.

G

X

X ^

G

Fig. 8.4:

Companded quantization.

Companded quantization

is

nonlinear

As shown above, a

compander

consists of a

compressor func-

tion

G

, a uniform quantizer, and an

expander function

G

The two commonly used companders are the

-law and

A

-law

companders.

Fig. 8.5:

Basic vector quantization procedure.

8.5 Transform Coding

The rationale behind transform coding

If

is the result of a linear transform