A Seminar on Multi-resolution analysis and Wavelet transform By Alok K Watve M.Tech. (IT)

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

2. 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 Multi-resolution analysis and wavelet transform

3. 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)* *http://documents.wolfram.com/applications/wavelet/ Multi-resolution analysis and wavelet transform

4. 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. Multi-resolution analysis and wavelet transform

5. 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. Multi-resolution analysis and wavelet transform

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

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

8. Image Pyramid http://imageprocessingplace.com 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 Multi-resolution analysis and wavelet transform

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

11. 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. Multi-resolution analysis and wavelet transform

12. 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. 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? Multi-resolution analysis and wavelet transform

14. Solution: Wavelet transform translation For energy normalization scaling Definition – The ‘continuous’ wavelet transform is defined as follows: 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 Multi-resolution analysis and wavelet transform

15. 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. 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 Multi-resolution analysis and wavelet transform

17. 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) Multi-resolution analysis and wavelet transform

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

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

20. Example Thus a signal (3,1,3,5) transforms to (6, -2, √2, -√2) 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)* *http://mathworld.wolfram.com/WaveletMatrix.html Multi-resolution analysis and wavelet transform

22. 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 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 Multi-resolution analysis and wavelet transform

24. 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. Multi-resolution analysis and wavelet transform

25. References • The wavelet tutorial by Dr. Robi Polikar http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html • http://mathworld.com • http://www.engmath.dal.ca/courses/engm6610/notes/node5.html • http://documents.wolfram.com/applications/wavelet/FundamentalsofWavelets/ • http://en.wikipedia.org/wiki/Discrete_wavelet_transform • Digital Image Processing – second edition: Rafael C. Gonzalez, Richard E. Woods – Pearson education Multi-resolution analysis and wavelet transform