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Wavelet Analysis Why? What? And How?

Wavelet Analysis Why? What? And How?. By L. Kiranmayi. Outline. Introduction History Fourier transforms Short Time Fourier transforms Wavelet transforms Wavelets Applications. Introduction. Data Time domain Frequency domain. Fourier Transforms.

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Wavelet Analysis Why? What? And How?

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  1. Wavelet Analysis Why? What? And How? By L. Kiranmayi

  2. Outline • Introduction • History • Fourier transforms • Short Time Fourier transforms • Wavelet transforms • Wavelets • Applications

  3. Introduction • Data • Time domain • Frequency domain

  4. Fourier Transforms • The Great Father Fourier - Fourier Transforms Any Periodic function can be expressed as linear combination of basic trigonometric functions (Basis functions used aresine and cosine)

  5. Cosine and Sine parts Real part and imaginary part t= -inf to inf

  6. How it works? • Harmonics • For a series with N points, ith harmonic means oscillation with N/i period i.e., fit a sine and cosine wave with that period • Series • Amplitudes of each sine and cosine wave at each frequency/period : series of amplitudes as a function of frequency/period

  7. Drawbacks • Integration from -inf to +inf • Gives frequency content of total time series but temporal information is lost! Stationary Non Stationary

  8. Short time Fourier Transforms Time series • Same as usual Fourier transforms, but data is modified by multiplication with a window function • Only part of data at a time is taken and processed Window function After multiplication

  9. Drawbacks of STFT • Frequency and time resolutions are fixed (Wider the window width, lesser the time resolution and more the frequency resolution and vice versa) • As frequency resolution increases, time resolution decreases – uncertainty principle Desired:Good time resolution at high frequencies and good frequency resolution at low frequencies!

  10. Wavelets • Automatic time and frequency resolution adjustments • Flexibility in choosing basic function • No need to confine to sine and cosines anymore!

  11. What are Wavelets? • A small wave • Extends to finite interval

  12. Some mathematical expressions x(t)actual time series (t)wavelet function Integrable and limited to finite region Total energy finite

  13. Some typical mother wavelets

  14. What exactly wavelet transform does? Scale (expand or contract) translate and the mother wavelet ((t-)/s) Multiply with the time series Sum this product for the total time series Scaled(expanded) Wavelet coefficient at time  and scale s Translated

  15. Typical picture

  16. Quantitative information • Scale and equivalent Fourier period p=const*s • Amplitude gives the local power • Summation of square over all the times gives equivalent Fourier power

  17. Some real life Applications Time series analysis • Intraseasonal Oscillations 60E 70E 90E 165E WT of filtered SST for 4 longitudinal belts from 10 to 12.5 N

  18. Filtering a non-stationary data Through reconstruction of original series from coefficients Take the coefficients of required scales alone, making others zero. Other applications

  19. Two dimensional Wavelet Transform • Definition • Useful to obtain time-space variations or spatial variations in 2D • Mainly used in Image processing

  20. Typical 2D wavelet

  21. Multiresolution analysis and discrete Wavelet Transforms • Varying resolution in time and frequency at different levels • Discrete transforms • Coefficients at discrete scales and time points

  22. Data compression • Image processing • Main contributions • FBI finger prints • JPEG2000 • Audio compression

  23. Denoising • Considering all the coefficients with amplitude less than a threshold value to be result of noise • Reconstruct the signal after removing the noise coefficients

  24. Many more… • Edge finding in images - defogging • Concealed object detection by fusion of images Cloud detection and tracking?

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