WaveletsWavelets

An Introductory ExpositionAn Introductory Exposition

Features, Analysis Structures, and Features, Analysis Structures, and

Selected ApplicationsSelected Applications

Entry of Wavelets into theEntry of Wavelets into the

Domain of SPDomain of SP

�� Most significant event in SP after FT.Most significant event in SP after FT.

�� Reference to Wavelet Transform: Reference to Wavelet Transform:

“Fourier Transform of 20“Fourier Transform of 20thth Century”Century”

�� Wavelets fills the missing link in signal Wavelets fills the missing link in signal

processing: link between time and processing: link between time and

frequency studiesfrequency studies

�� ““Little wave with big future”Little wave with big future”

Groups Active in Wavelet StudiesGroups Active in Wavelet Studies

Wavelets

Mathematics: Function

Analysis and Approximation

Engineering, Applied Scientists

In Signal Processing

Applications

Industrial Groups

Key Idea Key Idea

Alternative Signal Representations Alternative Signal Representations

and Transformationand Transformation

�� Signal representation in a suitable Signal representation in a suitable

domain for information extraction,….domain for information extraction,….

Examples:Examples:

�� Fourier Transform for spectral analysisFourier Transform for spectral analysis

�� Hilbert Transform in envelop detectionHilbert Transform in envelop detection

�� KLT(PCA) for optimal function approx.KLT(PCA) for optimal function approx.

�� Laguerre basis functionLaguerre basis function

�� Numerous other transforms( DCT,Radon,…)Numerous other transforms( DCT,Radon,…)

Two Domains for Signal Two Domains for Signal

Representation Representation

Different projection spaces different signal representaDifferent projection spaces different signal representa--

tionstions

Signal Domain

Projection space,

Basis functions

Transformed Domain

Transformation

Representation in Time

f(t)Representation in Freq

F(ω)

Time Domain Studies Frequency Domain Studies

Fourier Transform

Shortcomings of Fourier TransformShortcomings of Fourier Transform

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0- 0 . 0 2

0

0 . 0 2

0 . 0 4s i g n a l f ( t ) w i t h l o c a l i z e d c h a n g e s

f(t)

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0- 1

0

1

co

s(t)

S i n e a n d c o s i n e a s b a s i s f u n c t i o n s o f F o u r i e r T r a n s f o r m

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0- 1

0

1

sin

(t)

t i m e t

-- Basis function vary within Basis function vary within ±±±±±±±± ∞, no localization∞, no localization

-- Only one Basis function: Sine and Cosine or its complex formOnly one Basis function: Sine and Cosine or its complex form

-- Fourier Coefficients: Projection of function f(t) onto sine and Fourier Coefficients: Projection of function f(t) onto sine and

cosine bases functionscosine bases functions

-- Information about the entire range of a function is contained Information about the entire range of a function is contained

in the coefficients, no localizationin the coefficients, no localization

--

Fourier TransformFourier Transform

Loss of Local InformationLoss of Local InformationAn IllustrationAn Illustration

�� Information about singularities and sharp changes spread across many Information about singularities and sharp changes spread across many frequencies and many basic functionsfrequencies and many basic functions

�� Cost of computations for transients is highCost of computations for transients is high

x

freq ω

f(x)

F(ω)

Starting Point of WaveletsStarting Point of Wavelets

RealReal--world Signalsworld Signals

�� Dominance of transient and nonDominance of transient and non--

stationary signalsstationary signals

�� Information often reside in transients, Information often reside in transients,

changes changes

�� The need to develop tools for TThe need to develop tools for T--F F

analysis analysis

�� Shortcomings of STFT: Inability to Shortcomings of STFT: Inability to

model most of the nonstationary realmodel most of the nonstationary real--

world signalsworld signals

What Are WaveletsWhat Are Wavelets

�� Wavelets are Wavelets are wavelikewavelike oscilatory oscilatory

signals of finite bandwidth both signals of finite bandwidth both

in Time and in Frequencyin Time and in Frequency

�� Wavelets are Wavelets are basis functions of basis functions of

spacesspaces with certain propertieswith certain properties

Examples of WaveletsExamples of Wavelets

Db 4, Db 10 are from Daubechies family of wavelets.Db 4, Db 10 are from Daubechies family of wavelets.

