Some Essentials of Data Analysis with Wavelets Slides for the wavelet lectures of the course in data analysis at The Swedish National Graduate School of Space Technology Niklas Grip, Department of Mathematics, Luleå University of Technology Last update: 29--2
f() a ()
() ().8.8 6.6 6.6.4.4.2.2 -.2 -.2 -.4 -.4 -.6 -.6 -.8 -.8 - -.5.5.5 - -.5.5.5 Old approimation New approimation f() a () /2
() ().8.8 6.6 6.6.4.4.2.2 -.2 -.2 -.4 -.4 -.6 -.6 -.8 -.8 - -.5.5.5 - -.5.5.5 f() a 2 () Old approimation New approimation /4 2/4 3/4 /2
() ().8.8 6.6 6.6.4.4.2.2 -.2 -.2 -.4 -.4 -.6 -.6 -.8 -.8 - -.5.5.5 - -.5.5.5 f() a 3 () Old approimation New approimation /8 2/8 3/8 4/8 5/8 6/8 7/8 /4 2/4 3/4
() ().8.8 6.6 6.6.4.4.2.2 -.2 -.2 -.4 -.4 -.6 -.6 -.8 -.8 - -.5.5.5 - -.5.5.5 f() a 4 () Old approimation New approimation 2/6 4/6 6/6 8/6 /6 2/6 4/6 /8 2/8 3/8 4/8 5/8 6/8 7/8
() ().8.8 6.6 6.6.4.4.2.2 -.2 -.2 -.4 -.4 -.6 -.6 -.8 -.8 - -.5.5.5 - -.5.5.5 f() a 5 () Old approimation New approimation 4/32 8/32 2/32 6/32 2/32 24/32 28/32 2/6 4/6 6/6 8/6 /6 2/6 4/6
f() a 6 () 8/64 6/64 24/64 32/64 4/64 48/64 56/64
Wavelet bases { n/2 n j - k y = y - k } The Haar basis is of the type ( ), ( ) 2 (2 ). n, k Such a basis is called a wavelet basis with j and mother wavelet y. scaling function Consequence : The scaling function gives a large scale approimation and the wavelets adds finer details (illustrated in net slide).
Orthonormal bases Both the Haar basis and the usual Fourier basis is a set of building blocks { } with the following properties e k 2 Any function f Î L ( ) can be decomposed into a sum f = c e. There is a simple formula for computing the coefficients: Inner product c = f( ) e ( ) d = f, e ò k k k - ì if k = n The building blocks are orthonormal: ek, en = ï í ï if k ¹ n ïî Any such set of building blocks is called an ort honormal basis. å k k k
Good properites of the Haar wavelet basis : Orthonormal (just like the Fourier basis). Well localized Better suited for good approimation of small local details in a signal with a small number of terms (contrary to the Fourier basis). Usually less desirable properties of the Haar wavelet basis : Discontinuities ) Many terms needed for good approimation (=small "edges" in last slide ) of continuous signals. 2) Bad frequency localization (drawback in in time-frequency analysis (eplained soon)).
MRA adds smoothness Nt Natural question : Are there any way to contruct a continuous, or even " arbitrarily smooth" (say, k times differentiable), well localized and orthonormal wavelet basis? Answer : Yes. The construction is a generalization of the telescope sums in last lecture. Described in any wavelet book under the name multiresolution l analysis (MRA).
Etra bonus : It follows from the MRA theory that there is a special The fast wavelet transform algorithm for quick computation of the wavelet coefficients. The computation time is proportional to the signal length ( N ) and thus faster than the fast Fourier transform ( N log N).
Pyramid algorithm / filter banks / Mallat s algorithm
Daubechies scaling functions Eample : Daubechies n scaling functions, n=-2.5 n=.5 n=2.5 n=3.5 -.5 n=4 5.5.5 -.5 n=7 5.5.5 -.5 n= 5.5.5.5 -.5 n=5 5.5.5 -.5 n=8 5.5.5 -.5 n= 5.5.5.5 -.5 n=6 5.5.5 -.5 n=9 5.5.5 -.5 n=2 5.5.5 Nonzero only in the interval [,n-]. For any k and large enough n, the Daubechies n wavelet and scaling function is k times differentiable. -.5 5 -.5 5 -.5 5
Daubechies wavelets Corresponding Daubechies wavelets. n= n=2 n=3 - n=4 2 - n=5 2 - n=6 2 - n=7 2 - n=8 2 - n=9 2 - n= 2 - n= 2 - n=2 2-2 - 2-2
Spline wavelets Eample 2: Spline wavelets of degree 2 Translated scaling functions 2 2 3 2 Translated wavelets 2 3 2 2 3 2 Translated and dilated (with factor 2) wavelets 2 3 2 2 3 2 Translated and dilated (with factor 4) wavelets 2 3 2 2 3 2 2 3 Eponential decay (l (slower than Daubechies, but still fast). Spline wavelets of degree n is n times differentiable. nth degree polynomial in intervals [k,k+] (scaling function) and [k/2,(k+)/2] (wavelet).
