3.Wavelet Transform(Backup slide-3)

Wavelet Transformation

Department of Computer Science And Engineering

Shahjalal University of Science and Technology

Nashid AlamRegistration No: 2012321028

[email protected] -2 Presentation

(Backup Slides# 3)

Courtesy :

Prof. Fred HamprechtHeidelberg University

Source:https://www.youtube.com/watch?v=DGUuJweHamQ

http://www.sprawls.org/resources/CTIMG/module.htm

https://www.youtube.com/watch?v=DGUuJweHamQ

Wavelet

Wavelet

Working with wavelet:1. Convolve the signal with wavelet filter(h/g)2. Store the results in coefficients/frequency response

(Result in number is put in the boxes)3. Coefficients/frequency response:

- The representation of the signal in the new domain.

Properties:• Maximum frequency depends on the length of the signal.• Recursive partitioning of the lowest band in subjective to the application.

Details in upcoming slides

Good temper resolution in high frequencies

Good frequency resolution in low pass band

OBTAION:

Wavelet

A high pass filter

Temper resolution : A vertical high-resolutionFrequency resolution : The sample frequency divided by the number of samples

O/P of Low Pass Filter High Pass Filter = A Band Pass Result

1.A length 8 signal

3.Convolve the signal with the high pass filter

2.Split/divide the signal in two parts

Wavelet

To avoid redundancy

Down sample by 2

Wavelet

• For perfect low pass filter• Leave everything intact in 0 (zero)

Spectrodensity of the signal at this point

Unit cell

Unit cell is shrunk by half(1/2)

Wavelet

No information loss due to shrinking

First partitioning of lower and higher frequency band

Wavelet

Spectrodensity of the signal at this point

For perfect low pass filter For perfect high pass filter

This works even not for perfect high pass/low pass filter

Wavelet

Split the signalAnd

down-sample by 2In high frequency

Details at level 1

Wavelet

Split inthe low frequency

Details at level 2

Wavelet

Extra Split inthe low frequency

Details at level 3

Wavelet

Approximationat level 3



Wavelet

Works for Signals of 8 samples

23= 8, Sample=8, level=3.

Wavelet

Positive half of the

frequency axis

1

1 2 3 4

Wavelet

Positive half of the

frequency axis

2

1 21

1 2 3 4

Wavelet

Positive half

of

the frequency axis

31

2

1 21

1 2 3 4

Wavelet

Positive half

of

the frequency axis

Details at level 2

Details at level 3

Detailsat

level 1

Approximation

Good frequency resolution in low pass band

Wavelet

Filter response/Coefficientof

perfect bandpass filter

Wavelet Behaving

as bandpass

Wavelet


Practically used wavelet filter

Collect the low frequencies

High frequencies

Wavelet Behaving

as bandpass

Wavelet


Practically used wavelet filter

Modular square ofThese transfer

function Add up to 1.

Prevent Loosing

signal/energy

To

Wavelet Behaving

as bandpass

Code Fragments to do the task

% Extract the level 1 coefficients.

a1 = appcoef2(wc,s,wname,1);

h1 = detcoef2('h',wc,s,1);

v1 = detcoef2('v',wc,s,1);

d1 = detcoef2('d',wc,s,1);

% Display the decomposition up to level 1 only. ncolors = size(map,1); % Number of colors.

sz = size(X);

cod_a1 = wcodemat(a1,ncolors);

cod_a1 = wkeep(cod_a1, sz/2);

cod_h1 = wcodemat(h1,ncolors);

cod_h1 = wkeep(cod_h1, sz/2);

cod_v1 = wcodemat(v1,ncolors);

cod_v1 = wkeep(cod_v1, sz/2);

cod_d1 = wcodemat(d1,ncolors);

cod_d1 = wkeep(cod_d1, sz/2);

image([cod_a1,cod_h1;cod_v1,cod_d1]);

axis image; set(gca,'XTick',[],'YTick',[]);

title('Single stage decomposition')

colormap(map)

pause

% Here are the reconstructed branches

ra2 = wrcoef2('a',wc,s,wname,2);

rh2 = wrcoef2('h',wc,s,wname,2);

rv2 = wrcoef2('v',wc,s,wname,2);

rd2 = wrcoef2('d',wc,s,wname,2);

Wavelet

Wavelet

Transfer function of

The wavelets

Transfer function of

The Scaling function

Wavelet

Want to understand The effect of this label

Have to perform convolution

Understand The effect of each this label

Wavelet

Graph 01: Transfer functions of the wavelet transforms

Works for Signals more then 8 samples 23= 8, Sample=8, level=3.

