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)

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

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

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

Fourier Analysis

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

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.

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

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)

Examples

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

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

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

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

Continuous Wavelet transform for each Scale for each Position Coefficient (S,P) = Signal x Wavelet (S,P) end all time Coefficient Scale In contrast, the continuous wavelet transform is a function of different factors: namely, scale and position. So, the output of wavelet transforms are coeffs of scale and position, rater than sinusoids. We’ll investigate this further now. Here’s an algorithm to help you visualize the wavelet transform: consider a signal. Break the signal into small pieces, compare the first small piece to a given wavelet, determine how similar the piece of the input signal is to the wavelet. This value is a coefficient. Then, move on to the next signal piece, compare, etc. for all signal pieces. Finally, stretch (scale) the wavelet, and compare again. Position

Fourier versus Wavelets 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 Now that we know a little more about fourier and wavelets, here’s a final note about fourier versus wavelets. Fourier – wavelets cut a signal into small pieces and perform localized analysis on that piece this is called time-frequency analysis. As oppossed to Fourier – the scales (or frequency bands ) of wavelets are analyzed in much greater detail than with fourier.

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

Wavelet Transform Scale and shift original waveform Compare to a wavelet Assign a coefficient of similarity So – to sum up – wavelet transforms are … Then repeat this process for all possibles positions and scales.

Some wavelets – different shapes, different properties Mexican hat We’ve talked about wavelet analysis – but what’s a wavelet? Here’s an example of some wavelets. A wavelet is a small localized wave of particular shape and finite duration that has an average value of zero. Wavelets also tend to be irregular and asymmetric. Like the sinusoids in Fourier analysis, wavelets form bases that can decompose (analyze) and reconstruct (synthesize) signals and images. The choice of wavelets to use is relatively arbitrary. Some exhibit linear phase – that is needed for signal and image reconstruction. Orthogonal, bi-orthogonal Gauss Db3

Continuous Wavelet transform: shift wavelet and compare, … As mentioned before – wavelet transform is a matter of determining coefficients of position and scale. Part of the wavelet algorithm is to consider a waveform, and apply your wavelet to this piece, then shift. C = 0.0034

…then scale, and shift through positions Here’s a graphical example of what we mean by scaling

Scaling/stretching wavelet Same wavelet, different scales

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

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

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

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)

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

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

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

Wavelet synthesis Re-creates signal from coefficients Up-sampling required

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

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 BEG – Like I said CFT makes curve fitting in ML easy – the toolbox provides a GUI and … … We provide tools for: Previewing and preprocessing data Creating, comparing, and managing models Using a variety of standard and custom fitting equations Analyzing fits END - Here is an example of a data set that describes weather patterns (pretty much a blob of points) – CF methods helped fit models begin to attach a pattern to the data to help understand

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

Wavelet de-noising Thresholding for “zeroing” some detail coefficients Here’s an example of wavelet denoising. The Analysis filter bank performs a 3 level decomposition which produces three distinct types of data: a3, d3, d2, d1 (notice 4 traces) The thresholding is an important piece of wavelet denoising – we will “zero out” some detail coefficients with small absolute values. Wavelet denoising is said to exhiit perfect reconstruction because there is no aliasing introduced.

Wavelet de-noising

A demo

Wavelet Toolbox – Example Welcome, introduce

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 When we began, I said that my objective during this presentation was to get you comfortably using the Wavelet Toolbox. Now that we’re at the end, I trust that each of you now has enough information for you to be able to have the Wavelet Toolbox help you in your work. Thank you for your attention. Any Questions?