Wavelets and filterbanks. Mallat 2009, Chapter 7

Transcription

1 Wavelets ad filterbaks Mallat 2009, Capter 7

2 Outlie Wavelets ad Filterbaks Biortooal bases Te dual perspective: from FB to wavelet bases Biortooal FB Perfect recostructio coditios Separable bases (2D) Overcomplete bases Wavelet frames (aloritme à trous, DDWF) Curvelets

3 Wavelets ad Filterbaks Wavelet side Scali fuctio Desi (from multiresolutio priors) Sial approximatio Correspodi filteri operatio Coditio o te filter Couate Mirror Filter (CMF) Correspodi wavelet families Filterbak side Perfect recostructio coditios (PR) Reversibility of te trasform Equivalece wit te coditios o te wavelet filters Special case: CMFs Orooal wavelets Geeral case Biortooal wavelets

4 Wavelets ad filterbaks Te decompositio coefficiets i a wavelet ortooal basis are computed wit a fast aloritm tat cascades discrete covolutios wit ad, ad subsample te output Fast ortooal WT f() t a ϕ( t) V 0 0 { ϕ } Sice (t-) is a ortoormal basis * * 0 ( ), ϕ( ) ( ) ϕ ( ) ( ) ϕ ( ) * ϕ( ) ϕ() t ϕ( t) Ζ a f t t f t t dt f t t dt f

5 Liki te domais ω z e ω fˆ( ω) fˆ( e ) f ( z) ( ω π ) fˆ( ω π ) fˆ( e ) fˆ( e ω fˆ( ω) fˆ( e ) f ( z ) ˆ* f ( ω) fˆ( ω) f ( z ) ω ) f ( z) Switci betwee te Fourier ad te z-domai f f (z) k f z f (z) f f z () f f (z) f kz k uit delay ( ) reverse te order of te coefficiets eate odd terms Switci betwee te time ad te z-domai

6 Fast ortooal wavelet trasform Fast FB aloritm tat computes te ortooal wavelet coefficiets of a discrete sial a 0. Let us defie Sice { ϕ ( t ) } Ζ is ortoormal, te a f, ϕ d f, ψ ( ) f() t a ϕ t V 0 0 a 0 f( t), ϕ( t ) f ϕ( ),, sice ϕ is a ortoormal basis for V, A fast wavelet trasform decomposes successively eac approximatio PV f i te coarser approximatio PV f plus te wavelet coefficiets carried by PW f. I te recostructio, PV f is recovered from PV f ad PW f for decreasi values of starti from J (decompositio dept)

7 Fast wavelet trasform Teorem 7.7 At te decompositio At te recostructio p a a p p d p a a p p a p p p x x d a d p a p p a () (2) (4) xp x! " # $

8 (3) Proof: decompositio () ϕ p V V ϕ p ϕ p, ϕ ϕ (b) but t 2 p * t 2 ϕ p, ϕ ϕ ϕ dt (a) let t 2 p t ' t' 2 t2p t 2 t' 2 p t 2 p 2 t' 2 2 te t 2 p t ' ϕ ϕ 2 2 * t 2 * ϕ ϕ ( t' 2p) t' t t t' t p p t' 2p replaci ito (a) t' * t ϕ p, ϕ ϕ ϕ ( t ' 2 p ) dt ' ϕ, ϕ( t 2 p ) 2 p tus (b) becomes ϕ p 2 p ϕ

9 Proof: decompositio (2) qui Comi back to te proectio coefficiets a p f,ϕ,p f, 2 p ϕ, f 2 p ϕ * dt, 2 p f (t) ϕ * (t)dt 2 p, f,ϕ, 2 p a a p a 2 p Similarly, oe ca prove te relatios for bot te details ad te recostructio formula

10 Proof: decompositio (3) Details (3bis) ψ W V ψ ψ, ϕ ϕ, p, p,,, ' t 2 t 2p t ψ,, ϕ, ψ, ϕ( t 2p) 2p 2 2 ψ 2 p, p, ϕ f, ψ 2 p f, ϕ,, 2 d p p a

11 Proof: Recostructio Sice W is te ortoormal complemet of V i V, te uio of te two respective basis is a basis for V. Hece V V W ϕ ϕ, ϕ ϕ ϕ, ψ ψ, p, p,,, p,, but ϕ ϕ, ϕ p 2, p,, ψ p 2, p, (see (3) ad (3bis), te aaloous oe for ) tus ϕ p 2 ϕ p 2 ψ, p,, Taki te scalar product wit f at bot sides: a p p 2 a p 2 d a d x p 2 p x 0 2 p CVD

