Wavelets
Earl F. Glynn
Scientific Programmer
Bioinformatics
9 July 2003
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Wavelets
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History
Background
Denoising Yeast Cohesins Binding Data
Possible Applications of Wavelets
Wavelet References and Resources
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Wavelets: History
• Fourier analysis (~1807): classic, but limited, tool for
analyzing signals (problems with spikes, transients)
• Wavelets have ~15 distinct roots that trace back to
1930s
• Denis Gabor: windowed Fourier transform (1946)
• Jean Morlet: “cycle-octave transform” (1975)
• Mallat and Meyer: mathematical structure
“multiresolution analysis” (1987)
• Sweldens: “lifting technique” (1994)
(“2nd generation” wavelets used in JPEG2000, MPEG-4)
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Wavelets: History
Fourier Analysis
Windowed Fourier
Wavelets
www.maths.leeds.ac.uk/~kisilv/courses/wavelets-wt4-20.png
www.science.org.au/nova/029/029img/wave1.gif
www.engineering.uiowa.edu/~ahod/aip/hw2/hw2_files/image011.gif
www.cerm.unifi.it/EUcourse2001/Gunther_lecturenotes.pdf
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Wavelets: History
Fourier
Windowed-Fourier
Wavelet
Wavelets chop up data into frequency components, and analyze
each frequency component with a resolution matched to its scale.
From www.cerm.unifi.it/EUcourse2001/Gunther_lecturenotes.pdf, p. 10
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Wavelets: History
From ATMS 552 Notes, by D. L. Hartmann, p. 204
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Fourier Transform Background
Time Series
Fourier Transform
Filtered Series
Remove
High Frequency
Component
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Reconstruction
Decomposition
selected
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www.vectorsite.net/ttdsp1.html
Wavelet Transform Background
Data Series
Wavelet Transform
Decomposition
Filtered Data
Reconstruction
“Mother” Wavelet
Coefficients measure the variations of the field f(t)
about the point b, with the scale given by a.
From ATMS 552 Notes, by D. L. Hartmann, p. 204
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Wavelet Transform Background
“Mother” Wavelet
Dilations and translations of mother wavelet define an
orthogonal basis set. Any function may be represented as a
linear combination of these basis functions.
Haar wavelet
From ATMS 552 Notes, by D. L. Hartmann, p. 204
and Wavelets for Kids
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Continuous Wavelet Transform
User’s Guide
MatLab Wavelet Toolbox
P 1-13
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Discrete Wavelet Transform
Choose scales and positions based on powers of two, i.e.,
“dyadic” scales and positions, for efficiency. Mallat developed
a Discrete Wavelet Transform algorithm using filters in 1988,
which is called the Fast Wavelet Transform.
Approximation
Detail
User’s Guide, MatLab Wavelet Toolbox, p. 1-19
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Discrete Wavelet Transform
User’s Guide, MatLab Wavelet Toolbox, p. 1-20
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Approximation / Detail
Given: x and y
Approximation: A = (x + y)/2
(low pass filter – averaging or smoothing)
Detail: D = y - x
(high pass filter)
Points x and y can be recovered if you know A and D:
X = A – D/2
Y = A + D/2
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Approximation / Detail
For Haar Wavelet:
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Discrete Wavelet Transform
User’s Guide, MatLab Wavelet Toolbox, p. 1-21 and 2-33
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Denoising Yeast Cohesins Binding Data
Sample Cohesins Binding Data for Yeast Chromosome III from Jennifer Gerton’s Lab
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Denoising of Yeast Cohesins Binding Data
How many decomposition levels? How much detail to keep?
Type of thresholding to use? Which wavelet basis set?
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Denoising of Yeast Cohesins Binding Data
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Denoising of Yeast Cohesins Binding Data
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Denoising of Yeast Cohesins Binding Data
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Controlled Experiment
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Controlled Experiment
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Controlled Experiment
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Controlled Experiment
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Controlled Experiment
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Conclusions
• Wavelet denoising seems to be a powerful technique
• Several aspects of wavelets need further study
- exact algorithms used in Matlab analysis?
- choice of wavelet basis function?
- levels of decomposition (statistical test? entropy?)
- “thresholding” options: hard vs. soft?
- statistics behind “detail” noise to remove?
• Choice of wavelet basis function affects “compactness”
of wavelet decomposition
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Wavelets: Possible Applications
Revolutionary or just another analysis tool?
•Data Approximation
•Denoising
•Data compresssion
•Time-frequency analysis
•Image analysis
•Detecting patterns in DNA
sequences
• Protein structure investigation
• Microarray data analysis
• Modeling biological
microstruture
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Wavelet References and Resources
[1] Fischer, Patrick, et al, WavePred: A Wavelet-Based Algorithm for the Prediction of
Transmembrane Proteins, Comm. Math. Sci., Vol 1, No. 1, 2003, pp. 44-56
[2] Graps, Amara, An Introduction to Wavelets, IEEE Computational Science and
Engineering, Summer 1995, Vol 2, No. 2. www.amara.com/IEEEwave/IEEEwavelet.html.
Also, Amara’s Wavelet Page: www.amara.com/current/wavelet.html
[3] Hubbard, Barbara B., The World According to Wavelets: The Story of a
Mathematical Technique in the Making (2nd ed), Natick, MA: A K Peters, 1998.
[4] Liò, Pietro, Wavelets in bioinformatics and computational biology: state of art
and perspectives, Bioinformatics, Vol 19, No. 1, 2003, pp. 2-9.
[5] Vidaković, Brani and Peter Müller, Wavelets for Kids – A Tutorial Introduction,
Duke University, 1991, unpublished. http://www.isye.gatech.edu/~brani/wp/kidsA.pdf
[6] Wavelet Digest Site, www.wavelet.org (hosted by Swiss Federal Institute of
Technology)
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