Signal Analysis. Multi resolution Analysis (II)

Transcription

1 Multi dimensional Signal Analysis Lecture 2H Multi resolution Analysis (II) Discrete Wavelet Transform

2 Recap (CWT) Continuous wavelet transform A mother wavelet ψ(t) Define µ 1 µ t b ψ a,b (t) = p ψ a a Define thecontinuous wavelet transform (CWT) of f as W f (a, b) =h f ψ a,b i Lecture 2H, Klas Nordberg, LiU 2

3 Recap (CWT II) Continuing... f is a one variable function W f is a two variable function Scale and translation W f has an inverse transform (ICWT) iff Z Ψ(v) 2 0 < dv < Ψ is the FT of ψ v Lecture 2H, Klas Nordberg, LiU 3

4 Recap (CWT III) Example CWT a Signal b Mother wavelet Lecture 2H, Klas Nordberg, LiU 4

5 Recap (2chFB) The 2 channel filter bank Perfect reconstruction: s = s If and only if H 1 (u)h 0 (u)+g 1 (u)g 0 (u) =2 (FB1) H 1 (u)h 0 (u + π) + G 1 (u)g 0 (u + π) = 0 (FB2) Lecture 2H, Klas Nordberg, LiU 5

6 Recap (CMF) The conjugate mirror filter bank (CMF) We choose H (u) + H 1 (u + π) = 2 (O) H 0 (u) =H 1 (u) G 1 (u) =e iu H 1 (u + π) G 0 0( (u) = e iu H 1 1( (u + π) ) = G 1 1( (u) Leads to FB1 and FB2 being satisfied! This is the alternating flip g 1 [k] =( 1) 1 k h 1 [1 k] Lecture 2H, Klas Nordberg, LiU 6

7 Recap (SF I) Scaling function φ(t): must satisfy 1. φ(t k), k Z, is an orthogonal set of f i f f i V functions of some function space V 0 2. φ(t) can be written as a linear combination of the functions φ(2t k), k Z Lecture 2H, Klas Nordberg, LiU 7

8 Recap (SF II) 1. and 2. alone lead to Also 2 1/2 φ(2 t k) is an ON basis for V 1 V 0 Define sequence h as h[k] =hφ(t) 2 1/2 φ(2 t k)i Define H(u) = DFT{h} Then H(u) 2 + H(u+π) 2 = 2 Same as condition (O) in the CMF H here corresponds to H 1 in the CMF Lecture 2H, Klas Nordberg, LiU 8

9 Recap (SF III) Cont... Define sequence g as g[k] = ( 1) 1 k h[1 k] The alternating flip in CMF G here corresponds to G 1 in the CMF Define G(u) = DFT{g} Define a function ψ(t) with FT Ψ Given by Ψ(u) = 1 G 2 ³ u 2 Φ ³ u Lecture 2H, Klas Nordberg, LiU 9

10 Recap (SF IV) Cont... ψ(t k), k Z, is also an ON set Spans a subspace W 0 V 1 and W 0 V 0 In fact, V 1 = V 0 W 0 2 1/2 ψ(2 t k) is an ON basis of V 1 φ(t k) is an ON basis of V 0 ψ(t k) is an ON basis of W Lecture 2H, Klas Nordberg, LiU 10

11 Recap (SF V) Cont... An f V 1 can be written as f = f 0 + w 0 f 0 V 0 w 0 W 0 f V 1 has coordinates s[k] relative lti the ON basis in V 1 f V 1 has coordinates a[k] and d[k] relative the ON bases in V 0 and W 0, respectively Sequences s[k] or a[k] and d[k] are alternative representations of f V Lecture 2H, Klas Nordberg, LiU 11

12 Recap (SF VI) These representations can be mapped: s a & d a & d s Coordinates of f relative to basis in V 0 Coordinates of f relative to basis in V 1 Coordinates of f relative to basis in W Lecture 2H, Klas Nordberg, LiU 12

13 Recap (SF VII) The fact that V 1 is spanned by the ON basis 2 1/2 φ(2 t k) when V 0 V 1 is spanned by φ(t k) means: The space V 1 contains functions that can have smaller details than V 0 does Lecture 2H, Klas Nordberg, LiU 13