12 a p p 2 a a 0 a 2 a a Let's assume tat is symmetric a 0 a 2 a Grapically p

13 Grapically a a 0 2 a a

14 Summary a p a 2 p Te coefficiets a ad d are computed by taki every oter sample of te covolutio of a wit ad respectively. Te filter removes te ier frequecies of te ier product sequece a, wereas is a i-pass filter tat collects te remaii iest frequecies. Te recostructio is a iterpolatio tat iserts zeroes to expad a ad d ad filters tese sials, as sow i Fiure.

15 Filterbak implemetatio a a 2 2 a 2 d 2 d a 2 2 a 2 ă 0 2 d 2 d 2

16 Fast DWT Teorem 7.0 proves tat a ad d are computed by taki every oter sample of te covolutio o a wit ad respectively Te filter removes te ier frequecies of te ier product ad te filter is a badpass filter tat collects suc residual frequecies A ortoormal wavelet represetatio is composed of wavelet coefficiets at scales 2 2 plus te remaii approximatio at scale 2 J J { d}, J a J

17 Summary Aalysis or decompositio Sytesis or recostructio a 2 2 a d 2 2 a Teorem 7.2 (Mallat&Meyer) ad Teorem 7.3 Mallat&Meyer ω, ad ĥ(0) 2 ( ) 2 ĥ ( ω π ) 2 2 ĥ ω ω ˆ* ˆ( ω) ( ω π) ( ) e Te fast ortooal WT is implemeted by a filterbak tat is completely specified by te filter, wic is a CMF Te filters are te same for every

18 Filter bak perspective a ă 0 Taki as referece (wic amouts to coosi te sytesis low-pass filter) te followi relatios old for a ortooal filter bak: ( ) ( ) ( ) ( ) ( ) electi te uitary sift, as usually doe i applicatios ( ) ( ) ( )

19 Fiite sials Issue: sial extesio at borders Possible solutios: Periodic extesio Works wit ay kid of wavelet Geerates lare coefficiets at te borders Symmetryc/atisymmetric extesio, depedi o te wavelet symmetry More difficult implemetatio Haar filter is te oly symmetric filter wit compact support Use differet wavelets at boudary (boudary wavelets) Implemetatio by lifti steps

20 Wavelet raps

21 Ortooal wavelet represetatio A ortooal wavelet represetatio of a L < f,ϕ L, > is composed of wavelet coefficiets of f at scales 2 L <2 <2 J, plus te remaii approximatio at te larest scale 2 J : Iitializatio Let b be te discrete time iput sial ad let N - be te sampli period, suc tat te correspodi scale is 2 L N - Te: N - : discrete sample distace 2 L N - scale oriial cotiuous time sial discrete time sial iterpolatio fuctio

22 Iitializatio followi te defiitio: ϕ L, L t 2 ϕ L L 2 2 L /2 tn tn 2 N N ϕ L L, Nϕ ϕ ϕ L, N 2 N N N but t N f ( t) b ϕ b ϕl, ( t ) N N b t N f, ϕ f, ϕ L, al N N N al f, ϕ sice a f t N dt L L t N ϕ N ( ) ( ) a N f N Basis for V L by defiitio, te N - : discrete sample distace 2 L N - scale, if f is reular, te sampled values ca be cosidered as a local averae i te eiborood of f(n - ) L

23 Te filter bak perspective

24 Perfect recostructio FB Dual perspective: ive a filterbak cosisti of 4 filters, we derive te perfect recostructio coditios 2 a 2 ~ a 0 2 d 2 ~ a 0 Goal: determie te coditios o te filters esuri tat a 0 a 0

25 PR Filter baks Te decompositio of a discrete sial i a multirate filter bak is iterpreted as a expasio i l 2 (Z) sice te a l a * 2l a 2l a 2l ad te sial is recovered by te recostructio filter dual family of vectors tus poits to biortooal wavelets

26 Te two families are biortooal Tus, a PR FB proects a discrete time sials over a biortooal basis of l 2 (Z). If te dual basis is te same as te oriial basis ta te proectio is ortoormal.

