Transcript Wavelet Toolbox – Featured Product Anne Mascarin The MathWorks, Inc. Natick, MA [email protected] Copyright  2000by The MathWorks, Inc. www.mathworks.com.

Wavelet Toolbox – Featured
Product
Anne Mascarin
The MathWorks, Inc.
Natick, MA
[email protected]
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Copyright  2000by The MathWorks, Inc.
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MATLAB & SIMULINK
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Intro
 Introduction to wavelets
 Why use wavelets?

Wavelets versus Fourier
 Applications of wavelets
 The Wavelet Toolbox
 Demo of the Wavelet Toolbox
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What are wavelets?
 A “new” way of evaluating and processing signals
 Works on “non-stationary” data
 Useful in many types of applications
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Why use Wavelets?
 Traditional signal analysis (Fourier transform
based) doesn’t indicate when an “event” occurs
 trends, abrupt changes, loss of signal…
 “Time/frequency” aspect let you gain
information about frequency composition of a
signal at a particular time
 Fourier analysis doesn’t work well on
discontinuous, “bursty” data
 music, video, power, earthquakes,…
 Wavelet analysis – a different (non-Fourier) way
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of looking at signals
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Fourier transform
 Fourier transform:
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Continuous Wavelet transform
for each Scale
for each Position

Coefficient (S,P) = Signal x Wavelet (S,P)
end
all time
end
Coefficient
Scale
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

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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
10
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Continuous Wavelet transform:
shift 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”
f(t) = sin(t)
f(t) = sin(2t)
scale factor 2
f(t) = sin(3t)
scale factor 3
scale factor1
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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
Small scale
-Rapidly changing details,
-Like high frequency
Large scale
-Slowly changing
details
-Like low 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
 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

Metallurgy: characterization of rough surfaces

Finance: exploring variation of stock prices

Communications: wireless channel signals
 Feature extraction
 Trend detection:
 Perfect reconstruction
 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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Resources
 Product page and demos:

http://www.mathworks.com/products/wavelet/
 Availability:

Downloadable from:
 http://www.mathworks.com/products/wavelet/tryit.shtml
 Price: $900, individual PC
 Requires: MATLAB, Signal Processing Toolbox
recommended
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Copyright  2000by The MathWorks, Inc.
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Where to get 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
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