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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
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 2 / 8
Wavelet multiresolution decomposition
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 2 / 8
Wavelet multiresolution decomposition
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 2 / 8
Wavelet multiresolution decomposition
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 2 / 8
Wavelet multiresolution decomposition
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 2 / 8
Wavelet multiresolution decomposition
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 2 / 8
Wavelet multiresolution decomposition
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 2 / 8
Wavelet multiresolution decomposition
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 2 / 8
Wavelet multiresolution decomposition
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 2 / 8
Wavelet multiresolution decomposition
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 2 / 8
Wavelet examples
Haar
Scaling function φ(x) Mother wavelet ψ(x)
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 3 / 8
Wavelet examples
Daubechies 4
Scaling function φ(x) Mother wavelet ψ(x)
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 3 / 8
Wavelet examples
Daubechies 6
Scaling function φ(x) Mother wavelet ψ(x)
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 3 / 8
Wavelet examples
Daubechies 8
Scaling function φ(x) Mother wavelet ψ(x)
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 3 / 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.
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 4 / 8
Wavelet based image compression
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 5 / 8
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
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 6 / 8
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)
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 6 / 8
Time Frequency Representation of Signals
Given a discrete signal f (n), n = 0,1, . . ..
Fourier representation — total localization in frequency
f (n) =∑
j
f̂ (j)exp(inj2π/N)
Basis functions
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 6 / 8
Time Frequency Representation of Signals
Given a discrete signal f (n), n = 0,1, . . ..
Fourier representation — total localization in frequency
f (n) =∑
j
f̂ (j)exp(inj2π/N)
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 6 / 8
Time Frequency Representation of Signals
Given a discrete signal f (n), n = 0,1, . . ..
Wavelet representation — localization in time and frequency
f (n) =∑j,k
wj,kψj,k (n/2π)
Basis functions
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 6 / 8
Time Frequency Representation of Signals
Given a discrete signal f (n), n = 0,1, . . ..
Wavelet representation — localization in time and frequency
f (n) =∑j,k
wj,kψj,k (n/2π)
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 6 / 8
Time Frequency Representation of Signals
Given a discrete signal f (n), n = 0,1, . . ..
Wavelet representation — localization in time and frequency
f (n) =∑j,k
wj,kψj,k (n/2π)
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 6 / 8
Time Frequency Representation of SignalsExample
Time representation — total localization in time
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 7 / 8
Time Frequency Representation of SignalsExample
Fourier representation — total localization in frequency
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 7 / 8
Time Frequency Representation of SignalsExample
Wavelet representation — localization in time and frequency
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 7 / 8
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.
Olof Runborg (KTH) Introduction to Wavelets Austin, July 2008 8 / 8