27 Discrete Wavelet basis Questio: wy boter wit te costructio of wavelet basis if a PR FB ca do te same easily? Aswer: because couate mirror filters are most ofte used i filter baks tat cascade several levels of filteris ad subsamplis. Tus, it is ecessary to uderstad te beavior of suc a cascade N - : discrete sample distace 2 L N - scale t ϕ,, a f ϕ L N N for dept >L L ϕ N discrete sial at scale 2 L L,

28 Discrete wavelet basis

29 Perfect recostructio FB Teorem 7.7 (Vetterli) Te FB performs a exact recostructio for ay iput sial iif ĥ * ĥ * ( ω) ĥ(ω) ( ĝ* ω) ĝ(ω) 2 ( ω π ) ĥ(ω) ( ĝ* ω π ) ĝ(ω) 0 (alias free) Matrix otatios ˆ~ * ( ω) 2 ˆ( ω π ) ˆ~ * ( ω) Δ( ω) ˆ( ω π ) Δ( ω) ˆ( ω) ˆ( ω π ) ˆ( ω π ) ˆ( ω) We all te filters are FIR, te determiat ca be evaluated, wic yields simpler relatios betwee te decompositio ad te recostructio filters.

30 Cai te sampli rate Dowsampli Upsampli ( ) 2 ˆx ( ω ) ˆx ( ω π ) ŷ 2ω ˆx ω ( ) x! " 2 ŷ( ω) ˆx ( 2ω ) x! ê 2 " # $ e 2ω # $ e ω ( ) ˆx dow ( ω) ŷ( ω) ( ) # 2 ˆx # ω 2 ŷ ω! "! " $ % & ˆx! # ω " 2 π $ $ && %% ( ) ˆx é 2 up ω ˆx ω ( ) ŷ( ω)

31 ( ) y! 0 Subsampli: proof ŷ ω " # $ y! " # $ e ω y! " 2 # $ e 2ω x! " 0 # $ x! " 2# $ e ω x! " 4 # $ e 2ω tus ( ) x! 0 ŷ 2ω but " # $ x! " 2# $ e 2ω x! " 4 # $ e 4ω (ωπ ) ( e ) 0 x! " # $ e ω x! " # $ e (ωπ ) 0 2 x! " # $ e ω x! " # $ 2(ωπ ) ( e ) x! " 2 # $ e 2ω 2 x! " 2# $ e 2ω x! " 2 # $ tus ( ) x! 0 ŷ 2ω ŷ 2ω (ωπ ) ( e ) 2 x! 2 " # $ 2 x! " # $ e ω x! " # $ ( ) ( ) 2 ˆx ( ω ) ˆx ( ω π ) " # $ e 2ω x! " 2 # 2(ωπ ) ( $ e )

32 Perfect Recostructio coditios 2 a 2 a 0 2 d 2 a 0 ( ( ) ( ) ( ) ( )) ˆ ˆ a(2 ω) a0 ω ω a0 ω π ω π 2 sice ad are real ( ω) ˆ ˆ * ( ω) ( ω) ( ω) tus, replaci i te first equatio a ˆ ˆ (2 ω) a0 ω ω a0 ω π ω π 2 Similarly, for te i-pass brac d(2 ω) a0 ω ω ˆ 0 2 * * ( ( ) ( ) ( ) ( )) ( ) * * ( ) ˆ ( ) a ( ω π) ( ω π) â0 (ω) â (2ω) ĥ(ω) ˆd (2ω) ĝ(ω)

33 Perfect Recostructio coditios Putti all toeter â0 (ω) â (2ω) ĥ(ω) ˆd (2ω) ĝ(ω) 2 a ω 0 ( ( )ĥ* ω) a 0 ( ω π )ĥ* ω π â0 (ω)! # 2 " ( ( )) ĥ(ω) ( ) ( ĝ* ω) a 0 ( ω π ) ( ĝ* ω π ) ( ) ĝ(ω) 2 a 0 ω ĥ * ( ω) ĥ(ω) ( ĝ* ω) ĝ(ω) $ &a % 0 ( ω)! # ĥ * ( ω π ) ĥ(ω) 2 ĝ* ω π " 0 (alias-free) ( ) ĝ(ω) $ &a % 0 ( ω π ) ĥ * ĥ * ( ω) ĥ(ω) ( ĝ* ω) ĝ(ω) 2 ( ω π ) ĥ(ω) ( ĝ* ω π ) ĝ(ω) 0 (alias free) Matrix otatios ˆ~ * ( ω) 2 ˆ( ω π ) ˆ~ * ( ω) Δ( ω) ˆ( ω π ) Δ( ω) ˆ( ω) ˆ( ω π ) ˆ( ω π ) ˆ( ω)