Level 1details

Level 2details

Level 3details

Level 4details

Level 5details

Transfer functions of

Approximation:The low pass

result That we keep at

the end

Wavelet

Graph 01: Transfer functions of the wavelet transforms

Leveldetails

+ approximation= 1

Property of wavelet

Wavelet

Approximation is a sinc- A perfect low pass filter

sincA-sincBA=A frequencyB=A frequency

-A perfect bandpass filter

Wavelet

Signal withmore than

eight samplesScenario:

Temper resolution : A vertical high-resolutionFrequency resolution : The sample frequency divided by the number of samples

Temper resolution>Frequency resolution

Increasingfrequency resolution

Decreasestemporal resolution.

Discrete Wavelet Transform(DWT)


Requires a wavelet ,Ψ(t), such that:- It scales and shifts

from an orthonormal basis of the square integral function.

)2/)2((2

1)(, jt

jt n

jnj

Scale Shift

Denote Wavelet

j and n both are integer

nmjlmlnj ., ,, To offer an orthonormal basis:)(, tnj

Orthonormal basis: A vector space basis for the space it spans.

.

.


Basis Function

Wavelets,ΨBasis function : An element of a particular basis for a function space

Scaling Function,Ψ


With each label:By shifting-

+

+

-

Shift

Inter-product is zero

Wavelets are orthogonal


Details at level 1 Scale factor , j =2, 22 =4


Details at level 2

Scale factor , j =1, 21 =2


Details at level 3

Scale factor , j =0, 20 =1


ApproximationLow

frequency

No Scale factor

Daubchies’ Wavelet (DW)

Daubchies’ Wavelet (DW)

•H()=high pass filter•D4=Daubchies’ Tap 4 Filter•Not symmetrical

Initial shape

Backward transformation of Wavelets

Opposite of forward transformationMirror the forward transformation on the right hand sideReplace the down-sampling by up-sampling.

Signal

Wavelettransform

of the Signal

Wavelettransform

of the Signal

Signal

2D Wavelet Transform

Scaling function Wavelet

2Πk1 =ω1

2Πk2 =ω2

Low pass filter


High pass filter

Scaling function Wavelet

Use Separable Transform


Originalimage

hx = High pass filter(X-direction)

gx = low pass filter(X-direction)



hxy = High pass filter(y-direction)



gy = low pass filter(y-direction)





Further split



hy = High pass filter(y-direction)



hy = Low pass filter(y-direction)



Four region:

Blue= Diagonal Details at label 1

Green=Horizontal Details at label 1

Purple=vertical details at label 1

Yellow= Approximation at Label 1(Low pass in both x and y direction)



Doing the above steps recursively:Take the current approximation



Doing the above steps recursively:1. Take the current approximation2. And further split it up



New approximation

Doing the above steps recursively:1. Take the current approximation2. And further split it up3. Getting new approximation



Diagonal Details

Horizontal Details

vertical details

Approximation(can be furtherdecomposed)

In summary



In summary

Approximation(can be furtherdecomposed)



Visualization

Label ofapproximation

HorizontalDetails

HorizontalDetails

VerticalDetails

DiagonalDetails

VerticalDetails

DiagonalDetails



VisualizationLabel of approximation:• Very strong low pass filter• Few pixels



Visualization

Details in

Various Scale



Visualization

vertical details ->Shoulder

Horizontal Details ->Edges

Diagonal Details



More precise

Visualization

Original image:Gray square on a Black Background

Diagonal Details

Horizontal Details(row by row)

Vertical details(column by column)



Toy of original image



Decomposition at Label 4

Original image




Original image(with diagonal details areas indicated)

Diagonal Details



Vertical Details


Original image(with Vertical details areas indicated)

Experimental Results


DWT

1.Original Image(Malignent_mdb238) 2.Decomposition at Label 4

2.Decomposition at Label 1 3.Decomposition at Label 2 3.Decomposition at Label 3


DWT

1.Original Image(Malignent_mdb238) 2.Decomposition at Label 4


1.Original Image(Benign_mdb252)

2.Decomposition at Label 4


DWT


1.Original Image(Malignent_mdb253.jpg) 2.Decomposition at Label 4


CT vs. DWT

DWT Target Goal:1.Applying a DWT to decompose a digital mammogram into different subbands.

2.The low-pass wavelet band is removed (set to zero) and the remaining coefficients are enhanced.

3.The inverse wavelet transform is applied to recoverthe enhanced mammogram containing microcalcifications [7].

7. Wang T. C and Karayiannis N. B.: Detection of Microcalcifications in Digital Mammograms Using Wavelets, IEEETransaction on Medical Imaging, vol. 17, no. 4, (1989) pp. 498-509

The results obtained by the Contourlet Transformation (CT)are compared with

The well-known method based on the discrete wavelet transform

3.Wavelet Transform(Backup slide-3)

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