1. Multi-resolution analysis
A Seminar on
Multi-resolution analysis
and
Wavelet transform By
Alok K Watve
M.Tech. (IT)
November 11, 2018
Multi-resolution analysis and wavelet transform

2. Multi-resolution analysis and wavelet transform
Contents
What is MRA?
The need for MRA
Fourier transform
Short Term Fourier Transform (STFT)
Image pyramids and sub-band coding
Continuous and Discrete wavelet transform
Haar transform
Wavelet transform of 2D functions (e.g. images)
Conclusion
November 11, 2018
Multi-resolution analysis and wavelet transform

3. Multi-resolution analysis and wavelet transform
What is MRA
MRA(Multi-Resolution Analysis) is analysis of signals (more generally functions) simultaneously at varying levels of detail (known as resolutions).
fj(t) : Approximation of function at resolution level j
fj+1(t) : Approximation of function at resolution level j+1
fj+1(t) - fj(t) = dj(t): Details revealed by resolution level j+1
Hence, f(t) = fj(t) + Σk=j to ∞dj(t)* *
November 11, 2018
Multi-resolution analysis and wavelet transform

4. Multi-resolution analysis and wavelet transform
Need for MRA
Signals at lower resolution are suitable for compression, but are not suitable for analysis. On contrary, high resolution signals are suitable for analysis but have poor compression/communication capabilities
In many cases, a low resolution signal may satisfy the application requirement, but it is not affordable to lose the high resolution details altogether. In other words we need signals at several resolutions.
November 11, 2018
Multi-resolution analysis and wavelet transform

5. Multi-resolution analysis and wavelet transform
Fourier Transform
X(f) = ∫ -∞ to ∞ x(t).e-j2πftdt
x(t) = ∫ -∞ to ∞ X(f).ej2πftdf
Limitations
Fourier transform gives only spectral details of the signal without considering temporal properties
Hence not suitable for analyzing signals with time varying spectra (non-stationary signals).
It has fixed time and frequency resolution. i.e. 100% frequency information. 0% time information.
November 11, 2018
Multi-resolution analysis and wavelet transform

6. Multi-resolution analysis and wavelet transform
Some traditional Multi-Resolution Analysis techniques are:
Sub-band coding
Image pyramids
Short Term Fourier Transform (STFT)
November 11, 2018
Multi-resolution analysis and wavelet transform

7. Multi-resolution analysis and wavelet transform
Sub-band Coding
x(n)
HPF
LPF
Downsample .
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Multi-resolution analysis and wavelet transform

8. Multi-resolution analysis and wavelet transform
Image Pyramid
November 11, 2018
Multi-resolution analysis and wavelet transform

9. Image pyramid generation
level j-1
Approximation
Approximation filter
Downsampler 2
Upsampler 2
Interpolation filter
level j prediction residue
level j input image
November 11, 2018
Multi-resolution analysis and wavelet transform

10. Short Term Fourier Transform
STFT(t’, f) = ∫ t x(t).w*(t-t’).e-j2πftdt t
November 11, 2018
Multi-resolution analysis and wavelet transform

11. Multi-resolution analysis and wavelet transform
STFT contd…
In STFT, we use the window function to control the portion of the signal to be considered for fourier transform. By varying width and location of the window, signal spectra at various time instances can be analyzed.
Width of the window determines the time (and also frequency) resolution. Narrow windows give excellent time resolution but very poor frequency resolution and broad windows give good frequency resolution but poor time resolution.
November 11, 2018
Multi-resolution analysis and wavelet transform

12. Multi-resolution analysis and wavelet transform
Heisenberg’s uncertainty principle – It is impossible to locate position and momentum of a particle with 100% accuracy.
In DSP, this modifies to : It is impossible to locate frequency and time instance (at which that frequency is present) with 100 % accuracy. In other words, the more we locate a signal in the time domain, the less we can locate it in the frequency domain and vice versa. Hence, exact time-frequency representation of a signal is impossible.
Limitation of STFT – As we reduce the width of the window, we lose the spectral details. i.e. we can only know the range of frequencies present, not the exact frequencies that are present in the signal.
November 11, 2018
Multi-resolution analysis and wavelet transform

