Wavelets: Theory and Applications

1
Wavelets Theory and Applications

• Presented By Ernest Esposito
• Seminar Engineering Frontiers
• February 21, 2001 Spring 2001

2
Introduction

• Dominated journals in all mathematical and
  engineering related fields for the last decade
• Used in a plethora of applications

3
Objectives

• Complete understanding of how this technology
  emerged, and the associated theory
• Applications and why wavelets are beneficial to
  the following areas
• The impact wavelets have on society

4
History Lesson Fourier Series

• J. Fourier stated in 1807 that An arbitrary
  function, continuous or with discontinuities,
  defined in a finite interval by an arbitrarily
  capricious graph can always be expressed as a sum
  of sinusoids.

5
Fourier Series (Continued)

• Translation could superpose sines and cosines of
  different frequencies to represent other functions

6
Fourier Series (Continued)

• Implementing varying frequencies of sinusoids as
  building blocks for a signal, end up with that
  signals frequency content

7
Transform

• Mathematical operation that takes a function or
  sequence and maps it to another one
• Obtain further information that may not be
  obvious

8
Continuous Fourier Transform (CFT)

• Representation of the frequency amplitude
  relationship of a time domain signal
• Have access to only one type of information
• No frequency information in time
• No time information in frequency
• Informs us how much of a frequency, but not when

9
Stationary Signal

• x(t)cos(2pi10t)cos(2pi25t)
  cos(2pi50t)cos(2pi100t)

10
Non-Stationary Signal
11
Stationary and Non-Stationary Signals Spectra
12
Spectra (2)

• What do we notice?
• Conclusions
• Effective for stationary
• Ineffective for non-stationary

13
Why is that so?

• Signal x(t) is multiplied with a frequency and
  integrated over all times
• large sum major component
• small sum not a dominant frequency in the signal
• 0 means no frequency present

14
Conclusions for the CFT

• Fails for non-stationary signals
• Cannot distinguish between the two spectra
• Need some sort of time localization

15
Short Time Fourier Transform (STFT)

• Assumes we can look at the non-stationary part of
  a signal as stationary
• Done through looking at the signal through narrow
  windows
• STFT of a signal is the FT of a windowed signal
• How does the STFT work?

16
STFT (2)
17
Windowing

• Window most often used is a Gaussian Curve
• Problem with STFT is based on the window
• Narrow window Provide good time resolution, and
  poor frequency resolution
• Wide window Provide good frequency resolution,
  and poor time resolution

18
Windowing (2)

• Relation of the window is a result of the
  Heisenburg Uncertainty Principle

19
Heisenburg Uncertainty Principle

• Originally applied to the momentum and location
  of moving signals
• Can be applied to the time-frequency information
  of a signal
• Meaning
• Cannot know what spectral components exist at
  what instances in time

20
Example of STFT

• Perform it in the same non-stationary signal as
  before

21
Example of STFT (continued)

• Gaussian Window with different amounts of support

22
STFT with a 0.01 (narrowest window)

• Well separated in time
• Peaks correspond to a band of frequencies

23
STFT with a 0.0001(3rd widest window)

• Time resolution is becoming poorer
• Frequency resolution is improving

24
STFT with a 0.00001(widest window)

• Terrible time resolution
• Excellent frequency resolution

25
Time-Frequency Tiles STFT

• Single window used for all frequencies
• Resolution analysis is the same at all locations

26
Conclusions STFT

• Developed as an alternative to the FT
• STFT is application dependant
• Better time or frequency resolution?
• Be careful in the choice of window chosen

27
Multiresolutional Analysis

• Analyzing according to scale
• Designed to give
• Good time resolution and poor frequency
  resolution at high frequencies
• Good frequency resolution and poor time
  resolution at low frequencies

28
WAVELETS!

• Fundamental idea analyze according to scale
• Process data at different scales or resolutions
• CWT

29
Wavelets (2)Scale and Translation

• Scale used to dilate or contract wavelet for
  multiresolutional analysis
• Proportional to the frequency-1
• Low Scale
• If alt0lt1 wavelet contracts low scales
  correspond to high frequencies
• Large Scale
• If agt1 dilation occurs high scales
  correspond to low frequencies
• Translation corresponds to how wavelet is
  shifted in time

30
Some Common Wavelets
31
What does it take to be a wavelet?

• The wave part
• Shows it is oscillatory, characteristics of a
  wave
• The small part
• Shows it has finite energy, littlelet

32
Computation of a Wavelet

• Begins with a scale 1
• Wavelet is placed at beginning of the signal,
  inner product of signal and wavelet is taken, and
  integrated for all times
• Wavelet is shifted and process 2 is repeated
  until end of signal
• Steps 2-3 repeated for different scales