Db wavelets have no analytical expression, they are Db wavelets have no analytical expression, they are

constructed numericallyconstructed numerically

Db4 Db10

Examples of Wavelet FunctionsExamples of Wavelet Functions

0 20 40 60 80 100 120 140 160 180-4

-2

0

2

WP

3,5

0 20 40 60 80 100 120 140 160 180-1

0

1

2Coiflet

WP

1,5

0 20 40 60 80 100 120 140-4

-2

0

2

WP

2,5

Wavelet Functions Generated from Wavelet Functions Generated from

Bior 3.9Bior 3.9

0 50 100 150 200 250 300 350-10

-5

0

5

10bior3.9, wavelet packet W30

0 100 200 300 400 500 600 700-4

-2

0

2

4Bior3.9, wavelet packet W 10

Representation in Time

f(t)Representation in Freq

F(ω)

Time Domain Studies Frequency Domain Studies

Shortcomings of Fourier Transform

Loss of Time Information in FTLoss of Time Information in FT

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-1

-0.5

0

0.5

1

shift

ed

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-1

-0.5

0

0.5

1 SALAAM with switching the 1st 5000 samples with the tail segment

Ori

gin

al

Fourier TransformFourier TransformLoss of time informationLoss of time information

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

0 1000 2000 3000 4000 50000

1000

2000

3000

4000abs(fft) of SALAAM with shifting the 1st 5000 samples to the tail

Representation in Time

f(t)Representation in Freq

F(ω)

Time Domain Studies Frequency Domain Studies

Wavelets

An Important Property of Wavelets:

Wavelets Filling the Gap

Transient Nature of SignalsTransient Nature of Signals

��Most of the signals we deal areMost of the signals we deal are

�� Nonstationary: Nonstationary: ��NonNon--stationary refers to time variancy and stationary refers to time variancy and

spectral variation of the signals with timespectral variation of the signals with time

�� Unpredictable Unpredictable ��changes in statistics of the signals including changes in statistics of the signals including

changes in pdf function or statistical parameterschanges in pdf function or statistical parameters

��The degree of unpredictability varies for The degree of unpredictability varies for different signals and applicationsdifferent signals and applications

An Example of a Transient SignalAn Example of a Transient Signal

Speech SignalSpeech Signal

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-1

0

1SALAAM

0 500 1000 1500 2000 2500 3000-0.5

0

0.5

sAla

am

0 500 1000 1500-1

0

1

saL

aam

0 500 1000 1500 2000 2500 3000-1

0

1

saL

aam

Three segments of SALAAM

EEG Signals, Awake and AsleepEEG Signals, Awake and Asleep

EEG (EP)EEG (EP)

Examples of Transient SignalsExamples of Transient SignalsEngine VibrationsEngine Vibrations

500 1000 1500 2000 2500 3000

-2

0

2

Hea

lthy

500 1000 1500 2000 2500 3000

-2

0

2

Time (Samplings)

Fau

lty

An Example of RealAn Example of Real--world Signalsworld Signals

Engine VibrationsEngine Vibrations

Wavelet Functions, WPWavelet Functions, WP

0 20 40 60 80 100-40

-20

0

20

40Bior3. 1, wavelet packet W2

0 20 40 60 80 100-4

-2

0

2

4db4, wavelet packet W15

Main Stages in Signal AnalysisMain Stages in Signal Analysis

Signal domain Signal domain basis functionsbasis functions Transformed domain, CoeffsTransformed domain, Coeffs

