Wavelets: Theory and Applications

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Wavelets Theory and Applications

• Maria Elena Velasquez
• Idaho State University
• 3 May 2006
• Committee
• Dr. Ken Bosworth
• Dr. Don Cresswell
• Dr. Charles R. Peterson
• Dr. Habib Sadid
• Dr. Rob Van Kirk

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Fourier Analysis
Joseph Fourier (1768-1830) France

• 1780-Entered the Royal Military Academy
• French Revolution-joined the revolutionary
  committee of Auxerre
• After the revolution- Taught in Paris
• Accompanied Napoleon to Egypt
• 1802-Returned to France and become the prefect of
  Isare

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Fourier Analysis

• 1802-Returned to France and became the Prefect of
  Isere
• 1807-Began writing la Theorie Analytique de la
  Chaleur
• 1817-Was elected to the Academy of Science
• 1822-Published la Theorie Analytique de la
  Chaleur

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Fouriers Work
Mathematical statement Any periodic function can
be represented as a Sum of sines and cosines.
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Formalized Work- Fourier Analysis
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Theorem
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Problems with Fourier Transform

• Time to compute coefficients
• Nonlinear problems
• Fourier transform hides information about time
• Local characteristics of the signal become global
  characteristic of the transform
• Fourier transform is very vulnerable to errors

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Some Solutions To The Problems

• FFT Fast Fourier Transform
• WFT Window Fourier Transform
• Wavelet Theory

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What are Wavelets?
Wavelets are functions that satisfy certain
requirements Admissibility condition

• The function should integrate to zero (wavy)
• The function has to be well localized
• (diminutive connotation of wavelet)

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Daubechies wavelet family
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Other Wavelets
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Stretches and translations
Horizontal or vertical stretch
Translation
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Continuous Wavelet Transform
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Inverse Wavelet Transform
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Discrete Wavelet Transform
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MultiResolution Analysis in MRA
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How do we do it?
Define the first space as follows
Define the rest of the space as follows
Where
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Wavelets enter the picture
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(No Transcript)
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Wavelets enter the picture
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Steps for the Discrete Wavelet Transform

• Take wavelet and compare it to the beginning
  section of the original signal
• Calculate the coefficient, which represents how
  closely correlated the wavelet is with this
  section of the signal
• Shift wavelet to the right and repeat steps 1 and
  2 until you have covered the entire signal
• Scale (Stretch) the wavelet and repeat steps 1
  through 3
• Repeat steps 1 through 4 for all scales
• At the end we will have coefficients produced at
  different scales by different sections of the
  signal.

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Discrete Wavelet transform
Translation Localization on time, translations
of mother wavelet Scale Localization on
frequency (1/freq), dilations of mom
wavelet Scaling dilate large scale low
frequency global view compress small
scale high frequency detailed view The
scaling factor in the wavelets is in the
denominator.
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Discrete Wavelet and Filter Banks
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Similarities Between FT and WT

• FFT and DWT are both linear operations that
  generate log(n) segments of various lengths,
  usually filling and transforming it into a
  different data vector of length 2n
• The inverse transform matrix of both is the
  transpose of the original matrix.
• FFT is a rotation in discrete function space to a
  new domain spanned by the basis functions sines
  and cosines. DWT domain is spanned by more
  complicated functions scaled-shifted wavelets.
• In both cases the basis functions are localized
  in frequency.

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Dissimilarities Between FT and WT
The most interesting dissimilarity between the
Fourier Transform and the Wavelet Transform is
that individual wavelet functions are localized
in space. Fourier sine and cosine functions are
not. Therefore, The Wavelet transform is
localized in both time and frequency.
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Applications

1. Data compression (fingerprinting)
2. Signal processing
3. Transformational algebras
4. Analysis of spectra
5. PDE and Integral equations
6. Numerical solvers for dynamical systems
7. Imaging
8. Human vision
9. Speech recognition
10. Solutions for stochastic models applied to
    Biology
11. Seismic Theory
12. Geosciences
13. Non-destructive assay techniques

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Detection of Single Isotopes in Composite
Gamma-ray Spectrum

• Existing software applications for spectra
  analysis
• can be described as follows
• Preprocess (calibration)
• Peak finding and fitting
• Peak analysis (calculation of areas under peaks)
• Peak verification (calculation of activities)

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Wavelet signatures for Gamma-raySpectra
Phase I construction and storage of the
signatures Construction of the wavelet
approximation Construction of the
non-parametric local smooth Detection of
significant peaks Phase II Construction of
the signature for the composite Ranking
detection of the single isotopes Verification
of detection
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Wavelet Decomposition db1 level 4
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Non-Parametric Smoothing
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Detection of the Significant Peaks
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Signature Example
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Detection of single isotopes
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Automated Photographic Identification of
Amphibians

• Method Description
• Image Pre-processing
• Signature Construction
• Signature Comparison

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Image Pre-processing
Best bounding ellipse (user can adjust
manually). Recommended white or gray
background. Automatic edge detection for
irrelevant white-zones. Orientation to
major/minor axes.
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Warping the frog
Impose a polar-elliptical coordinate system
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Signature and Mask Construction
The stardarized image was transformed using db4
at Level 3. Coefficients with magnitude below a
threshold (0.1) were ignored and a mask created
(only significant coefficients are
compared). The resulting coefficients were
encoded using a binary code.
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Signature Construction
Semisoft threshold (0.1)
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Signature Construction
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Signature Comparison
Fractional Hamming distance is a measure of
dissimilarity between two binary vectors
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Proof of Concept Gamma-ray
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ADVVS Output File
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Proof of Concept frog identification
Identifrog