Download pdf - Introduction to Wavelets - web.ma.utexas.eduIntroduction to Wavelets Olof Runborg Numerical Analysis, School of Computer Science and Communication, KTH RTG Summer School on Multiscale

Introduction to Wavelets

Olof Runborg

Numerical Analysis,School of Computer Science and Communication, KTH

RTG Summer School on Multiscale Modeling and AnalysisUniversity of Texas at Austin

2008-07-21 – 2008-08-08

Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 1 / 8

Wavelet multiresolution decomposition




















Wavelet examples

Haar

Scaling function φ(x) Mother wavelet ψ(x)


Wavelet examples

Daubechies 4



Wavelet examples

Daubechies 6



Wavelet examples

Daubechies 8



Wavelet based image compression

Wavelets successful in image compression. Eg: JPEG 2000standard (Daubechies (9,7) biorthogonal wavelets), FBI fingerprintdatabase, . . .Consider image as a function and use 2D wavelets.

Compress by thresholding + coding + quantization.


Wavelet based image compression


Time Frequency Representation of Signals

Given a discrete signal f (n), n = 0,1, . . ..

Time representation — total localization in time

f (n) =∑

k

f (k)δ(n − k)

Basis functions





f (n) =∑

k

f (k)δ(n − k)




Fourier representation — total localization in frequency

f (n) =∑

j

f̂ (j)exp(inj2π/N)

Basis functions





f (n) =∑

j

f̂ (j)exp(inj2π/N)




Wavelet representation — localization in time and frequency

f (n) =∑j,k

wj,kψj,k (n/2π)

Basis functions





f (n) =∑j,k

wj,kψj,k (n/2π)





f (n) =∑j,k

wj,kψj,k (n/2π)


Time Frequency Representation of SignalsExample









Wavelets – some contributors

Strömberg – first continuous waveletMorlet, Grossman – "wavelet"Meyer, Mallat, Coifman – multiresolution analysisDaubechies – compactly supported waveletsBeylkin, Cohen, Dahmen, DeVore – PDE methods using waveletsSweldens – lifting, second generation wavelets

. . . and many, many more.