33
Pictorial Illustration for the Computation of a
Wavelet
34
CWT Example

• Non-stationary Signal

35
CWT Example

• Poor scale resolution at high scales
• Interpreted as good frequency resolution at
  lower scales

36
Time-frequency Tiles CWT

• Image recaps everything we have gone through
• Areas of the boxes are also determined by
  Heisenburgs Uncertainty Principle

37
Current Researchers Organizations

• Lawrence Livermore National Laboratory
• Bell Labs
• Rice University
• Schools such as
• Yale
• Berkley
• Dartmouth

38
Individual Researchers

• Ingrid Daubechies still exploring the area of
  wavelets
• Chris Brislawn current developer for the wavelet
  compression used by the FBI
• Ali N. Akansu working at the NJCMR working on
  wavelets and subband coding

39
Applications

• Applications explored will be
• Denoising of noisy signals
• Image Compression
• Signal Processing and how it pertains to the
  Medical Industry

40
Denoising of Noisy Signals
41
Denoising of Noisy Signals (continued)
42
Showing How in Image is Compressed and
Reconstructed
43
Demonstration the Quality of a Compressed Image

• Original left side
• Compression Ratio of 1001 no degradation is
  visible

44
Demonstration the Quality of a Compressed Image
(2)

• Comparison of compression tools
• Wavelet compressor 801 left inset
• JPEG 801 right inset

45
Signal Processing Biomedical Aspect

• Analysis of non-stationary signals
• Electrocardiograph (EEG)
• Electroencephalograph (EEG)
• Biomedical Imaging
• Accentuate image features
• Detection of Microcalcifications

46
Impacts

• Scientific Community
• Government
• Implementing wavelets as the compression
  technique for the database of fingerprints

47
Digital Fingerprinting

• Since 1924, FBI has accumulated over 30 million
  sets of fingerprints
• With 40,000 arriving daily
• 5,000 new prints to be saved
• 15,000 repeat prints
• 20,000 security checks to return and check

48
Putting the Problem in Perspective

• Fingerprint is digitized at a resolution of 500
  pixels/inch
• 256 levels of gray scale per pixel
• 1 fingerprint 700,000 pixels needing about .6
  Mbytes
• Pair of hands 6 Mbytes
• Total archive will require 200 terabytes

49
Digital Fingerprints and Society

• Director Louis J. Freeh of the FBI said, IAFS is
  the latest in a recent series of major
  technological advancements that will
  revolutionize law enforcements ability to better
  serve and protect the American people.

50
Market Impact

• Market opportunities will arrive mostly in image
  and video compression software
• Development of software
• Development of video systems
• Surveillance Kallix Corporation
• Courtroom

51
Typical Systems

• Pentium Based 750Mhz, 60Gbyte HD
• Compression Real Time 50-60 Kbytes/frame
• 4 Cameras, 8 hour business day, Record up to 17.6
  Days
• Postprocessing Archive Time 6 Kbytes/frame
• 4 Cameras, 8 hour business day, Record up to 17.6
  Days

52
Courtroom Drama

• Wavelet compressed images are authentic
• Motion JPEG looks at differences between frames
• inadmissible

53
Current Challenges and Constraints

• In the beginning people tried wavelets on
  everything in sight, recollects Ingrid
  Daubechies, one of the key contributors to the
  theory. We know by now on what wavelets work
  and on what they dont work, better than we did a
  couple of years ago. And I think there are some
  very nice success stories.

54
Current Challenges and Constraints (2)

• Signals Understanding wavelets and how to
  implement them
• Choosing Mother Wavelet
• Image and Video Lack of standards

55
Projections

• Used in voice recognition applications
• Iris identification
• Implementing possibly in types of media

56
Summary

• Looked at why wavelets developed
• Impacts on the scientific community and the
  broader society
• Diverse Applications
• Revolutionizing Signal Processing Industry
• Applications seem boundless

57
For Further Information

• http//engineering.rowan.edu/polikar/WAVELETS/WTt
  utorial.html (Wavelet tutorial)
• www.wavelet.org (wavelet digest)
• www.dsp.rice.edu (publications in the area of
  signal processing)
• http//www.mt.mevis.de/products/MT-WICE/ (for
  high quality wavelet image compression software)
• http//www.cosy.sbg.ac.at/uhl/wav.html
  (excellent link for references, papers, java
  applets)
• http//www.amara.com/current/wavelet.html
  (excellent resource page of links)

58
Questions

• ?