Wavelet Basics. (A Beginner s Introduction) J. S. Marron Department of Statistics University of North Carolina
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1 Wavelet Bascs (A Beger s Itroducto) J. S. Marro Departmet o Statstcs Uversty o North Carola Some reereces: Marro, J. S. (999) Spectral vew o wavelets ad olear regresso, Bayesa Ierece Wavelet-Based Models, Müller, P. ad Vdaovc, B. Eds., Lecture Notes Statstcs No. 4, Sprger, New Yor, 9-3. Strag, G. (989) Wavelets ad dlato equatos: a bre troducto, SIAM Revew, 3, For deeper mathematcs, but cocsely preseted: Chps. ad o: Beedetto, J. J. ad Frazer, M. W. (994) Wavelets: Mathematcs ad Applcatos, CRC Press, Boca Rato, Florda.
2 Two Worlds World : Eucldea vector space, y R M : y,..., y R y World : L (Hlbert) Fucto Space, { } ( x) : ( x) dx < Coecto: va dgtzato For equally spaced x < L < x, Relate (x) to ( x ) M ( x )
3 Ier Product Structure World : z y z z y y z y,, M M World : ) ( ) (, dx x g x g Cosequeces:. dstace b a b a,. agle : b a, b a Coecto: Rema Summato
4 Lear Bases {,,... } ψ ψ s a bass meas every member has a lear represetato: θψ A bass s orthoormal whe: ψ,ψ (all orthogoal to each other, wth legth )
5 Orthoormal Bases Cosequeces: - Compute θ, ψ - R, θ θ M θ s the trasorm - trasorm s a rotato operato (legths ad agles preserved)
6 Example : Ut vector bass u R M M,,..., Notes: - orthoormal - or y y y M, trasorm has y θ - detty rotato
7 Example : Fourer Bass Show FourerBass.ps, wth s s ad cos s. World : Dscrete Fourer Bass World : Cotuous Fourer Bass Exactly orthoormal both (taes trgoometry) Fourer Trasorm: Rotato that decomposes to perodctes
8 Example 3: Haar Wavelet Bass Show HaarFullBass.ps Up ad Dow step uctos, ψ, doubly dexed by: - scale - locato / dlato orm : x ( ψ ( ) x ), ψ Exactly orthoormal both worlds Dyadc structure, very smlar to cascades Hstogram Vew: successve dereces Show HaarHsto.ps
9 Example 4: Smoother Wavelet Bases Daubeches 4: Cotuous but rough Show Daub4Bass.ps Symmlet 8: much smoother, stll local Show Symm8Bass.ps
10 Applcato : Sgal Compresso Idea: represet y by trasorm θ, ad hope that may θ - lossless compresso, wat θ - approxmate compresso, replace θ by whe close Ma Cocept: Good Compresso more θ
11 Qualty o approxmato: Measure by Eergy sgal: E y y or E ( x) dx - lossless compresso: E y Eθ (Parseval Idetty) - Good approxmato: E y Eθ - Bad approxmato: E y >> Eθ
12 Approxmato Follore: Ut vectors: terrble or terestg sgals Fourer bass: good or smooth ad perodc Wavelet bases: allow some umps may varatos, ad ways o coog up good bases Show ExactRsEGs.ps ad CompressoEG.ps
13 Applcato : Deosg Goal: rom data y s + try to recover sgal s rom ose, (e.g...d. mea ) Trasorm approach: - d rotato wth good compresso o sgal - zero out small θ - vert trasorm
14 Deosg Examples Show WaveDNFourer.eps, StepDNFourer.eps ad WaveStepDNHaar.eps Wave Target: - Fourer bass: Excellet - Haar bass: Poor Step Target: - Fourer bass: Terrble - Haar bass: Excellet Note: drve by sgal compresso
15 Fast Computato o trasorm: θ y, ψ,,...,. Naïve mplemetato: O ( ) matrx multplcato. Fast Fourer Trasorm: O( log) usg trgoometrc propertes 3. Fast wavelet Trasorm: O () usg smple pyramd algorthm
16 Haar Pyramd Algorthm, I Notato: ) ( M, ) ( M mothers : ) (, ψ Show HaarFullBass.ps aga
17 Haar Pyramd Algorthm, II athers : + ) (, ϕ Show HaarFathers.ps Note: ather vectors are also a bass (but ot orthoormal) Ca mx ad match mothers ad athers Show HaarPartBass.ps
18 Haar Pyramd Algorthm, III Relatos across scales:. Magcato (dlato): ϕ + s hal wdth o ϕ ψ + s halwdth o ψ. Father Mother, Father ψ ( ϕ ), +!, + ϕ +!, ϕ ( ϕ ), +!, + + ϕ +!,
19 Haar Pyramd Algorthm, IV Apply er product to get: θ, ( ) +, + +, ( ), +, + + +, where, ϕ,, y Start wth log ( ), y, ad terate up through scales, to get O () algorthm
20 Haar Pyramd Algorthm, V Overall Structure: lo pass ( avg) h pass( avg), +, +,..., ' s θ ' s,..., θ...,, +,,..., θ,
21 Haar Pyramd Algorthm, VI Notes:. each level s eergy preservg : + +,, + θ,. Eergy o costats passed to s 3. At-costat eergy passed to θ s Aga vst ExactRsEGs.ps ad CompressoEG.ps 4. Eergy ssues are ANOVA style decomposto o sums o squares
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