Image Analysis Using

Wavelet Transformation- From Instrumentation Research & Development Establishment, DRDO,

Dehradun

By : Nidhi GopalB.tech (EC)- M.tech(VLSI)

JV-B/09/1179JVWU, Jaipur

PROJECT ON :-

Digital image processing : a Brief Overview

Image processing involves a set of computational techniques for analyzing, enhancing, compressing, and reconstructing images. This is a wide field of processing images in a digital form.

Terminologies :•Image segmentation : Extracting information for set of segments/pixels.•Image enhancement and restoration : Reconstruction from degraded image.• Image sharpening and smoothening : Edge and corner detection.• Grey scale Transformation : Deals with contrast transformation, or, setting up the grey level values i.e. RGB values.

Figure : Displaying the word “BASIC” in the form of pixels.

Image : A set of Pixels

For defense (Army, Navy & Air force) purposes, Image compression is of basic necessity , as-

1.Security : It does not allow the other party, to copy, or to steal information from the opposite one, through secure data transmission.

2.Small target detection.3.Tracking missiles, Vehicle navigation.4.Automatic Target Recognition.. etc

Objectives of my ProjectWhy wavelet instead of Fourier / STFT?

Algorithms for wavelet transformation

Image Compression& Transmission

About Fourier Analysis

)sin()cos(21)(

110 x

Tnbx

Tnaaxf

mm

mm

An infinite series of sine and cosine functions. It is an explanation for the periodic series. Used in various fields like, optics, electrical engineering, quantum mechanics, image processing etc. Fourier transform, obtained by Fourier series, though, has many disadvantages.. …….

Fourier Transform

Localized in frequency(gives information according to frequencies

only)

Analysis of periodic signals is possible

Not efficient for Continuous & moving signals

Sinusoids of infinite energy

Another Approach : STFT (Short Term Fourier Transform) or Windowing

Time/Frequency localization depends on window size.

Once particular window size is chosen, it will be the same for all frequencies.

Many signals require a more flexible approach - vary the window size to determine more accurately either time or frequency.

Comparison of the three: time-frequency graph

FOURIER TRANSFORM

WAVELET TRANSFORMSTFT

Overcomes the preset resolution problem of the STFT by using a variable length window:◦ Use narrower windows at high frequencies

for better time resolution.◦ Use wider windows at low frequencies for

better frequency resolution.

Resolution Analysis :

Wide windows do not provide good localization at high frequencies.

Solution :

Use narrower windows at high frequencies.

Narrow windows do not provide good localization at low frequencies.

Use wider windows at low frequencies.

Solution:

Introduction to wavelet analysis What are Wavelets? Wavelets are functions

that “wave” above and below the x-axis, have

(1) varying frequency, (2) limited duration,

and (3) an average value of

zero.

This is in contrast to sinusoids, used by FT, which have infinite energy.

Sinusoid Wave

Example - Haar Wavelets (cont’d) Start by averaging the pixels together

(pairwise) to get a new lower resolution image:

To recover the original four pixels from the two averaged pixels, store some detail coefficients.

Example - Haar Wavelets (cont’d)

Repeating this process on the averages gives the full decomposition:

Example - Haar Wavelets (cont’d)

The Harr decomposition of the original four-pixel image is:

We can reconstruct the original image to a resolution by adding or subtracting the detail coefficients from the lower-resolution versions.

Wavelet expansionEfficient wavelet decompositions

involves a pair of waveforms (mother wavelets):

The two shapes are translated and scaled to produce wavelets (wavelet basis) at different locations and on different scales.

φ(t) ψ(t)

φ(t-k) ψ(2jt-k)

encode lowresolution info

encode details orhigh resolution info

Where .. K= scaling/low pass coeff and j= translation coeff/high pass coeff

Wavelet expansion (cont’d)

f(t) is written as a linear combination of φ(t-k) and ψ(2jt-k) :

Note: in Fourier analysis, there are only two possible values of k ( i.e., 0 and π/2); the values j correspond to different scales (i.e., frequencies).

( ) ( ) (2 )jk jk

k k j

f t c t k d t k

scaling function wavelet function

Wavelet families

Figure : Wavelet families (a) Haar (b) Daubechies (c) Coiflet1 (d) Symlet2 (e) Meyer (f) Morlet (g) Mexican Hat.