14 Recap (SF VII) The fact that V 1 = V 0 W 0 means: W 0 contains the details that are missing in V 0 to make up V 1 Also: V 0 contains an approximation of V 1 without the details that are in W Lecture 2H, Klas Nordberg, LiU 14

15 The story continues... There is an obvious relation between a CMFbank and the results derived from 1. and 2. of the scaling function φ: φ and ψ define sequences h and g which we can identify with the reconstructing filters h 1 and g 1 in the 2 channel CMF bank Any φ that satisfies 1. and 2. generates a 2 channel lcmf bank And vice versa (!) Lecture 2H, Klas Nordberg, LiU 15

16 CMF In the CMF bank, the signal space V is the space where the discrete input signal a lives The filter bank decomposes s into two components u V and v in W components u 2 V 0 and v 2 in W 0 With V = V 0 W 0 a[n] are the coordinates of u 2 [k] relative h 1 [k 2 n] d[n] are the coordinates of v 2 [k] relative g 1 [k 22 n] Lecture 2H, Klas Nordberg, LiU 16

17 Scaling function In the case of the scaling function: The signal space is V 1 which hosts functions of a continuous variable: f(t) The discrete signal a[k] are the coordinates of f 1 V 1 relative to the ON basis 2 1/2 φ(2 t k) The discrete signals a[k] and d[k] are the coordinates of f V 1 relative to the ON bases φ(t k) and ψ(t k) Together they span V 1 = V 0 W Lecture 2H, Klas Nordberg, LiU 17

18 Looking ahead In what follows, we will take the second view: Any discrete time function holds the coordinates of some continuous time function relative to some ON basis of some space s[k] are the coordinates of f 1 V 1 relative 2 1/2 φ(2 t k) a[k] are the coordinates of f 0 V 0 relative φ(t ( k) d[k] are the coordinates of w 0 W 0 relative ψ(t k) f 1 = f 0 + w 0 V 0 W Lecture 2H, Klas Nordberg, LiU 18

19 Using 1. and 2. again Consider the set of functions given by 2 φ(4 t k), k Z It relates to the set 2 1/2 φ(2 t k) in the same way as that set relates to φ(t k) (why?) All the results derived dfrom 1. and 2. between V 1 and V 0 applies!! Lecture 2H, Klas Nordberg, LiU 19

20 The space V 2 Consequently: The set of functions given as 2 φ(4 t k) forms an ON basis of a space V 2 V 1 V 0 The space V 2 contains functions of even finer details than V 1 does And finer still than V 0 does Lecture 2H, Klas Nordberg, LiU 20

21 The space W 1 Furthermore: We can define a difference space W 1 such that V 2 = V 1 W 1 and W 1 V 1 2 φ(4 t k) is an ON basis for V 2 2 1/2 φ(2 t k) is an ON basis for V 1 2 1/2 ψ(2 t k) is an ON basis for W 1 (why?) Lecture 2H, Klas Nordberg, LiU 21

22 Decomposition of V 2 Furthermore, V 2 = V 1 W 1 means that any f 2 V 2 can be decomposed into f 2 = f 1 + w 1 where f 1 V 1 and w 1 W Lecture 2H, Klas Nordberg, LiU 22

23 Coordinates of f 2 V 2 The coordinates of f 2 are: s[k] relative the ON basis 2φ(4 t k) in V 2 Alternatively: a[k] and d[k] relative the ON bases 2 1/2 φ(2 t k) in V 1/2 1 and 2 ψ(2 t k) in W Lecture 2H, Klas Nordberg, LiU 23

24 Coordinates of f V 2 a and d are obtained from a as a[k] =(s[ ] h[ ])[2 k] Convolve and skip every second sample (the odd ones) d[k] =(s[ ] g[ ])[2 k] s is obtained from a and d as Insert zeros between every sample and convolve s[k] = X (a[n] h[k 2 n]+d[n] g[k 2 n]) n= Lecture 2H, Klas Nordberg, LiU 24

25 Putting things together (I) We have already seen that: f 1 V 1 can be decomposed as f 1 = f 0 + w 0 where f 0 V 0 and w 0 W Lecture 2H, Klas Nordberg, LiU 25

