Induction to Wavelet Transform and Image Compression

8/6/2019 Induction to Wavelet Transform and Image Compression

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Introduction to Wavelet

Transform and Image

Compression

Student: Kang-Hua Hsu

Advisor: Jian-Jiun Ding

E-mail: [email protected] Institute of Communication Engineering

National Taiwan University, Taipei, Taiwan, ROC

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Outline (1) Introduction

Multiresolution Analysis (MRA)- Subband Coding- Haar Transform- Multiresolution Expansion

Wavelet Transform (WT)

- Continuous WT- Discrete WT- Fast WT- 2-D WT

Wavelet Packets

Fundamentals of Image Compression- Coding Redundancy- Interpixel Redundancy- Psychovisual Redundancy- Image Compression Model

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Outline (2) Lossless Compression

- Variable-Length Coding

- Bit-plane Coding

- Lossless Predictive Coding

Lossy Compression

- Lossy Predictive Coding- Transform Coding

- Wavelet Coding

Conclusion

Reference

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Introduction(1)-WT v.s FTBases of the

FT: time-unlimited weighted sinusoids with different

frequencies. No temporal information.

WT: limited duration small waves with varying frequencies,

which are called wavelets. WTs contain the temporal time

information.

Thus, the WT is more adaptive.

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Introduction(2)-WT v.s TFA Temporal information is related to the time-frequency

analysis.

The time-frequency analysis is constrained by the

Heisenberg uncertainty principal.

Compare tiles in a time-frequency plane (Heisenberg cell):

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Introduction(3)-MRA It represents and analyzes signals at more than one

resolution.

2 related operations with ties to MRA:

Subband coding

Haar transform

MRA is just a concept, and the wavelet-based

transformation is one method to implement it.

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Introduction(4)-WT The WT can be classified according to the of its input

and output.

Continuous WT (CWT)

Discrete WT (DWT)

1-D 2-D transform (for image processing)

DWT Fast WT (FWT)

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recursive relation of the coefficients


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MRA-Subband Coding(1)

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Since the bandwidth of the resulting subbands is smaller

than that of the original image, the subbands can be

downsampled without loss of information.

We wish to select so that the

input can be perfectly reconstructed.

Biorthogonal

Orthonormal

0 1 0 1, , ,h n h n g n g n


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Biorthogonal filter bank:

Orthonormal (its also biorthogonal) filet bank:

: time-reversed relation,where 2K denotes the number of coefficients in each filter.

The other 3 filters can be obtained from one prototype filter.

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? A ? A ? A

? A ? A

? A ? A ? A

? A ? A

0 0

0 1

1 1

1 0

, 2

, 2 0

, 2

, 2 0

g k h n k n

g k h n k

g k h n k n

g k h n k

H

H

! !

!

!

1 0( ) ( 1) (2 1 )

( ) (2 1 ), {0,1}

n

i i

g n g K n

h n g K n i

!

! !


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1-D to 2-D: 1-D two-band subband coding to the rows andthen to the columns of the original image.

Where a is the approximation (Its histogram is scattered, andthus lowly compressible.) and d means detail (highlycompressible because their histogram is centralized, and thuseasily to be modeled).

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FWT can be implemented by subband coding!


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Haar Transform

will put the lower frequency components of X

at the top-left corner of Y. This is similar to the

DWT.

This implies the resolution (frequency) and location (time).

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1 / 2 1 / 2 1 / 2 1 / 2

1 / 2 1 / 2 1 / 2 1 / 2

1 / 2 1 / 2 0 0

0 0 1 / 2 1 / 2

H

!

TY H X H !


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Multiresolution Expansions(1) , : the real-valued expansion coefficients.

, : the real-valued expansion functions.

Scaling function : span the approximation of the

signal.

: this is the reason of its name.

If we define , then

, : scaling function coefficients

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( ) ( )k kk

f x xE J! kE

( )k xJ

xJ

/2

, ( ) 2 (2 )j j

j k x x kJ J!

_ a, ( )j j kk

V span xJ!0 1 2... ....V V V

2 2n

x h n x nJ

J J! h nJ


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Multiresolution Expansions(2)

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4 requirements of the scaling function:

The scaling function is orthogonal to its integer translates.

The subspaces spanned by the scaling function at low scales

are nested within those spanned at higher scales.

The only function that is common to all is .

Any function can be represented with arbitrary coarse

resolution, because the coarser portions can be represented

by the finer portions.

jV _ a0 f x V g! !


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The wavelet function : spans difference between any

two adjacent scaling subspaces, and .

span the subspace .

x]

jV 1jV

2, 2 2j

j

j k x x k] ]! jW


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,

: wavelet function coefficients

Relation between the scaling coefficients and the wavelet

coefficients:

This is similar to the relation between the impulse responseof the analysis and synthesis filters in page 11. There is

time-reverse relation in both cases.

( ) ( ) 2 (2 )n

x h n x n]

] J!

