Introduction to Multiresolution Analysis (MRA)

Transcription

1 Outline Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation References Introduction to Multiresolution Analysis (MRA) R. Schneider F. Krüger TUB - Technical University of Berlin November 22, 2007 R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

2 Outline Introduction and Example Multiresolution Analysis Discrete Outline Wavelet Transform (DWT) Finite Calculation References Outline Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

3 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

4 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References goal approximation of functions (e.g. signals, images, orbitals) idea coarse approximation (trend) + fine improvement (detail) with detail << trend imagination building a house. start with big pieces and fill in with middle sized and at the end with little pieces R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

5 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References step 1 step detail R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

6 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References step 2 step detail R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

7 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References step 3 step detail R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

8 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References step 4 step detail R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

9 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References step 5 step detail R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

10 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References Details 2 x 10 3 detail R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

11 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References In the example a very simple set of basis functions is used: { 1, x [0, 1) ϕ(x) =1 [0,1) (x) = 0, else 1.5 Haar function R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

12 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References ϕ j k (x) =2j/2 1 [2 j k,2 j (k+1))(x) = =2 j/2 ϕ(2 j x k). { 2 j/2, x [2 j k, 2 j (k + 1)) 0, else 2.5 Haar function, j = 2, k = R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

13 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References The details are given by the functions ψ j k (x) =2j/2 ψ(2 j x k) with 1, x [0, 0.5) ψ(x) = 1, x [0.5, 1) 0, else. 1.5 Haar wavelet 2.5 Haar wavelet, j = 2, k = R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

14 Outline Introduction and Example Multiresolution Analysis Discrete Introduction Wavelet and Transform Example (DWT) Finite Calculation References Refinement Equation ϕ is called Haar function and ψ is called the Haar wavelet. They satisfy the so-called refinement equations : ϕ j k (x) =2 1/2 (ϕ j+1 2k (x) + ϕj+1 ψ j k (x) =2 1/2 (ϕ j+1 2k (x) ϕj+1 2k+1 (x)) 2k+1 (x)) 1.5 Refinement for the Haar function φ 2 1/2 1 φ Refinement for Haar wavelet ψ 2 1/2 φ /2 φ /2 φ R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

15 Outline Introduction and Example Multiresolution Analysis Discrete Multiresolution Wavelet Transform Analysis (DWT) Finite Calculation References Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

16 Outline Introduction and Example Multiresolution Analysis Discrete Multiresolution Wavelet Transform Analysis (DWT) Finite Calculation References Multiresolution Analysis Given a function ϕ L 2 (R) = {f f 2 = ( f(x) 2 dx ) 1/2 < }. R We consider the shifts and dilatations of ϕ : We write ϕ j k (x) = 2j/2 ϕ(2 j x k), j, k Z. V j = span{ϕ j k k Z}. R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

17 Outline Introduction and Example Multiresolution Analysis Discrete Multiresolution Wavelet Transform Analysis (DWT) Finite Calculation References If every f L 2 (R) can be arbitrarily accurately approximated by ϕ j k s, i.e. j j 0 V j = L 2 (R) holds and ϕ fulfills a refinement equation ϕ = k Z h k ϕ 1 k then we say that ϕ or the V j s, respectively, build a multi resolution analysis (MRA). R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

18 Outline Introduction and Example Multiresolution Analysis Discrete Multiresolution Wavelet Transform Analysis (DWT) Finite Calculation References Orthonormal Wavelets Because of the refinement equation it holds V j V j+1 for every j Z. Therefore there exists the orthogonal space W j of V j in V j+1. Therefore V j W j, V j W j = V j+1. W j is called the detail space or the wavelet space for V j. A function ψ that satisfies 1. R ψ(x)dx = 0 2. {ψ( k) k Z} is an orthonormal basis of W 0 is called (orthonormal) wavelet or mother wavelet for the function ϕ. ϕ is also called scaling function or generator function or father wavelet. The Haar wavelet is a wavelet for the Haar function, for example. R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

19 Outline Introduction and Example Multiresolution Analysis Discrete Multiresolution Wavelet Transform Analysis (DWT) Finite Calculation References If the translations of ψ are not orthonormal we need biorthogonal wavelets. But we do not go into details for this case. R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

20 Outline Introduction and Example Multiresolution Analysis Discrete Multiresolution Wavelet Transform Analysis (DWT) Finite Calculation References Multi Levels Let J be the level at which we want to approximate, i.e. we project into the space V J. Then we have V J =V J 1 W J 1 =V J 2 W J 2 W J 1. J 1 =V 0 W j. j=0 R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

21 Outline Introduction and Example Multiresolution Analysis Discrete Multiresolution Wavelet Transform Analysis (DWT) Finite Calculation References If we go to infinity we have L 2 (R) = V 0 W j. j=0 We can imagine that we start at the very coarse level 0 and improve the result by successively adding the finer becoming details. R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