26 Putting things together (II) This means: any f 2 V 2 can be decomposed into f 2 = f 0 + w 0 + w 1 where f 0 V 0 and w 0 W 0 and w 1 W Lecture 2H, Klas Nordberg, LiU 26

27 Putting things together (III) This also means: V 2 = V 0 W 0 W 1 V 0 W 0 W 1 (they are mutually orthogonal) (why?) Lecture 2H, Klas Nordberg, LiU 27

28 Putting things together (IV) We can see f 1 V 1 as an approximation of f 2 V 2, and f 0 V 0 as a coarser approximation of f 2 The details that are missing in f 0 to get to f 2 are found in W 0 and W 1 V 0 W 0 V 1 W 1 V Lecture 2H, Klas Nordberg, LiU 28

29 Analysis in two steps Coordinates of f 1 V 1, relative 2 1/2 φ(2 ( t k) Coordinates of f 0 V 0, relative φ(t k) Coordinates of f 2 V 2, relative lti 2φ(4 t k) Coordinates of w 1 W 1, relative 2 1/2 ψ(2 t k) Coordinates of w 0 W 0, relative ψ(t k) Lecture 2H, Klas Nordberg, LiU 29

30 Reconstruction on two steps Coordinates of f 0 V 0, relative φ(t k) ) Coordinates of f 1 V 1, relative 2 1/2 φ(2 t k) Coordinates of f 2 V 2, relative 2φ(4 t k) Coordinates of w 0 W 0, Coordinates of w 1 W 1, relative ψ(t k) relative 2 1/2 ψ(2 t k) Lecture 2H, Klas Nordberg, LiU 30

31 The full view Coordinates of Coordinates of f 0 V 0 Coordinates of f 1 V 1 f 1 V 1 Coordinates of f 2 V 2 Coordinates of f 2 V 2 Coordinates of w 1 W 1 Coordinates of w 0 W Lecture 2H, Klas Nordberg, LiU 31

32 Generalisation Consider the set of functions given by 2 p/2 φ(2 p t k), k Z, p Z, p 1 It relates to the set 2 (p 1)/2 φ(2 (p 1) t k) in the same way as that setrelates to 2 (p 2)/2 φ(2 (p 2) t k) And so on all the way to the set φ(t k) Lecture 2H, Klas Nordberg, LiU 32

33 The space V p 2 p/2 / φ(2 p t k) is an ON basis for a space V p V p can be decomposed as V p = V 0 W 0 W 1... W p Lecture 2H, Klas Nordberg, LiU 33

34 The space V p All spaces V 0, W 0, W 1,..., W p 1 are mutually orthogonal V 0 is spanned by φ(t k), k Z W j is spanned by 2 j/2 ψ(2 j t k), k Z,, j Z, 0 j p Lecture 2H, Klas Nordberg, LiU 34

35 The space V p Any f p V p can be decomposed as f p = f 0 + w 0 + w w p 1 A very, very coarse approximation of f p Finer and finer details added to f 0 in order to construct f p f 0 V 0 and w j W j Lecture 2H, Klas Nordberg, LiU 35

36 Decomposition of spaces Lecture 2H, Klas Nordberg, LiU 36

37 Decomposition of spaces V p contains functions with details of some minimal size or scale Then V p 1 contains function that have twice the size or scale at minimum compared to V p And V p 2 contains functions that have 4 times the size or scale at minimum compared to V p And so on Lecture 2H, Klas Nordberg, LiU 37

38 The space V 0 We can choose p (a positive integer) as large as we want Or is practically useful or necessary This specifies a certain scale of V 0 relative V p A factor 2 p larger V 0 contains the coarsest approximation of f p that is reasonable for a particular application Lecture 2H, Klas Nordberg, LiU 38

39 Multi level level decomposition In terms of coordinates! Here: p = 5: V p 5 = V 0 approximation of f p in V p 1 approximation of f p in V p 2 approximation of f p in V p 3 approximation of f p in V p 4 approximation of f p in V p 5 dtil details of f p details of f p in W f fp V f p 44 p details of fp in W p 5 details of f p details of fp in W p 2 2 in W p 3 in W p Lecture 2H, Klas Nordberg, LiU 39