( )h n]

( ) ( 1) (1 )n

h n h n] J!


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CWT

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The definition of the CWT is

Continuous input to a continuous output with 2 continuous

variables, translation and scaling.

Inverse transform:

Its guaranteed to be reversible if the admissibility criterion is

satisfied.

Hard to implement!

1

,

| |

xW s f x dt

ss

J

XX ]

g

g

!

2

0

1,

xf x W s d ds

sC s s]

]

XX ] X

g g

g

!

2( ) |

| |

fC df

f

]

=! g


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DWT(1) wavelet series expansion:

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0 0

0

, ,( ) ( ) ( ) ( ) ( )

j j k j j k

k j j k

f x c k x d k xJ ]g

!

!

: arbitrary starting scale0j

0( )jc k

( )jd k

: approximation or scaling coefficients

: detail or wavelet coefficients

0 0 0, ,( ) ( ), ( ) ( ) ( )j j k j kc k f x x f x x dxJ J! ! % %

, ,( ) ( ), ( ) ( ) ( )j j k j kd k f x x f x x dx] ]! ! % %

This is still the continuous case. If we change the integral

to summation, the DWT is then developed.


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DWT(2)

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0

0

0 , ,

1 1

( ) ( , ) ( ) ( , ) ( )j k j kk j j kf x W j k x W j k xM MJ ]J ]

g

!! % %

0

1

0 ,

0

1( , ) ( ) ( )

M

j k

x

W j k f x xM

J J

!

! %

1

,0

1

( , ) ( ) ( )

M

j kx

W j k f x xM] ]

!! %

The coefficients measure the similarity (in linear algebra,

the orthogonal projection) of with basis functions

and .

f x

0 ,( )j k xJ

%, ( )j k x]%


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FWT(1)

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2 2n

x h n x nJJ J! ( ) ( ) 2 (2 )

n

x h n x n]] J!

By the 2 relations we mention in subband coding,

We can then have

2 , 0( , ) ( 2 ) ( 1, ) ( ) ( 1, )

n k km

W j k h m k W j m h n W j n] ] J ] J ! u! !

2 , 0( , ) ( 2 ) ( 1, ) ( ) ( 1, )

n k km

W j k h m k W j m h n W j nJ J J J J ! u! !


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FWT(2)

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When the input is the samples of a function or an image, we canexploit the relation of the adjacent scale coefficients to obtain all

of the scaling and wavelet coefficients without defining the

scaling and wavelet functions.

2 , 0( , ) ( 2 ) ( ) ( 1,, )( 1 ) n k km

W j k h m k W j m h n W j n] ] ] JJ ! u! ! 2 , 0

( , ) ( 2 ) ( ) ( 1,, )( 1 )n k k

m

W j k h m k W j m h n W j nJ J J J J ! u! !


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FWT(3)

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FWT resembles the two-band subband coding scheme!

1 :FWT

The constraints for perfect reconstruction is the same

as in the subband coding.

0

1

( ) ( )

( ) ( )

h n h n

h n h n

J

]

!

!

0 0

1 1

( ) ( ) ( )

( ) ( ) ( )

g n h n h n

g n h n h n

J

]

! !

! !


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2-DWT(1)

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( , ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

H

V

D

x y x y

x y x y

x y y x

x y x y

J J J

] ] J

] J ]

] ] ]

!

!

! !

2-D1-D (row)

1-D (column)

These wavelets have directional sensitivity naturally.


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2-DWT(2)

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Note that the upmost-leftmost subimage is similar to the

original image due to the energy of an image is usually

distributed around lower band.


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Wavelet Packets

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A wavelet packet is a more flexible decomposition.


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Fundamentals of Image

Compression(1)

3 kinds of redundancies in an image:

Coding redundancy

Interpixel redundancy

Psychovisual redundancy

Image compression is achieved when the redundancies

were reduced or eliminated.

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Goal: To convey the same information with

least amount of data (bits).


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Compression(2) Image compression can be classified to

Lossless(error-free, without distortion after

reconstructed)

Lossy

Information theory is an important tool .

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Data Information{ : information is carried by the data.


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Compression(3)

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1

2

R

nC

n!

11

DC

!

Evaluation of the lossless compression:

Compression ratio :

Relative data redundancy :

Evaluation of the lossy compression:

root-mean-square (rms) error

121 1

0 0

1

, ,

M N

rms

x ye f x y f x yMN

! !

!


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Coding Redundancy

We can obtain the probable information from the histogram

of the original image.

Variable-length coding: assign shorter codeword to more

probable gray level.

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If there is a set of codeword to represent theoriginal data with less bits, the original data is

said to have coding redundancy.


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Interpixel Redundancy(1)

B

ecause the value of any given pixel can bereasonably predicted from the value of its neighbors,

the information carried by individual pixels is

relatively small.

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Interpixel redundancy is resulted from the

correlation between neighboring pixels.


