This document discusses image analysis using wavelet transformation. It provides an overview of digital image processing and compares Fourier transforms, short-term Fourier transforms, and wavelet transforms. Wavelet transforms provide better time-frequency localization than Fourier transforms. The document demonstrates Haar wavelets and how they can be used to decompose an image into different frequency subbands. It discusses applications of wavelet transforms such as image compression, denoising, and feature extraction. The document includes MATLAB code for performing wavelet decomposition on an image.

1. Image Analysis
Using
Wavelet Transformation
- From Instrumentation Research &
Development Establishment, DRDO,
Dehradun
By : Nidhi Gopal
B.tech (EC)- M.tech(VLSI)
JV-B/09/1179
JVWU, Jaipur
PROJECT ON
:-

2. 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

3. 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

4. Objectives of my Project
Why wavelet
instead of
Fourier /
STFT?
Algorithms for
wavelet
transformation
Image
Compression&
Transmission

5. About Fourier Analysis
)sin()cos(
2
1
)(
11
0 x
T
n
bx
T
n
aaxf
m
m
m
m







 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.. …….

6. 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
Wavelet Transform
Localized in time and
frequency both
Analysis of non-
periodic signals are
also possible
Efficient for
continuous & moving
signals
Waves of finite
energy

7. 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.

8. Comparison of the three: time-frequency
graph
FOURIER
TRANSFORM
WAVELET
TRANSFORM
STFT

9.  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.

10. Solution :
Use narrower windows at high frequencies.

11. Narrow windows do not provide good localization at
low frequencies.

12. Use wider windows at low frequencies.
Solution:

13. 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

14. 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.

15. Example - Haar Wavelets (cont’d)
 Repeating this process on the averages
gives the full decomposition:

16. 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.

17. Wavelet expansion
 Efficient 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 low
resolution info
encode details or
high resolution info
Where .. K= scaling/low pass coeff and
j= translation coeff/high pass coeff

18. 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 )j
k jk
k k j
f t c t k d t k     
scaling function wavelet function

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

20. 1D Haar Wavelets
 Haar scaling and wavelet functions:
computes average computes details
φ(t) ψ(t)

21. Haar wavelet transform & its functions
 Mother scaling function:
• Father Scaling Function
:

22. Example - Haar basis (revisited)

23. [9 7 3 5]
low-pass,
down-sampling
high-pass,
down-sampling
(9+7)/2 (3+5)/2 (9-7)/2 (3-5)/2

24. [9 7 3 5]
high-pass,
down-sampling
low-pass,
down-sampling
(8+4)/2 (8-4)/2

25. Convention for illustrating
1D Haar wavelet decomposition
x x x x x x … x x
detail
average
…
re-arrange:
re-arrange:

26. Standard Haar wavelet
decomposition (cont’d)
x x x … x
x x x … x
… … .
x x x ... x
(1) row-wise Haar decomposition:
…
detail
average
…
… … .
…
…
… … .
re-arrange terms

27. Standard Haar wavelet
decomposition (cont’d)
(1) row-wise Haar decomposition:
…
detail
average
…
…
… … .
…
… … .
…
row-transformed resultfrom previous slide:

28. Standard Haar wavelet
decomposition (cont’d)
(2) column-wise Haar decomposition:
…
detail
average
…
…
… … .
…
…
…
… … .
…
row-transformed result column-transformed result

29. Example
…
…
…
… … .
row-transformed result
…
… … .
re-arrange terms

30. Example (cont’d)
…
…
…
… … .
column-transformed result

31. Applications of Wavelet
Transformation
Wavelet
Analysis
Speech
Compression
De-Noising
Edge
Detection
Feature
Extraction
Speech
Recognit
ion
Echo
cancellation
Audio & Video
compression
Frequency
Division
Multiplexing
Biomedical
Imaging

32. 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)

33.  [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])

34. Output :
ORIGINAL IMAGE IN .JPG

35. LEVEL ONE
DECOMPOSITION

36. LEVEL TWO DECOMPOSITION

37.  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