Transformation

Analysis

Reconstruction

Signals

Recon Signal

basis functions

Modified

Coefficients

Information

extraction

Examples of Processing in Examples of Processing in

Coefficient DomainCoefficient Domain

��Coding for communication, transmissionCoding for communication, transmission

��Compression and Data StorageCompression and Data Storage

��Detection and Pattern RecognitionDetection and Pattern Recognition

��Modification for Enhancement purposesModification for Enhancement purposes

��Noise ReductionNoise Reduction

��WatermarkingWatermarking

��Signal SeparationSignal Separation

��ModelingModeling

Two Main Tasks of Wavelet Two Main Tasks of Wavelet

AnalysisAnalysis

��Decomposition:Decomposition:

�� Information Reside in Signal Constituents / Information Reside in Signal Constituents /

ComponentsComponents

��TimeTime--Frequency RepresentationFrequency Representation

�� Transformation of a signal into timeTransformation of a signal into time--

frequency representation frequency representation

�� Different basis and transformations result in Different basis and transformations result in

different constituents and Tdifferent constituents and T--f informationf information

Key Concern in SPKey Concern in SP

Resolution/Localization in Signal Resolution/Localization in Signal

ProcessingProcessing��Resolution: Information Extraction at a Resolution: Information Extraction at a

Narrow Band of Signal Span.Narrow Band of Signal Span.

��Localization both in Time and in FrequencyLocalization both in Time and in Frequency

High freq

Low freq

Time/space

freq

Resolution/LocalizationResolution/Localization

in Timein Time

33

3

Resolution/Localization in Resolution/Localization in

FrequencyFrequency

0 2000 4000 6000 8000 10000 12000 140000

50

100

150Power Spectrum, Vibrat ion data at combustion zone, original and denois ed

Ori

gin

al

0 2000 4000 6000 8000 10000 12000 140000

50

100

150

de

no

ise

d,

soft

Freq. Hz

22

Need for Joint TimeNeed for Joint Time--Frequency Frequency

Analysis of High ResolutionAnalysis of High Resolution

��Why Joint TWhy Joint T--F Analysis:F Analysis:

�� Transient signal information reside in Transient signal information reside in

different bands of time and frequency different bands of time and frequency

domaindomain

�� The need for the study of signal at the The need for the study of signal at the

limits of resolutionlimits of resolution determined by determined by

Heisenberg Uncertainty LimitsHeisenberg Uncertainty Limits

Heisenberg Uncertainty PrincipleHeisenberg Uncertainty Principle

�� Heisenberg uncertainty principle defined in Heisenberg uncertainty principle defined in quantum physics and is applicable to signal quantum physics and is applicable to signal processing problems as well. processing problems as well.

�� Position and momentum of a particle can not Position and momentum of a particle can not be determined simultaneouslybe determined simultaneously

�� Sets a limit given as follows:Sets a limit given as follows:

�� ∆t. ∆f ≥ 1/4π ∆t. ∆f ≥ 1/4π

�� ∆x∆x22=⌠{(x=⌠{(x--µµm m ))22 |f(x)||f(x)|2 2 dxdx /⌠|f(x)|/⌠|f(x)|2 2 dxdx

µµm m is center of mass of the function f(x): is center of mass of the function f(x):

µµm=m=⌠{x| f(x)|⌠{x| f(x)|2 2 dx dx /⌠|f(x)|/⌠|f(x)|2 2 dxdx

Second ConcernSecond ConcernAlternative TimeAlternative Time––Frequency TilingFrequency Tiling

�� Information about Different signal behavior reside at Information about Different signal behavior reside at

different Tdifferent T--F BandsF Bands

�� Need to have alternative TNeed to have alternative T--F tiling and cell structuresF tiling and cell structures

Time/scale

Freq/scale

Third ConcernThird Concern

Adaptive Adaptive MultiMulti--Resolution StudyResolution Study

�� Different information reside at different resolutions. Different information reside at different resolutions.

