Understanding Fourier Analysis and Wavelets: Advanced Concepts Explained

1. CIS750 – Seminar in Advanced Topics in Computer ScienceAdvanced topics in databases – Multimedia Databases V. Megalooikonomou Preliminaries (some slides are based on notes from “Searching multimedia databases by content” by C. Faloutsos and notes from Anne Mascarin)

2. General Overview • Fourier analysis • Discrete Cosine Transform (DCT) • Wavelets • Karhunen-Loeve • Singular Value Decomposition

3. Fourier Analysis • Fourier’s Theorem: • Every continuous function can be considered as a sum of sinusoidal functions • Discrete case – n-point Discrete Fourier Transform of a signal is defined to be a sequence of n complex numbers given by where j is the imaginary unit ( ) • We denote a DFT pair as

4. Fourier Analysis • The signal can be recovered by the inverse transform: is a complex number with the exception of which is real if the signal is real

5. Fourier Analysis

6. Fourier Analysis • Main Idea of DFT: decompose a signal into sine and cosine functions of several frequencies, multiples of the basic frequency 1/n • DFT as a matrix operation: where is an n x n matrix with

7. Fourier Analysis • The matrix A is column-orthonormal, i.e., its column vectors are unit vectors, mutually orthogonal (also row-orthonormal since it is a square matrix) where I is the (n x n) identity matrix and A* is the conjugate-transpose (‘hermitian’) of A that is DFT corresponds to a matrix multiplication with A and since A is orthonormal the matrix A performs a rotation (no scaling) of the vector x in n-d complex space. As a rotation, it does not affect the length of the original vector nor the Euclidean distance between any pair of points.

8. Properties of DFT • Parseval Theorem: Let be the Discrete Fourier Transform of the sequence . Then we have • The DFT also preserves the Euclidean distance (proof?) • Any transformation that corresponds to an orthonormal matrix A also enjoys a theorem similar to Parseval’s theorem for the DFT. Examples: DCT, DWT

9. Properties of DFT • A shift in the time domain changes only the phase of the DFT coefficients, but not the amplitude • For real signal we have so we only need to plot the amplitudes up to the middle, q, if n=2q+1 or q+1 if the duration is n=2q • The resulting plot of |Xf| vs f is called the amplitude spectrum (or spectrum) of the given time sequence; its square is the energy spectrum (or power spectrum) • The DFT requires O(nlogn) computation time. Straightforward computation requires O(n2), however, FFT exploits regularities of the function achieving O(nlogn)

10. Examples

11. Discrete Cosine Transform (DCT) • Objective: to concentrate the energy into a few coefficients as possible • DFT is helpful to highlight periodicities in the signal through its amplitude spectrum • When successive values are correlated DCT is better than DFT • DCT avoids the ‘frequency leak’ that DFT has when the signal has a ‘trend’ • DCT’s coefficients are always real (as opposed to complex) • DCT reflects the original sequence in the time axis around the last point and takes DFT on the twice-as-long (symmetric) sequence -> all the coefficients are reals, their amplitute is symmetric along the middle (Xf=X2n-f), thus only the first n need to be kept

12. Discrete Cosine Transform (DCT) • The formulas for DCT: • For the inverse DCT: • The complexity of DCT is also O(nlogn)

13. m-Dimensional DFT/DCT (JPEG) • m=2, gray scale images • m=3, MRI brain volumes • We do the transformation along each dimension (DFT on each row, then DFT on each column) • For a n1 x n2 array where is the value of the position (i1,i2) of the array and f1, f2 are the spatial frequencies ranging from 0 to (n1-1) and (n2-1) • The 2-d DCT is used in the JPEG standard for image and video compression

14. Wavelets • It is believed that it avoids the ‘frequency leak’ problem of DFTeven better than DCT • Short Window Fourier Transform (SWFT): restricted frequency leak • In the time domain each values gives full information about that instant (no info about f) • DFT’s coefficients give full info about a given f but it needs all frequencies to recover the value at a given instant in time • SWFT is in between • SWFT: how to choose the width w of the window? • Discrete Wavelet Transform: let w be variable

15. all time Scale Coefficient Continuous Wavelet transform for each Scale for each Position Coefficient (S,P) = Signal x Wavelet (S,P) end end Position

16. Fourier versus Wavelets • Fourier • Loses time (location) coordinate completely • Analyses the whole signal • Short pieces lose “frequency” meaning • Wavelets • Localized time-frequency analysis • Short signal pieces also have significance • Scale = Frequency band

17. Wavelets Defined “The wavelet transform is a tool that cuts up data, functions or operators into different frequency components, and then studies each component with a resolution matched to its scale” Dr. Ingrid Daubechies, Lucent, Princeton U

18. Wavelet Transform • Scale and shift original waveform • Compare to a wavelet • Assign a coefficient of similarity

19. Some wavelets – different shapes, different properties Mexican hat Gauss Db3

20. Continuous Wavelet transform:shift wavelet and compare, … C = 0.0004 C = 0.0034

21. …then scale, and shift through positions

22. Scaling/stretching wavelet Same wavelet, different scales

23. f(t) = sin(2t) scale factor 2 f(t) = sin(3t) scale factor 3 f(t) = sin(t) scale factor1 Wavelet transform: Scaling – value of “stretch”

24. More on scaling • It lets you either narrow down the frequency band of interest, or determine the frequency content in a narrower time interval • Scaling = frequency band • Good for non-stationary data

25. Small scale -Rapidly changing details, -Like high frequency Large scale -Slowly changing details -Like low frequency Scale is (sort of) like frequency

26. Discrete Wavelet Transform • “Subset” of scale and position based on power of two • rather than every “possible” set of scale and position in continuous wavelet transform • Behaves like a filter bank: signal in, coefficients out • Down-sampling necessary (twice as much data as original signal)

27. Discrete Wavelet transform signal lowpass highpass filters Approximation (a) Details (d)

28. Results of wavelet transform: approximation and details • Low frequency: • approximation (a) • High frequency • Details (d) • “Decomposition” can be performed iteratively

29. Levels of decomposition • Successively decompose the approximation • Level 5 decomposition = a5 + d5 + d4 + d3 + d2 + d1 • No limit to the number of decompositions performed

30. Wavelet synthesis • Re-creates signal from coefficients • Up-sampling required

31. Multi-level Wavelet Analysis Multi-level wavelet decomposition tree Reassembling original signal

32. The Wavelet Toolbox (Matlab) • The Wavelet Toolbox contains graphical tools and command-line functions for analysis, synthesis, de-noising, and compression of signals and images. These tools work particularly well in “non-stationary data” • These tools are used for de-noising, compression, feature extraction, enhancement, pattern recognition in MANY types of applications and industries

33. Applications of wavelets • Pattern recognition • Biotech: to distinguish the normal from the pathological membranes • Biometrics: facial/corneal/fingerprint recognition • Feature extraction • Metallurgy: characterization of rough surfaces • Trend detection: • Finance: exploring variation of stock prices • Perfect reconstruction • Communications: wireless channel signals • Video compression – JPEG 2000

34. Wavelet de-noising • Thresholding for “zeroing” • some detail coefficients

35. Wavelet de-noising

36. A demo

37. Wavelet Toolbox – Example

38. Wavelets: more information • References • Wavelets and Filter Banks by Gilbert Strang and Truong Nguyen • A Friendly Guide to Wavelets by Gerald Kaiser • Web Resources • Wavelet Digest http://www.wavelet.org/ • Amara’s Wavelet Page http://www.amara.com/current/wavelet.html