34 Perfect recostructio biorooal filters Teorem 7.8. Perfect recostructio filters also satisfy ĥ * ( ω) ĥ(ω) ĥ* ω π ( ) ĥ(ω π ) 2 Furtermore, if te filters ave a fiite impulse respose tere exists a i R ad l i Z suc tat ˆ( ω) ae ˆ~ ( ω) e a i(2l ) ω i(2l ) ω ˆ~ * ( ω π ) ˆ* ( ω π ) a, l0 ĝ(ω) e ω ĥ* (ω π ) ĝ(ω) e ω * (ω π ) Correspodily () () Couate Mirror Filters: ĥ ( ω ) 2 ĥ ( ω π ) 2 2

35 Perfect recostructio biortooal filters qui Give ad ad setti a ad l0 i (2) te remaii filters are ive by te followi relatios Te filters ad are related to te scali fuctios φ ad ~φ via te correspodi two-scale relatios, as was te case for te ortooal filters (see eq. ). Switci to te z-domai Sial domai ) ( ˆ ) ˆ~ ( ) ( ˆ~ ) ˆ( * * π ω ω π ω ω ω ω e e i i ) ( ~ ~ ) ( ) ( ) ~ ( ) ( ~ ) ( z z z z z z ~ ~ (3)

36 Biortooal filter baks A 2-cael multirate filter bak covolves a sial a 0 wit a low pass filter ad a i pass filter ad sub-samples te output by 2 A recostructed sial ã 0 is obtaied by filteri te zero-expaded sials wit a dual low-pass ad i pass filter Imposi te PR coditio (output sialiput sial) oe ets te relatios tat te differet filters must satisfy (Teorem 7.7) a d a a ~ ~ ~ ~ ~ 0 p p p x x y d a a

37 Revisiti te ortooal case (CMF) a ă 0 Taki as referece (wic amouts to coosi te aalysis low-pass filter) te followi relatios old for a ortooal filter bak: sytesis low-pass (iterpolatio) filter: reverse te order of te coefficiets ( ) eate every oter sample

38 Ortooal vs biortooal PRFB 2 a 2 a 0 2 d 2 a 0 ĥ * ( ω) ĥ(ω) ĥ* ω π ĝ(ω) e ω ĥ* (ω π ) ĝ(ω) e ω * (ω π ) Biortooal PRFB ( ) ĥ(ω π ) 2 ( ) 2 ĥ ( ω π ) 2 2 ĥ ω Ortooal PRFB I te sial domai () ()

39 Fast BWT Two differet sets of basis fuctios are used for aalysis ad sytesis PR filterbak ~ ~ 2 2 d a a a d a a 0 0 a a a 0 ~ ~ ă 0 ) ( ~ ~ ) (

40 Be careful wit otatios! I te simplified otatio were is te aalysis low pass filter ad is te aalysis bad pass filter, as it is te case i most of te literature; te delay factor is ot made explicit; Te relatios amo te filters modify as follows 2 2 ~ a ~ ă 0 a 0 a0 ~ ( ) Slitly differet formulatio: te ~ ( ) i pass filters are obtaied by te low pass filters by eati te odd terms

41 Biortooal bases Ortoormal basis Bi-ortooal basis {e } N : basis of Hilbert space {e } N : liearly idepedet Ortooality coditio < e, e p >0 p y H, A>0 ad B>0 : y H, Tere exists a sequece λ y, e y λ e : λ y, e : y λ e ~ y B 2 Biortooality coditio: λ 2 y A 2 e 2 orto-ormal basis e e p e, e~ p δ p y, ~ f e e AB ortooal basis f, e e~

42 Biortooal bases If ad are FIR ˆ Φ ˆ ( 2 ( ω) p ω) ˆΦ(0), ˆΦ ω 2 p ( ) p ĥ( 2 p ω) 2 Tou, some oter coditios must be imposed to uaratee tat φ^ ad φ^tilde are FT of fiite eery fuctios. Te teorem from Coe, Daubecies ad Feaveau provides sufficiet coditios (Teorem 7.0 i M999 ad Teorem 7.3 i M2009) ˆΦ(0) Te fuctios ˆφ ad ˆφ satisfy te biortooality relatio ϕ(t), ϕ(t ) δ Te two wavelet families ψ, { } (,) Z 2 ad { ψ }, (,) Z 2 are Riesz bases of L 2 (R) ψ, ψ, δ 'δ ' ', ' Ay f L 2 ( R) as two possible decompositios i tese bases f f,ψ, ψ, f, ψ, ψ,,,