13. Extending capabilities of MRA
Signals are analyzed in two domains:
Time Domain : signal is expressed as a function of time. i.e. y = x(t)
Frequency domain : Signal is expressed as a function of frequency. i.e. Y = X(f)
Higher frequencies are better resolved in time domain and lower frequencies are better resolved in frequency domain
Can We analyze spectral as well as temporal properties of the signal simultaneously without losing on resolution?
November 11, 2018
Multi-resolution analysis and wavelet transform

14. Solution: Wavelet transform
Definition – The ‘continuous’ wavelet transform is defined as follows:
For energy normalization
scaling
translation
All the windows used for wavelet transform are scaled and/or shifted versions of ‘mother wavelet ψ’ .
An example of wavelet transform using gaussian mother wavelet
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Multi-resolution analysis and wavelet transform

15. Multi-resolution analysis and wavelet transform
Important terms
Wavelet : A small wave (window function) of finite length
Mother wavelet : The original wavelet which is translated and scaled and then correlated with the signal to get the transform
Scale : Degree of dilation of the mother wavelet. High scale corresponds to low details and low scale corresponds to high details.
Translation : Refers to position of the scaled wavelet.
November 11, 2018
Multi-resolution analysis and wavelet transform

16. Discrete Wavelet Transform
DWT is defined as:
It can be observed that DWT is a sampled version of CWT. Samples are taken for scales s=2j and translations k.2j
DWT samples CWT while preserving nyquist criteria
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Multi-resolution analysis and wavelet transform

17. Multi-resolution analysis and wavelet transform
WT of Discrete Signals
Where,
h[n] is low pass filter and g[n] high pass filter which are quadrature mirror filters
x[n] Input discrete signal
y[n] Transformed signal
The entire process of filtering is recursively applied on ylow until we get wavelet transform up to any desired level (bounded by input signal)
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Multi-resolution analysis and wavelet transform

18. Multi-resolution analysis and wavelet transform
Haar Transform
Haar scaling function
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Multi-resolution analysis and wavelet transform

19. Multi-resolution analysis and wavelet transform
Haar Transform
Haar wavelet function
The wavelets are scaled using haar scaling function for energy normalization
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Multi-resolution analysis and wavelet transform

20. Multi-resolution analysis and wavelet transform
Example
Thus a signal (3,1,3,5) transforms to (6, -2, √2, -√2)
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Multi-resolution analysis and wavelet transform

21. Features of wavelet transform
Varying time and frequency resolutions
Good time but poor frequency resolution at higher frequencies
Poor time but good frequency resolution at lower frequencies
Suitable for analyses of non-stationary signals
Wavelet matrix can be computed in O(n) compared to fourier matrix which takes O(nlgn)* *
November 11, 2018
Multi-resolution analysis and wavelet transform

22. Multi-resolution analysis and wavelet transform
Wavelet Transform in 2D
Wavelet transform of 2D functions is based on 1D transform. To get wavelet transform of a 2D signal f(x,y), 1D transform is taken first along x axis and then along y axis. As images can be represented as 2D functions this procedure is commonly used to get WT of images.
Wavelet transform can be taken recursively for multiple levels. Number of levels is bounded by the number of samples in the input signal
Example of 2D transform using Haar wavelets
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Multi-resolution analysis and wavelet transform

23. Applications in image processing
A 2D wavelet transform is very good way of analyzing image properties as it gives texture details of the image.
Certain image features can extracted from 2D WT of an image which can serve for the purpose of matching or identifying images
Wavelet transform can also be used for image compression
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Multi-resolution analysis and wavelet transform

24. Multi-resolution analysis and wavelet transform
Conclusion
Multi-Resolution analysis is a different approach of signal processing that gives coarse as well as detailed information at the same time.
Wavelet transform is extension of MRA which resolves signals in domain best suitable for analysis.
As wavelet transform not only gives more information that fourier transform but it is also computationally more efficient, it is expected to get more attention in future.
November 11, 2018
Multi-resolution analysis and wavelet transform

25. Multi-resolution analysis and wavelet transform
References
The wavelet tutorial by Dr. Robi Polikar
Digital Image Processing – second edition: Rafael C. Gonzalez, Richard E. Woods – Pearson education
November 11, 2018
Multi-resolution analysis and wavelet transform