Some threshold techniques
Caruso wa roll eample Source: http://www.fmah.com ) Original, 2) Single pass denosied, 3) Removed noise, 4), second pass denoised seeking decorrelation between the noise model and the original file
L H H L L H
256 256 piels, 256256 i l 256 greyscale l Whi noise White i added dd d Restored, daub4, reduced to,8 % of the original file size
FBI fingerprint eample
Original image Image size: 847683 piels 24 bit colours =66MB.66
Compression: JPEG 65.8 times JPEG-compressed image
Compression: JPEG2 3 times JPEG2-compressed image
Movie eample Original: Denoised: Removed noise: Source: http://www.fmah.com
Digital subscriber lines
ADSL vs. VDSL Multicarrier transmission eamples: ADSL: Out now. About 2-8.5 megabits per second (Mbps) VDSL: (Originally) planned for 2. From 5 Mbps in 5 m long wires up to about 6 Mbps in 4 m long wires. (5 Mbps is enough for, for eample, 8 digital TV channels or 2-4 high definition TV channels.)
Maimum delay restrictions
Choice of basis functions Each symbol is built up of N basis functions The transmitted information f s (t)= () å c f () t kl, k N å l = k, l k, l must be well localized in time (because too long symbols introduce unacceptable delays). Wavelets can be used, but for this particular application, the short time Fourier transform has some advantages and is used in VDSL.
Railway bridge strains
Channel A ( m/m), 7 level decomposition with haar wavelet. Channel A2 ( m/m), 7 level decomposition with haar wavelet. 5-5 - 5 5 2 25 3 35 4 45 2 - -2-3 5 5 2 25 3 35 4 45 Channel A5 ( m/m), 7 level decomposition with haar wavelet. Channel A4 ( m/m), 7 level decomposition with haar wavelet. Channel A5 ( m/m) 7 level decomposition with haar wavelet 2 - -2-3 5 5 2 25 3 35 4 45 3 2 - Channel A6 ( m/m), 7 level decomposition with haar wavelet. 5 5 2 25 3 35 4 45 Channel A8 ( m/m), 7 level decomposition with haar wavelet. 5-5 - -5 5 5 2 25 3 35 4 45 Channel A7 ( m/m), 7 level decomposition with haar wavelet. 3 2 - -2 5 5 2 25 3 35 4 45 Channel R ( m/m), 7 level decomposition with haar wavelet. 5 5 5-5 -5 5 5 2 25 3 35 4 45 5 5 2 25 3 35 4 45
5-5 Channel A ( m/m), 7 level decomposition with db2 wavelet. - 5 5 2 25 3 35 4 45 2 - -2-3 3 2 - Channel A4 ( m/m), 7 level decomposition with db2 wavelet. 5 5 2 25 3 35 4 45 Channel A6 ( m/m), 7 level decomposition with db2 wavelet. 5 5 2 25 3 35 4 45 Channel A8 ( m/m), 7 level decomposition with db2 wavelet. 2 - -2-3 -5 - -5 Channel A2 ( m/m), 7 level decomposition with db2 wavelet. 5 5 2 25 3 35 4 45 5 3 2 Channel A5 ( m/m), 7 level decomposition with db2 wavelet. 5 5 2 25 3 35 4 45 Channel A7 ( m/m), 7 level decomposition with db2 wavelet. - -2 5 5 2 25 3 35 4 45 Channel R ( m/m), 7 level decomposition with db2 wavelet. 5 5 5-5 -5 5 5 2 25 3 35 4 45 5 5 2 25 3 35 4 45
5-5 Channel A ( m/m), 7 level decomposition with coif3 wavelet. - 5 5 2 25 3 35 4 45 2 - -2-3 3 2 - Channel A4 ( m/m), 7 level decomposition with coif3 wavelet. 5 5 2 25 3 35 4 45 Channel A6 ( m/m), 7 level decomposition with coif3 wavelet. 5 5 2 25 3 35 4 45 Channel A8 ( m/m), 7 level decomposition with coif3 wavelet. 2 - -2-3 -5 - -5 Channel A2 ( m/m), 7 level decomposition with coif3 wavelet. 5 5 2 25 3 35 4 45 5 3 2 Channel A5 ( m/m), 7 level decomposition with coif3 wavelet. 5 5 2 25 3 35 4 45 Channel A7 ( m/m), 7 level decomposition with coif3 wavelet. - -2 5 5 2 25 3 35 4 45 Channel R ( m/m), 7 level decomposition with coif3 wavelet. 5 5 5-5 -5 5 5 2 25 3 35 4 45 5 5 2 25 3 35 4 45
Some applications Wavelets were developed independently in the fields of mathematics, quantum physics, electrical engineering and siesmic geology. Some application areas are Data compressision Astronomy Acoustics Nuclear engineering Sub-band coding Signal and image processing Neurophysiology py Music Magnetic resonance imaging Speech discrimination Optics Fractals Turbulence Earthquake-prediction Radar Human vision Mathematical analysis Partial differential equations Numerical analysis Statistics Econometrics Communication theory Computer graphics