43 Remider

44 Summary of Biortooality relatios ~ (, ),(, ~ ) A ifiite cascade of PR filter baks yields two scali fuctios ad two wavelets wose Fourier trasform satisfy ˆΦ ( 2ω ) 2 ĥ ( ω ) ˆΦ ( ω) ϕ t 2 # % & $ ' ( ϕ t ( ) (i) Φ ˆ ( 2ω ) 2 ˆ ( ω) ˆΦ ω ( ) ϕ t 2 ˆΨ ( 2ω ) 2 ĝ ( ω ) ˆΦ ( ω) ψ t 2 Ψ ˆ ( 2ω ) 2 ĝ ( ω ) ˆΦ ( ω ) ψ t 2 # % & $ ' ( ϕ t ( ) # % & $ ' ( ϕ t ( ) # % & $ ' ( ϕ t ( ) (ii) (iii) (iv)

45 Properties of biortooal filters Imposi te zero averae coditio to ψ i equatios (iii) ad (iv) ˆΨ(0) ˆΨ(0) 0 ĝ(0) ĝ(0) 0 replaci ito te relatios (3) (also sow below) ĝ(ω) e iω ĥ* (ω π ) ĝ(ω) eiω ĥ * (ω π ) ĥ* (π ) ĥ(π ) 0 Furtermore, replaci suc values i te PR coditio () ĥ * (ω) ĥ(ω) ĝ* (ω) ĝ(ω) 2 ĥ* (0) ĥ(0) 2 It is commo coice to set ĥ * (0) ĥ(0) 2

46 Biortooal bases If te decompositio ad recostructio filters are differet, te resulti bases is oortooal Te cascade of J levels is equivalet to a sial decompositio over a o-ortooal basis { 2 } { } J ϕ J k, ψ k2, Te dual bases is eeded for recostructio Ζ J Ζ

47 Example: bior3.5

48 Example: bior3.5

49 Biortooal bases

50 Biortooal bases

51 CMF : ortooal filters PR filter baks decompose te sials i a basis of l 2 (Z). Tis basis is ortooal for Couate Mirror Filters (CMF). Smit&Barwell,984: Necessary ad sufficiet coditio for PR ortooal FIR filter baks, called CMFs Imposi tat te decompositio filter is equal to te recostructio filter ~, eq. () becomes ĥ * (ω) ĥ(ω) ĥ* (ω π ) ĥ(ω π ) 2 () ĥ * (ω)ĥ(ω) ĥ* (ω π )ĥ(ω π ) 2 ĥ(ω) 2 ĥ(ω π ) 2 2 Correspodily ~ ~ ( )

52 Summary PR filter baks decompose te sials i a basis of l 2 (Z). Tis basis is ortooal for Couate Mirror Filters (CMF). Smit&Barwell,984: Necessary ad sufficiet coditio for PR ortooal FIR filter baks, called CMFs Imposi tat te decompositio filter is equal to te recostructio filter ~, eq. () becomes Correspodily 2 ) ˆ( ) ˆ( 2 ) ˆ( ) ( ˆ ) ˆ( ) ( ˆ 2 ) ( ˆ~ ) ( ˆ ) ( ˆ~ ) ( ˆ 2 2 * * * * π ω ω π ω π ω ω ω π ω π ω ω ω ) ( ~ ~

53 Properties Support, ~ are FIR scali fuctios ad wavelets ave compact support Vaisi momets Te umber of vaisi momets of Ψ is equal to te order p~ of zeros of ~ i π. Similarly, te umber of vaisi momets of ~ ψ is equal to te order p of zeros of i π. Reularity Oe ca sow tat te reularity of Ψ ad φ icreases wit te umber of vaisi momets of ~ ψ, tus wit te order p of zeros of i π. Viceversa, te reularity of ad icreases wit te umber of vaisi momets of Ψ, tus wit te order p~ of zeros of ~ i π. ψ ~ ϕ ~ Symmetry It is possible to costruct bot symmetric ad ati-symmetric bases usi liear pase filters I te ortooal case oly te Haar filter is possible as FIR solutio.