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Interpixel Redundancy(2)

To reduce interpixel redundancy, the original imagewill be transformed to a more efficient and nonvisual

format. This transformation is called mapping.

Run-length coding. Ex. 10000000 1,111

Difference coding.

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7 0s


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Psychovisual Redundancy

For example, the edges are more noticeable for us.

Information loss!

We truncate or coarsely quantize the gray levels (or

coefficients) that will not significantly impair the perceived

image quality.

The animation take advantage of the persistence of vision to

reduce the scanning rate.

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Humans dont respond with equal importanceto every pixel.


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Image Compression Model

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The quantizer is not necessary.

The mapper would

1.reduce the interpixel redundancy to compress directly,

such as exploiting the run-length coding.

or

2.make it more accessible for compression in the later

stage, for example, the DCT or the DWT coefficients are

good candidates for quantization stage.


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Lossless Compression

No quantizer involves in the compression procedure.

Generally, the compression ratios range from 2 to 10.

Trade-off relation between the compression ratio and the

computational complexity.

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It can be reconstructed without distortion.


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Variable-Length Coding

It merely reduces the coding redundancy.

Ex. Huffman coding

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It assigns fewer bits to the more probable gray levels than

to the less probable ones.


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Bit-plane Coding

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A monochrome or colorful image is decomposed into a series ofbinary images (that is, bit planes), and then they are compressed

by a binary compression method.

It reduces the interpixel redundancy.


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Lossless Predictive Coding

It reduces the interpixel redundancies of closely spaced

pixels.

The ability to attack the redundancy depends on the

predictor.

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It encodes the difference between the actual and predictedvalue of that pixel.


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Lossy Compression

It exploits the quantizer.

Its compression ratios range from 10 to 100 (much more

than the lossless cases).

Trade-off relation between the reconstruction accuracy and

compression performance.

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It can not be reconstructed without distortiondue to the sacrificed accuracy.


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Lossy Predictive Coding

It exploits the quantizer.

Its compression ratios range from 10 to 100 (much more

than the lossless cases).

The quantizer is designed based on the purpose for

minimizing the quantization error.

Trade-off relation between the quantizer complexity and less

quantization error.

Delta modulation (DM) is an easy example exploiting the

oversampling and 1-bit quantizer.

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It is just a lossless predictive coding containinga quantizer.


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Transform Coding(1)

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Most of the information is included among a small number

of the transformed coefficients. Thus, we truncate or coarsely

quantize the coefficients including little information.

The goal of the transformation is to pack as much informationas possible into the smallest number of transform coefficients.

Compression is achieved during the quantization of the

transformed coefficients, not during the transformation.


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Transform Coding(2)

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More truncated coefficients Higher compression ratio, butthe rms error between the reconstructed image and the original

one would also increase.

Every stage can be adapted to local image content.

Choosing the transform:

Information packing ability

Computational complexity needed

KLT WHT DCT

Information packing ability Best Not good Good

Computational complexity High Lowest Low

Practical!


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Transform Coding(3)

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Disadvantage: Blocking artifact when highly compressed(this causes errors) due to subdivision.

Size of the subimage:

Size increase: higher compression ratio, computational

complexity, and bigger block size.

How to solve the blocking artifact problem? Using the WT!

?

?

?

?

??

?


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Wavelet Coding(1)

No subdivision due to:

Computationally efficient (FWT)

Limited-duration basis functions.

Avoiding the blocking artifact!

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Wavelet coding isnot only

the transforming coding

exploiting the wavelet transform------No subdivision!


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Wavelet Coding(2)

We only truncate the detail coefficients.

The decomposition level: the initial decompositions would

draw out the majority of details. Too many decompositions is

just wasting time.

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Wavelet Coding(3)

Quantization with dead zone threshold: set a threshold to

truncate the detail coefficients that are smaller than the

threshold.

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ConclusionThe WT is a powerful tool to analyze signals. There are

many applications of the WT, such as image

compression. However, most of them are still not

adopted now due to some disadvantage. Our future

work is to improve them. For example, we could

improve the adaptive transform coding, including the

shape of the subimages, the selection of transformation,

and the quantizer design. They are all hot topics to be

studied.

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Reference[1] R.C Gonzalez, R.E Woods, Digital I age Processing,

2nd edition, Prentice Hall, 2002.[2] J.C Goswami, A.K Chan, Funda entals ofWavelets,

John Wiley & Sons, New York, 1999.

[3] Contributors of the Wikipedia, Arithmetic coding,

available inhttp://en.wikipedia.org/wiki/Arithmetic_coding.

[4] Contributors of the Wikipedia, Lempel-Ziv-Welch,available in http://en.wikipedia.org/wiki/Lempel-Ziv-

Welch.[5] S. Haykin, Co unication Syste , 4th edition, John

Wiley & Sons, New York, 2001.

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Documents

Induction to Wavelet Transform and Image Compression