�� Redundancy in Signal RepresentationRedundancy in Signal Representation

Time/scale

Freq/scale

Redundancy

in tiling

High Frequency

Low Frequency

Wavelets Wavelets

Tools for TTools for T--F AnalysisF AnalysisWavelets allow:Wavelets allow:

�� High resolutionHigh resolution and focused study of signals in time and focused study of signals in time

and in frequencyand in frequency

�� Low resolution, Coarse Low resolution, Coarse and more general picture and more general picture

and trend analysis and trend analysis

Comment:Comment:

Wavelet analysis resembles human Wavelet analysis resembles human mental activitymental activity i.e i.e

capability for a detail focusing as well as general and capability for a detail focusing as well as general and

a broadlya broadly--based data analysis based data analysis

Fourier vs Wavelet TransformFourier vs Wavelet Transform

FourierFourier TransformTransform WaveletWavelet TransformTransform

Numerous Analysis structures:

� CWT,

�DWT(2 Band and M Band),

�WP

�DDWT, SWT,

�Adaptive Signal Transform, Numerous Best

basis selection algorithms

�Frame structure

Analysis Structures:

- FS( periodic functions only)

- FT and DFT

Low computational costsComputational Cost High

Many Basis Functions Single Basis Function

Joint Time and Frequency InformationFrequency Information only,

Time/space information is lost

Nonstationary Transient signal AnalysisStationary signal Analysis

Examples of WaveletsExamples of Wavelets

0 5 10 15 20 25-0.2

0

0.2

0.4

0.6

0.8

1

1.2c oif4 : phi

0 5 10 15 20 25-1

-0.5

0

0.5

1

1.5c oi f4 : psi

Scaling function Wavelet function

Coiflet 4

Wavelet Functions Generated from Wavelet Functions Generated from

HAAR functionHAAR function

0 5 10 15 20 25 30 35-2

-1

0

1

2Haar, wavelet packet W13

0 5 10 15 20 25 30 35-2

-1

0

1

2dmey , wavelet packet W10

Gabor 2D WaveletGabor 2D Wavelet

Gabor Wavelet at different scale Gabor Wavelet at different scale and Orientationand Orientation

Laguerre Gaussian WaveletLaguerre Gaussian Wavelet

Laguerre Gaussian WaveletLaguerre Gaussian Wavelet

Wavelet TransformWavelet Transform

��Wavelet Transform is defined as in other Wavelet Transform is defined as in other

transforms:transforms:

�� W W = <= <f(t),f(t),ψ(t)ψ(t)> = > = ∫∫f(t) ψ*(t)dtf(t) ψ*(t)dt

�� wwjkjk = <= <f(t),f(t),ψψjj,,kk(t)(t)> = > = ∫∫f(t) ψf(t) ψj,kj,k*(t)dt*(t)dt

ψj,k(t) is a shifted and scaled wavelet function, j is ψj,k(t) is a shifted and scaled wavelet function, j is

scaling and k translation parameters, J,K can be any scaling and k translation parameters, J,K can be any

scalars. In DWT they assume integer valuesscalars. In DWT they assume integer values..

jkkjkj wtf*

,, )( ψ=

Physical Interpretation of Physical Interpretation of

Wavelet TransformWavelet Transform

��Correlation (extent of matching) of Correlation (extent of matching) of ff(t) with (t) with

the wavelet the wavelet ψψjkjk(t(t) at scale j and location ) at scale j and location k. k.

��It carries signal information at scale j, It carries signal information at scale j, and location k. It gives localized and location k. It gives localized information of a signal.information of a signal.

