Wavelet Toolbox – Featured Product Anne Mascarin The MathWorks, Inc. Natick, MA [email protected] Copyright 2000by The MathWorks, Inc. www.mathworks.com.
Download
Report
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]
1
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
Intro
Introduction to wavelets
Why use wavelets?
Wavelets versus Fourier
Applications of wavelets
The Wavelet Toolbox
Demo of the Wavelet Toolbox
2
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
What are wavelets?
A “new” way of evaluating and processing signals
Works on “non-stationary” data
Useful in many types of applications
3
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
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
4
of looking at signals
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
Fourier transform
Fourier transform:
5
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
Continuous Wavelet transform
for each Scale
for each Position
Coefficient (S,P) = Signal x Wavelet (S,P)
end
all time
end
Coefficient
Scale
Position
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
6
MATLAB & SIMULINK
The MathWorks
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
Copyright 2000by The MathWorks, Inc.
significance
Scale = Frequency band
7
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
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
8
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
Wavelet Transform
Scale and shift original
waveform
Compare to a wavelet
Assign a coefficient of
similarity
9
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
Some wavelets – different shapes,
different properties
Mexican hat
Gauss
Db3
10
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
Continuous Wavelet transform:
shift wavelet and compare, …
C = 0.0004
C = 0.0034
11
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
…then scale, and shift through
positions
12
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
Scaling/stretching wavelet
Same wavelet, different scales
13
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
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
14
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
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
15
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
Scale is (sort of) like frequency
Small scale
-Rapidly changing details,
-Like high frequency
Large scale
-Slowly changing
details
-Like low frequency
16
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
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)
17
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
Discrete Wavelet transform
signal
lowpass
highpass
filters
Approximation
(a)
Details
(d)
18
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
Results of wavelet transform:
approximation and details
Low frequency:
approximation (a)
High frequency
Details (d)
“Decomposition”
can be performed
iteratively
19
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
Levels of decomposition
Successively decompose the
approximation
Level 5 decomposition =
a5 + d5 + d4 + d3 + d2 + d1
No limit to the number of
decompositions performed
20
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
Wavelet synthesis
•Re-creates signal from coefficients
•Up-sampling required
21
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
Multi-level Wavelet Analysis
Multi-level wavelet
decomposition tree
Reassembling original
signal
22
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
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
23
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
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
24
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
Wavelet de-noising
•Thresholding for “zeroing”
some detail coefficients
25
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
Wavelet de-noising
26
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
The MathWorks
MATLAB & SIMULINK
A demo
27
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
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
28
Copyright 2000by The MathWorks, Inc.
www.mathworks.com
MATLAB & SIMULINK
The MathWorks
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
29
Copyright 2000by The MathWorks, Inc.
www.mathworks.com