Introduction to Wavelet Transform and Image

Compression

Student: Kang-Hua Hsu 徐康華Advisor: Jian-Jiun Ding 丁建均E-mail: r96942097@ntu.edu.tw

Graduate Institute of Communication Engineering

National Taiwan University, Taipei, Taiwan, ROC

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Outline (1) IntroductionMultiresolution Analysis (MRA)

- Subband Coding- Haar Transform- Multiresolution Expansion

Wavelet Transform (WT)- Continuous WT- Discrete WT- Fast WT- 2-D WT

Wavelet PacketsFundamentals 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

ConclusionReference

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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

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.BiorthogonalOrthonormal

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

MRA-Subband Coding(2)

• Biorthogonal filter bank:

• Orthonormal (it’s 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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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

1 0( ) ( 1) (2 1 )

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

n

i i

g n g K n

h n g K n i

MRA-Subband Coding(3)• 1-D to 2-D: 1-D two-band subband coding to the rows and

then to the columns of the original image.

• Where a is the approximation (Its histogram is scattered, and thus lowly compressible.) and d means detail (highly compressible because their histogram is centralized, and thus easily to be modeled).

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FWT can be implemented by subband coding!

Haar Transform

will put the lower frequency components of X at the top-left corner of Y. This is similar to the DWT.

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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

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 it’s name.

• If we define , then

• , : scaling function coefficients

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( ) ( )k kk

f x x k( )k x

x

/2, ( ) 2 (2 )j j

j k x x k , ( )j j k

kV span x

0 1 2... ....V V V

2 2n

x h n x n h n

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 0f x V

Multiresolution Expansions(3)

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• The wavelet function : spans difference between any two adjacent scaling subspaces, and .

• span the subspace .

xjV

1jV

2, 2 2

jj

j k x x k jW

Multiresolution Expansions(4)

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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 response of the analysis and synthesis filters in page 11. There is time-reverse relation in both cases.

( ) ( ) 2 (2 )n

x h n x n ( )h n

( ) ( 1) (1 )nh n h n

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:

It’s guaranteed to be reversible if the admissibility criterion is satisfied.

• Hard to implement!

1,

| |

xW s f x dt

ss

2

0

1,

xf x W s d ds

sC s s

2| ( ) || |

fC df

f

DWT(1)• wavelet series expansion:

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0 0

0

, ,( ) ( ) ( ) ( ) ( )j j k j j kk j j k

f x c k x d k x

: 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 dx

, ,( ) ( ), ( ) ( ) ( )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.

DWT(2)

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0

0

0 , ,

1 1( ) ( , ) ( ) ( , ) ( )j k j k

k j j k

f x W j k x W j k xM M

0

1

0 ,0

1( , ) ( ) ( )

M

j kx

W j k f x xM

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 x, ( )j k x

FWT(1)

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2 2n

x h n x n ( ) ( ) 2 (2 )

n

x h n x n

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

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

n k km

W j k h m k W j m h n W j n

FWT(2)

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When the input is the samples of a function or an image, we can exploit 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

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

n k km

W j k h m k W j m h n W j n

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

0 0

1 1

( ) ( ) ( )

( ) ( ) ( )

g n h n h n

g n h n h n

2-D WT(1)

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( , ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

H

V

D

x y x y

x y x y

x y y x

x y x y

2-D1-D (row)

1-D (column)

These wavelets have directional sensitivity naturally.

2-D WT(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.

Wavelet Packets

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A wavelet packet is a more flexible decomposition.

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).

Fundamentals of Image 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.

Fundamentals of Image Compression(3)

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1

2R

nC

n

11D

R

RC

•Evaluation of the lossless compression:Compression ratio : Relative data redundancy :

•Evaluation of the lossy compression:root-mean-square (rms) error

1

21 1

0 0

1 ˆ , ,M N

rmsx y

e f x y f x yMN

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 the original data with less bits, the original data is said to have coding redundancy.

Interpixel Redundancy(1)

• Because the value of any given pixel can be reasonably 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.

Interpixel Redundancy(2)• To reduce interpixel redundancy, the original image

will 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

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 don’t respond with equal importance to every pixel.

Image Compression Model

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•The quantizer is not necessary.•The mapper would1.reduce the interpixel redundancy to compress directly, such as exploiting the run-length coding.or2.make it more accessible for compression in the later stage, for example, the DCT or the DWT coefficients are good candidates for quantization stage.

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.

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.

Bit-plane Coding

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A monochrome or colorful image is decomposed into a series of binary images (that is, bit planes), and then they are compressed by a binary compression method.

•It reduces the interpixel redundancy.

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 predicted value of that pixel.

Lossy Compression

• It exploits the quantizer.

• Its compression ratios range from 10 to 100 (much more than the lossless case’s).

• Trade-off relation between the reconstruction accuracy and compression performance.

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It can not be reconstructed without distortion due to the sacrificed accuracy.

Lossy Predictive Coding

• It exploits the quantizer.

• Its compression ratios range from 10 to 100 (much more than the lossless case’s).

• 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 containing a quantizer.

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 information as possible into the smallest number of transform coefficients.•Compression is achieved during the quantization of the transformed coefficients, not during the transformation.

Transform Coding(2)

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•More truncated coefficients Higher compression ratio, but the 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 abilityComputational complexity needed

KLT WHT DCTInformation packing ability Best Not good Good

Computational complexity High Lowest Low

Practical!

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!

?

?

?

?

??

?

Wavelet Coding(1)

• No subdivision due to: Computationally efficient (FWT) Limited-duration basis functions.

Avoiding the blocking artifact!

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Wavelet coding is not only the transforming coding exploiting the wavelet transform------No subdivision!

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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Conclusion The 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 Image

Processing, 2nd edition, Prentice Hall, 2002.[2] J.C Goswami, A.K Chan, Fundamentals of Wavelets,

John Wiley & Sons, New York, 1999.[3] Contributors of the Wikipedia, “Arithmetic coding”,

available in http://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, Communication System, 4th edition, John Wiley & Sons, New York, 2001.

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