��Representation of a signal in a domain Representation of a signal in a domain described by wavelet basis functionsdescribed by wavelet basis functions

Wavelet TransformWavelet Transform

Given a function f(t) єL^2Given a function f(t) єL^2

f(t) wab

Ψab(t)= Ψ(atΨab(t)= Ψ(at--b)b)

Function Domain

single dimensional

Transformed Domain:

two dimensional,

paramters a,b

Translation b

Scale a /freq

f(t)

Time/space

coeffs

Illustration of TranslationIllustration of Translation

by two different Waveletsby two different Wavelets f(t)

t

φ0k(t) shifted wavelets

t

φ0k(t) Haar

shifted Haar wavelet

t

Illustrative Examples of WT Illustrative Examples of WT Signal f(t)

t

φ0k(t) shifted wavelets

t t

φ0k(t) Haar φjk(t)

shifted Haar wavelet scaled Haar wavelet t t t

Wavelet TransformWavelet TransformWindow Function InterpretationWindow Function Interpretation

�� Different information are extracted form the same Different information are extracted form the same signal using different wavelets. The need to access signal using different wavelets. The need to access multitude of transforms and wavelet basesmultitude of transforms and wavelet bases for for extraction of different information of a given extraction of different information of a given signalsignal

�� Wavelets as window functions may be considered Wavelets as window functions may be considered as lenses having different resolutions. Different as lenses having different resolutions. Different information are extracted by different lenses at information are extracted by different lenses at different scales and different locationsdifferent scales and different locations of a given of a given signalsignal

�� STFT has only one window function at a given STFT has only one window function at a given scalescale

Wavelet TransformWavelet Transform, , Translation of Window FunctionTranslation of Window Function

Illustration of Wavelet Illustration of Wavelet

Scaling/DilationScaling/Dilation

Common WaveletsCommon Wavelets� Old Wavelets:

� Haar function� Gabor function and wavelet(1D,2D) � Morlet wavelets� Shannon function� 1st and 2nd Derivatives of Gaussian function(mexican hat)� Truncated and lapped Sine or Cosine functions

� Recent Wavelets � Db Wavelets (orthogonal, biorthogonal), � Coiflets, � Symlets� Biorthgonal wavelets� Mallat Wavelet (1D,2D)� Bathlets� Curvelets� Ridgelets� Meyer wavelets� Banana Wavelets� Your Wavelets

� Second Generation Wavelets( wavelets constructed by lifting scheme)

Commonly Used Numerical Commonly Used Numerical

WaveletsWavelets

��Daubechies WaveletsDaubechies Wavelets

��Biorthogonal WaveletsBiorthogonal Wavelets

��CoifletsCoiflets

��SymmletsSymmlets

��BathletsBathlets

��DmeyerDmeyer

How Wavelets are GeneratedHow Wavelets are Generated

��A few number of wavelets are constructed A few number of wavelets are constructed

using known analog functions. Some are using known analog functions. Some are

expressed in analog function form, e.g. expressed in analog function form, e.g.

Morlet waveletsMorlet wavelets

��Many others are designed and constructed Many others are designed and constructed

numericallynumerically. Db wavelets are examples of . Db wavelets are examples of

these waveletsthese wavelets

�� New wavelets are introduced by individuals New wavelets are introduced by individuals

each yeareach year

Categorization of WaveletsCategorization of Wavelets

�� Analytical or Numerical Analytical or Numerical

�� Real or ComplexReal or Complex

�� Symmetric, antisymmetric, asymmetric (e.g. db)Symmetric, antisymmetric, asymmetric (e.g. db)

�� Compactly supported or not compactCompactly supported or not compact

�� Causal, NonCausal, Non--causal waveletscausal wavelets

�� Do they have efficient computation algorithms such as FWTDo they have efficient computation algorithms such as FWT

�� Wavelets with special features e.g. complex wavelet with real part Wavelets with special features e.g. complex wavelet with real part (high freq) and complex part carry low frequency content(high freq) and complex part carry low frequency content

�� Maximally flat waveletes (Flatness of spectrum, Rate of decay at ω=pi Maximally flat waveletes (Flatness of spectrum, Rate of decay at ω=pi and origin)and origin)

�� Have Orthogonal, biorthogonal analysis system Have Orthogonal, biorthogonal analysis system

�� Wavelets of high Resolution in timeWavelets of high Resolution in time--frequency domain: frequency domain: ∆t ∆ω. ∆t ∆ω. Gabor: Gabor: H=0.5, Haar: H=0.58 in normalized frequency (0,1). H=0.5, Haar: H=0.58 in normalized frequency (0,1).

