Wavelet Transform

A very brief look

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Wavelets vs. Fourier Transform

In Fourier transform (FT) we represent a signal in terms of sinusoids

FT provides a signal which is localized only in the frequency domain

It does not give any information of the signal in the time domain

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Wavelets vs. Fourier Transform

Basis functions of the wavelet transform (WT) are small waves located in different times

They are obtained using scaling and translation of a scaling function and wavelet function

Therefore, the WT is localized in both time and frequency

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Wavelets vs. Fourier Transform

In addition, the WT provides a multiresolution system

Multiresolution is useful in several applications

For instance, image communications and image data base are such applications

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Wavelets vs. Fourier Transform

If a signal has a discontinuity, FT produces many coefficients with large magnitude (significant coefficients)

But WT generates a few significant coefficients around the discontinuity

Nonlinear approximation is a method to benchmark the approximation power of a transform

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Wavelets vs. Fourier Transform

In nonlinear approximation we keep only a few significant coefficients of a signal and set the rest to zero

Then we reconstruct the signal using the significant coefficients

WT produces a few significant coefficients for the signals with discontinuities

Thus, we obtain better results for WT nonlinear approximation when compared with the FT

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Wavelets vs. Fourier Transform

Most natural signals are smooth with a few discontinuities (are piece-wise smooth)

Speech and natural images are such signals Hence, WT has better capability for representing

these signal when compared with the FT Good nonlinear approximation results in

efficiency in several applications such as compression and denoising

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Series Expansion of Discrete-Time Signals

Suppose that is a square-summable sequence, that is

Orthonormal expansion of is of the form

Where is the transform of

The basis functions satisfy the orthonormality constraint

[ ]x n2[ ] ( )x n Z

[ ]x n

[ ] [ ], [ ] [ ] [ ] [ ]k k kk k

x n l x l n X k n

Z Z

*[ ] [ ], [ ] [ ] [ ]k kl

X k l x l n x l [ ]x n

k

[ ], [ ] [ ]k ln n k l

2 2x X

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Haar expansion is a two-point avarage and difference operation

The basis functions are given as

It follows that

Haar Basis

21 2 , 2 , 2 1[ ]

0, otherwisekn k kn

2 1

1 2 , 2

[ ] 1 2 , 2 10, otherwise

k

n k

n n k

2 0[ ] [ 2 ],k n n k 2 1 1[ ] [ 2 ]k n n k

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The transform is

The reconstruction is obtained from

Haar Basis

21

[2 ] , [2 ] [2 1] ,2

kX k x x k x k

[ ] [ ] [ ]kk

x n X k n

Z

2 11

[2 1] , [2 ] [2 1]2

kX k x x k x k

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Two-Channel Filter Banks

Filter bank is the building block of discrete-time wavelet transform

For 1-D signals, two-channel filter bank is depicted below

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Two-Channel Filter Banks

For perfect reconstruction filter banks we have

In order to achieve perfect reconstruction the filters should satisfy

Thus if one filter is lowpass, the other one will be highpass

x̂ x

0 0

1 1

[ ] [ ][ ] [ ]

g n h ng n h n

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Two-Channel Filter Banks

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Two-Channel Filter Banks

To have orthogonal wavelets, the filter bank should be orthogonal

The orthogonal condition for 1-D two-channel filter banks is

Given one of the filters of the orthogonal filter bank, we can obtain the rest of the filters

1 0[ ] ( 1) [ 1]ng n g n

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Haar Filter Bank

The simplest orthogonal filter bank is Haar The lowpass filter is

And the highpass filter

0

1, 0, 1

[ ] 20, otherwise

nh n

1

1, 021

[ ] , 12

0, otherwise

n

h n n

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Haar Filter Bank

The lowpass output is

And the highpass output is

0 0 02

1 1[ ] [ ]* [ ] [ ] [2 ] [2 ] [2 1]

2 2n kl

y k h n x n h l x k l x k x k

1 1 12

1 1[ ] [ ]* [ ] [ ] [2 ] [2 ] [2 1]

2 2n kl

y k h n x n h l x k l x k x k

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Haar Filter Bank

Since and , the filter bank implements Haar expansion

Note that the analysis filters are time-reversed versions of the basis functions

since convolution is an inner product followed by time-reversal

0[ ] [2 ]y k X k 1[ ] [2 1]y k X k

0 0[ ] [ ]h n n 1 1[ ] [ ]h n n

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Discrete Wavelet Transform

We can construct discrete WT via iterated (octave-band) filter banks The analysis section is illustrated below

Level 1

Level 2

Level J

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Discrete Wavelet Transform

And the synthesis section is illustrated here If is an orthogonal filter and , then we have an

orthogonal wavelet transform

0V

1V

2V

JV

1W

2W

JW

[ ]ih n [ ] [ ]i ig n h n

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Multiresolution

We say that is the space of all square-summable sequences if

Then a multiresolution analysis consists of a sequence of embedded closed spaces

It is obvious that

0V

0 2 ( )V

2 1 0 2 ( )JV V V V

0 20

( )J

jj

V V

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Multiresolution

The orthogonal component of in will be denoted by :

If we split and repeat on , , …., , we have

1jV

1 1j j jV V W

jV

1jW

1 1j jV W

0V 1V 2V 1JV

0 1 1 J JV W W W V

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2-D Separable WT

For images we use separable WT First we apply a 1-D filter bank to the rows of the

image Then we apply same transform to the columns of

each channel of the result Therefore, we obtain 3 highpass channels

corresponding to vertical, horizontal, and diagonal, and one approximation image

We can iterate the above procedure on the lowpass channel

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2-D Analysis Filter Bank

1h

0h

1h

1h

0h

0h

x diagonal

vertical

horizontal

approximation

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2-D Synthesis Filter Bank

x̂diagonal

vertical

horizontal

approximation

1g

1g

1g

0g

0g

0g

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2-D WT Example

Boats image WT in 3 levels

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WT-Application in Denoising

Boats image Noisy image (additive Gaussian noise)

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WT-Application in Denoising

Boats image Denoised image using hard thresholding

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Reference

Martin Vetterli and Jelena Kovacevic, Wavelets and Subband Coding. Prentice Hall, 1995.