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. 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

16. 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.

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
So – to sum up – wavelet transforms are …
Then repeat this process for all possibles positions and scales.

19. 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

20. 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 =

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

22. Scaling/stretching wavelet
Same wavelet, different scales

23. 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

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

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
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.

35. Wavelet de-noising

36. A demo

37. Wavelet Toolbox – Example
Welcome, introduce

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
Amara’s Wavelet Page
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?