�� Balanced Wavelets (Bathlets) are designed under a criteria based on Balanced Wavelets (Bathlets) are designed under a criteria based on balancing balancing ∆t and ∆ω.∆t and ∆ω.

Two Common Analysis StructuresTwo Common Analysis Structures

11-- Standard Two Band Discrete Wavelet TransformStandard Two Band Discrete Wavelet Transform

High Pass

Low Pass

Low pass

High pass

Decimated Wavelet Transform-Analysis Stage

High Freq Details

Low freq Approx

Signal

2

2

MultiresolutionMultiresolution

Pyramidal Signal Analysis StructurePyramidal Signal Analysis Structure

��Multiresolution as a subband coding of a Multiresolution as a subband coding of a

signal, signal,

��Signal components at different subbands Signal components at different subbands

are extractedare extracted

��Different analyzing wavelets results in Different analyzing wavelets results in

different components at different bands different components at different bands

An Illustration of MultiresolutionAn Illustration of Multiresolution

Signal DecompositionSignal Decomposition

Binary Tree of DWTBinary Tree of DWT

Common Analysis StructuresCommon Analysis Structures

Wavelet PacketsWavelet Packets

High Pass

Low Pass

2

2

2

Low pass

High pass

Wavelet Packets -Analysis Stages

2

High freq details

High pass

Low pass 2

2

V space of the signal

Wavelet Packets TreeWavelet Packets Tree

Standard Two Channel DWT and Standard Two Channel DWT and

WP TWP T--F TilingF Tiling

TimeTime--Freq. TilingFreq. Tiling

Standard DWT Standard WP

Perfect Reconstruction FilterbankPerfect Reconstruction Filterbank

Filterbank implementation of DWTFilterbank implementation of DWT

High Pass

Low Pass Low pass

High pass

Perfect Reconstrution Filter bank

Wavelet

Subspace W

Alternative Signal Analysis Alternative Signal Analysis

ArchitecturesArchitectures

Wavelets are rich in analysis structures and Wavelets are rich in analysis structures and

algorithmsalgorithms

Main Analysis StructuresMain Analysis Structures

�� ORTHOGONALORTHOGONAL

�� BIORTHOGONALBIORTHOGONAL

�� REDUNDANT WAVELET TRANSFROMREDUNDANT WAVELET TRANSFROM

�� FRAMEFRAME--BASED REDUNDANT BASED REDUNDANT

TRANSFORMTRANSFORM

Beyond Orthogonality: Compression Beyond Orthogonality: Compression

RequirementsRequirements

Slow rate of decay Fast rate of decay

Coeffs 2Coeffs 1

Orthogonal Expansion Non-orthogonal Expansion

Primary Features of Wavelets used Primary Features of Wavelets used

in Signal Processingin Signal Processing

1.1. TimeTime-- frequency Localizationfrequency Localization (useful for transient (useful for transient data analysis)data analysis)

2.2. Sparsity: Sparsity: Coefficients are often grouped into high Coefficients are often grouped into high and low amplitudeand low amplitude

3.3. Function Approximation: Function Approximation: near Optimal Function near Optimal Function ApproximationApproximation

4.4. Noise Reduction, Noise Reduction, No wavelet can model white noise No wavelet can model white noise leading to small amplitude coefficientsleading to small amplitude coefficients

5.5. Decorrelation: Signal components( coefficients) in Decorrelation: Signal components( coefficients) in wavelet domain have a lower degree of correlation wavelet domain have a lower degree of correlation