1D Haar Wavelets Haar scaling and wavelet

functions:

computes average computes details

φ(t) ψ(t)

Haar wavelet transform & its functions

Mother scaling function:

• Father Scaling Function :

Example - Haar basis (revisited)

[9 7 3 5]

low-pass,down-sampling

high-pass, down-sampling

(9+7)/2 (3+5)/2 (9-7)/2 (3-5)/2

[9 7 3 5]

high-pass, down-sampling

low-pass,down-sampling

(8+4)/2 (8-4)/2

Convention for illustrating 1D Haar wavelet decomposition

x x x x x x … x x

detailaverage

…

re-arrange:

re-arrange:

Standard Haar wavelet decomposition (cont’d)

x x x … xx x x … x

… … .x x x ... x

(1) row-wise Haar decomposition:

…

detailaverage

…

… … .

…

…

… … .

re-arrange terms

Standard Haar wavelet decomposition (cont’d)

(1) row-wise Haar decomposition:

…

detailaverage

…

…… … .

…

… … . …

row-transformed resultfrom previous slide:

Standard Haar wavelet decomposition (cont’d)

(2) column-wise Haar decomposition:

…

detailaverage

…

…… … .

……

…… … .

…

row-transformed result column-transformed result

Example

……

…… … .

row-transformed result

…

… … .

re-arrange terms

Example (cont’d)

……

…… … .

column-transformed result

Applications of Wavelet Transformation

Wavelet Analysis

Speech Compressio

n

De-Noising

Edge Detection

Feature Extraction

Speech RecognitionEcho

cancellation Audio &

Video compressio

nFrequency Division

MultiplexingBiomedical Imaging

Programs for wavelet analysis clear all; %close all; clc; x=imread('test.jpg'); imshow(x) [xar,xhr,xvr,xdr] = dwt2(x(:,:,1),'haar'); [xag,xhg,xvg,xdg] = dwt2(x(:,:,2),'haar'); [xab,xhb,xvb,xdb] = dwt2(x(:,:,3),'haar'); xa(:,:,1)=xar; xa(:,:,2)= xag; xa(:,:,3)=xab ; xh(:,:,1)=xhr; xh(:,:,2)= xhg; xh(:,:,3)=xhb ; xv(:,:,1)=xvr; xv(:,:,2)= xvg; xv(:,:,3)=xvb ; xd(:,:,1)=xdr; xd(:,:,2)= xdg; xd(:,:,3)=xdb ; x1 = [xa*0.003 log10(xv)*0.3 ;log10(xh)*0.3 log10(xd)*0.3]; figure(); imshow(x1)

[xaar,xhhr,xvvr,xddr] = dwt2(xa(:,:,1),'haar'); [xaag,xhhg,xvvg,xddg] = dwt2(xa(:,:,2),'haar'); [xaab,xhhb,xvvb,xddb] = dwt2(xa(:,:,3),'haar'); xaa(:,:,1)=xaar; xaa(:,:,2)= xaag; xaa(:,:,3)=xaab ; xhh(:,:,1)=xhhr; xhh(:,:,2)= xhhg; xhh(:,:,3)=xhhb ; xvv(:,:,1)=xvvr; xvv(:,:,2)= xvvg; xvv(:,:,3)=xvvb ; xdd(:,:,1)=xddr; xdd(:,:,2)= xddg; xdd(:,:,3)=xddb ; x11 = [xaa*0.001 log10(xvv)*0.3 ;log10(xhh)*0.3

log10(xdd)*0.3]; figure(); imshow(x11) [r,c,s] =size(xv); figure(); imshow( [x11(1:r,1:c,:) xv*0.05 ; xh*0.05 xd*0.05])    

Output :

ORIGINAL IMAGE IN .JPG

LEVEL ONE DECOMPOSITION

LEVEL TWO DECOMPOSITION

The algorithm used in programming was transferred into an FPGA chip, and then to the computers. Further, the scaling and translating coefficients (instead of whole image) were sent to the receiving end, then, by inverse wavelet transformation, the original image was reconstructed.

Consequently, this helped in feature extraction and de-noising of image.

CONCLUSION

Any Queries ??