22 Outline Introduction and Example Multiresolution Analysis Discrete Multiresolution Wavelet Transform Analysis (DWT) Finite Calculation References Filters Because of the fact that V j V j+1 and W j V j+1 in a MRA we have the refinement equations ϕ j k = l ψ j k = l h l ϕ j+1 2k+l g l ϕ j+1 2k+l. (h l ) l Z and (g l ) l Z are called filters. If ϕ has compact support h has finite length. If additionally ψ has compact support g has also finite length. R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

23 Outline Introduction and Example Multiresolution Analysis Discrete Multiresolution Wavelet Transform Analysis (DWT) Finite Calculation References Reconstruction If {ϕ j k k Z} and {ψj k k Z} are orthonormal bases, i.e. ϕ j k, ϕj l = ϕ j k (x)ϕj l (x)dx =δ k,l ψ j k, ψj l =δ k,l we have the reconstruction R ϕ j+1 k = l h k 2l ϕ j l + l g k 2l ψ j l. There is also a very simple correlation between g and h: g k = ( 1) 1 k h 1 k R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

24 Outline Introduction and Example Multiresolution Analysis Discrete Discrete Wavelet Wavelet Transform Transform (DWT) (DWT) Finite Calculation References Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

25 Outline Introduction and Example Multiresolution Analysis Discrete Discrete Wavelet Wavelet Transform Transform (DWT) (DWT) Finite Calculation References Orthogonal Projection Assume that {ϕ k k Z} and {ψ k k Z} are orthonormal. Given f L 2 (R). We consider the orthogonal projections P J onto V J and Q J onto W J. Let λ j k = ϕj k, f = ϕ j k (x)f(x)dx and Then it holds µ j k = ψj k, f. R f J = P J f = k Q J f = (P J+1 P J )f = k λ J k ϕj k V J µ J k ψj k W J. R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

26 Outline Introduction and Example Multiresolution Analysis Discrete Discrete Wavelet Wavelet Transform Transform (DWT) (DWT) Finite Calculation References We can easily get the coefficients in the coarser levels and the detail spaces by using successively the synthese equation. f J = k λ J k ϕj k = k λ J k ( l h k 2l ϕ J 1 l + l g k 2l ψ J 1 l ) = l λ J 1 l ϕ J 1 l + l µ J 1 l ψ J 1 l. = l J 1 λ 0 l ϕ0 l + j=0 l µ j l ψj l R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

27 Outline Introduction and Example Multiresolution Analysis Discrete Discrete Wavelet Wavelet Transform Transform (DWT) (DWT) Finite Calculation References The occurring coefficients are given by λ j l = k µ j l = k λ j+1 k h k 2l and (1) λ j+1 k g k 2l, j = 0,..., J 1. (2) (1) and (2) can be written as ( ) λ j = T λ j+1. µ j R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

28 Outline Introduction and Example Multiresolution Analysis Discrete Discrete Wavelet Wavelet Transform Transform (DWT) (DWT) Finite Calculation References Discrete Inverse Wavelet Tansform (DIWT) Given the coefficients λ j k and µj k by we get the cofficients λj+1 k back λ j+1 k = l h k 2l λ j l + l g k 2l µ j l. This describes again a linear transformation ( ) ( ) λ j+1 = T 1 λ j = T T λ j. µ j µ j R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

29 Outline Introduction and Example Multiresolution Analysis Discrete Discrete Wavelet Wavelet Transform Transform (DWT) (DWT) Finite Calculation References Note: Because of the D(I)WT it is not really necessary to know explicitly the functions ϕ and ψ. It is sufficient to know the filters h and g. R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

30 Outline Introduction and Example Multiresolution Analysis Discrete Discrete Wavelet Wavelet Transform Transform (DWT) (DWT) Finite Calculation References Schemes DWT DIWT λ J λ 0 µ 0 λ J 1 µ J 1 λ 1 µ 1 λ J 2 µ J 2 λ 2 µ λ 1 µ 1 λ J 1 µ J 1 λ 0 µ 0 λ J R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

31 Outline Introduction and Example Multiresolution Analysis Discrete Finite Wavelet Calculation Transform (DWT) Finite Calculation References Introduction and Example Multiresolution Analysis Discrete Wavelet Transform (DWT) Finite Calculation R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

32 Outline Introduction and Example Multiresolution Analysis Discrete Finite Wavelet Calculation Transform (DWT) Finite Calculation References In the previous chapter we still had infinite many coefficients λ j k, µj k. For practical calculations we have to make their size finite. There are mainly two ways to do this. 1. Zeropadding: You consider only a finite region and assume that all coefficients out of this region are zero. 2. Periodizing: You assume that your data is periodic and you calculate only on one period. Then the D(I)WT is a finite transform if the filters are finite. The number of arithmetic operations for one transformation is O(N) if N is the size of the input data. To transform on J levels you have O(J N) arithmetic operations. R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33

33 Outline Introduction and Example Multiresolution Analysis Discrete References Wavelet Transform (DWT) Finite Calculation References References S. Mallat, A Wavelet Tour of Signal Processing, 2nd. ed., Academic Press, 1999 R. Schneider F. Krüger Introduction to Multiresolution Analysis (MRA) November 22, / 33