Categorization ofCategorization of

Wavelet ApplicationsWavelet Applications

Applications: Methodology/Algorithm IntensiveApplications: Methodology/Algorithm Intensive

�� Change Detection ( of all kinds)Change Detection ( of all kinds)

�� CompressionCompression

�� Feature Extraction and Pattern Recognition with Numerous ApplicationsFeature Extraction and Pattern Recognition with Numerous Applications

�� Image Enhancement Image Enhancement

�� Noise reduction with Numerous ApplicationsNoise reduction with Numerous Applications

�� Diagnostics (such as NDT, NDE)Diagnostics (such as NDT, NDE)

�� Signal Separation, ICA using WaveletsSignal Separation, ICA using Wavelets

�� Solution of Differential and Partial Differential EquationsSolution of Differential and Partial Differential Equations

�� Analysis of FractalsAnalysis of Fractals

�� System Identification in Control systemsSystem Identification in Control systems

�� Data and Image FusionData and Image Fusion

�� WatermarkingWatermarking

Applications, Industry Applications, Industry

CategorizationCategorization

Application Areas: Industry IntensiveApplication Areas: Industry Intensive

�� Power Systems, Transient data Analysis, Component Diagnosis, Signal Power Systems, Transient data Analysis, Component Diagnosis, Signal detectiondetection

�� NDT, Ultra sonic Flaw Detection, DiagnosisNDT, Ultra sonic Flaw Detection, Diagnosis

�� Communications, Coding, TransmissionCommunications, Coding, Transmission

�� Medical Applications: Compression, Noise Reduction, signal separation, Medical Applications: Compression, Noise Reduction, signal separation, signal classification, image enhancementsignal classification, image enhancement

�� Machine Vision, Robotics, geometry and spatial edge detectionMachine Vision, Robotics, geometry and spatial edge detection

�� Industrial Machinery and Machine Diagnosis,Industrial Machinery and Machine Diagnosis,

�� Remote Sensing, Satellite Image Analysis, Image segmentation, fusionRemote Sensing, Satellite Image Analysis, Image segmentation, fusion

�� Biometrics: Fingerprint, Iris and Facial IdentificationBiometrics: Fingerprint, Iris and Facial Identification

�� Nuclear Reactors, Boiler Pipe fault analysis, Nuclear Reactors, Boiler Pipe fault analysis,

�� Oil Industry, Oil ExplorationOil Industry, Oil Exploration

�� Transportation Industry (railroad engine diagnosis, Aero jet engine and Transportation Industry (railroad engine diagnosis, Aero jet engine and Helicopter drive and gear diagnosisHelicopter drive and gear diagnosis

�� Seismic Data analysisSeismic Data analysis

�� GeneticsGenetics

Applications, Fields of WorkApplications, Fields of Work

Applications in different fieldsApplications in different fields�� Physics, Geophysics, Physics, Geophysics,

�� Mathematics, function approximation, signal Mathematics, function approximation, signal processing, Harmonic analysis, Differential processing, Harmonic analysis, Differential Equation, Frames, Function Space AnalysisEquation, Frames, Function Space Analysis

�� Electrical Engineering, numerous aplicationsElectrical Engineering, numerous aplications

�� Biomedical Engineering, EEG, ECG, MRIBiomedical Engineering, EEG, ECG, MRI

�� Finance and Financial Data AnalysisFinance and Financial Data Analysis

�� Mechanical, Civil, Ocean Science and EngineeringMechanical, Civil, Ocean Science and Engineering

�� Remote Sensing, ForestryRemote Sensing, Forestry

�� Nontechnical: Historial document retrievalNontechnical: Historial document retrieval

Details of ApplicationsDetails of Applications

�� To Be Presented in a Separate To Be Presented in a Separate

�� Slide Presentation Slide Presentation