Introduction to Wavelets
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1 Introduction to Wavelets Olof Runborg Numerical Analysis, School of Computer Science and Communication, KTH RTG Summer School on Multiscale Modeling and Analysis University of Texas at Austin Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
2 Wavelet multiresolution decomposition Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
3 Wavelet multiresolution decomposition Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
4 Wavelet multiresolution decomposition Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
5 Wavelet multiresolution decomposition Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
6 Wavelet multiresolution decomposition Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
7 Wavelet multiresolution decomposition Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
8 Wavelet multiresolution decomposition Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
9 Wavelet multiresolution decomposition Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
10 Wavelet multiresolution decomposition Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
11 Wavelet multiresolution decomposition Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
12 Wavelet examples Haar Scaling function φ(x) Mother wavelet ψ(x) Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
13 Wavelet examples Daubechies 4 Scaling function φ(x) Mother wavelet ψ(x) Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
14 Wavelet examples Daubechies 6 Scaling function φ(x) Mother wavelet ψ(x) Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
15 Wavelet examples Daubechies 8 Scaling function φ(x) Mother wavelet ψ(x) Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
16 Wavelet based image compression Wavelets successful in image compression. Eg: JPEG 2000 standard (Daubechies (9,7) biorthogonal wavelets), FBI fingerprint database,... Consider image as a function and use 2D wavelets. Compress by thresholding + coding + quantization. Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
17 Wavelet based image compression Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
18 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 / 8
19 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 / 8
20 Time Frequency Representation of Signals Given a discrete signal f (n), n = 0, 1,.... Fourier representation total localization in frequency f (n) = j ˆf (j)exp(inj2π/n) Basis functions Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
21 Time Frequency Representation of Signals Given a discrete signal f (n), n = 0, 1,.... Fourier representation total localization in frequency f (n) = j ˆf (j)exp(inj2π/n) Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
22 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 w j,k ψ j,k (n/2π) Basis functions Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
23 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 w j,k ψ j,k (n/2π) Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
24 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 w j,k ψ j,k (n/2π) Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
25 Time Frequency Representation of Signals Example Time representation total localization in time Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
26 Time Frequency Representation of Signals Example Fourier representation total localization in frequency Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
27 Time Frequency Representation of Signals Example Wavelet representation localization in time and frequency Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
28 Wavelets some contributors Strömberg first continuous wavelet Morlet, Grossman "wavelet" Meyer, Mallat, Coifman multiresolution analysis Daubechies compactly supported wavelets Beylkin, Cohen, Dahmen, DeVore PDE methods using wavelets Sweldens lifting, second generation wavelets... and many, many more. Olof Runborg (KTH) Introduction to Wavelets Austin, July / 8
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