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

• Fouriers 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

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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

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Fourier Analysis
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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

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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 Parsevals theorem for the DFT.
  Examples DCT, DWT

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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 n2q1 or q1 if the duration
  is n2q
• 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)

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Examples
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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
• DCTs 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 -gt all the
  coefficients are reals, their amplitute is
  symmetric along the middle (XfX2n-f), thus only
  the first n need to be kept

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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)

• m2, gray scale images
• m3, 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

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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)
• DFTs 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

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Continuous Wavelet transform
for each Scale for each Position
Coefficient (S,P) Signal x Wavelet (S,P)
end end
Position
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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

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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

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Wavelet Transform

• Scale and shift original waveform
• Compare to a wavelet
• Assign a coefficient of similarity

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Some wavelets different shapes, different
properties
Mexican hat
Gauss
Db3
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Continuous Wavelet transformshift wavelet and
compare,
C 0.0004
C 0.0034
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then scale, and shift through positions
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Scaling/stretching wavelet
Same wavelet, different scales
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Wavelet transform Scaling value of
stretch
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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

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Scale is (sort of) like frequency
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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)

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Discrete Wavelet transform
signal
lowpass
highpass
filters
Approximation (a)
Details (d)
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Results of wavelet transform approximation and
details

• Low frequency
• approximation (a)
• High frequency
• Details (d)
• Decomposition
• can be performed
• iteratively

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Levels of decomposition

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

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Wavelet synthesis

• Re-creates signal from coefficients
• Up-sampling required

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

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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

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Wavelet de-noising

• Thresholding for zeroing
• some detail coefficients

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Wavelet de-noising
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A demo
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Wavelet Toolbox Example
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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/
• Amaras Wavelet Page http//www.amara.com/current/
  wavelet.html