Spatial and Temporal Data Mining

1. Spatial and Temporal Data Mining 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