40 Multi level level reconstruction approximation in V 0 approximation of f 4 in V 1 approximation of f 4 in V 2 details in W 0 approximation of f 4 in V 3 Details in W 1 approximation of f 4 in V 4 Details in W 2 Details in W 3 Details in W 4 reconstruction of f 5 in V Lecture 2H, Klas Nordberg, LiU 40

41 Coordinates in W j 1 With f j = f j 1 + w j 1 V j : coordinates of w j 1 relative to the ON basis in W j 1 are given as d j 1 [k] = h f j 2 (j 1)/2 ψ(2 (j 1) t k) i Lecture 2H, Klas Nordberg, LiU 41

42 Coordinates in W j 2 With f j 1 = f j 2 + w j 2 V j 1 : coordinates of w j 2 relative to the ON basis in W j 22 are given as d j 2 [k] = h f j 1 2 (j 2)/2 ψ(2 (j 2) t k) i Lecture 2H, Klas Nordberg, LiU 42

43 Coordinates in W j 2 But f j 1 = f j w j 1 where w j 1 W j 1 Furthermore: W j 1 W j 2 w j 1 W j 2 2 (j 2)/2 ψ(2 (j 2) t k) is an ON basis of W j 2 w j 1 to all 2 (j 2)/2 ψ(2 (j 2) t k) d j 2 [k] = h f j 2 (j 2)/2 ψ(2 (j 2) t k) i Lecture 2H, Klas Nordberg, LiU 43

44 Coordinates in W j In general, the coordinates of w j Wj is given by h f j+1 2 j/2 ψ(2 j t k) i = h f p 2 j/2 ψ(2 j t k) i (why?) Lecture 2H, Klas Nordberg, LiU 44

45 Wavelets ψ(x) is a wavelet Satisfies the wavelet condition Not thoroughly proven here but follows from the special properties of φ The details in terms of coordinates relative to the ON basis in W j are computed as d j [k] = D f p (t) 2 j/2 ψ(2 j t k) E Lecture 2H, Klas Nordberg, LiU 45

46 Continuous Wavelet Transform (CWT) In the continuous wavelet transform we compute W f (a, b) = h f fp p (t) ψ a,b (t) i ψ a,b (t) = 1 µ t b p ψ a a Scaling and translation of the mother wavelet Mother wavelet Lecture 2H, Klas Nordberg, LiU 46

47 Discrete Wavelet Transform (DWT) Consequently, we can see the details of f p in the different spaces W j as a sampling of the continuous wavelet transform given by ψ: D E d j [k] = f p (t) 2 j/2 ψ(2 j t k) = = µ t 2 f p (t) 2 j/2 j k ψ 2 jj À W f j j fp (2, 2 k), j, k Z, j 0 = Lecture 2H, Klas Nordberg, LiU 47

48 DWT = Sampling of CWT Scale parameter a 1 Space W 0 1/2 Space W 1 1/4 Space W 2 Space W 1/ Translation parameter b Sampling pattern of the CWT to get the DWT Lecture 2H, Klas Nordberg, LiU 48

49 DWT In practice the DWT computes these samples of the CWT and the approximation at level V 0 Together they can completely reconstruct the function f p in terms of its coordinates relative to the ON basis in V p The reconstruction in V p is made by a set of ON basis functions DWT is a transform based on an ON basis CWT is based on a frame Lecture 2H, Klas Nordberg, LiU 49

50 Multi resolution analysis This approach of decomposing a function f p V p into coarser and coarser approximations in together with the corresponding details is referred to as multi resolution analysis (MRA) Formulated by Stéphane Mallat, 1989 Ingrid Daubechies described first ψ for MRA 1987 Filter banks have been around since the Lecture 2H, Klas Nordberg, LiU 50

51 Practical implementation of DWT N samples = coordinates of f p in V p a d N/2 samples = coordinates of f p-1 in V p-1 a d N/2 samples = coordinates of w p-1 in W p-1 N/4 samples = coordinates of f p-2 in V p-2 N/4 samples = coordinates of w p-2 in W p Lecture 2H, Klas Nordberg, LiU 51

52 An observation Since V j 1 W j 1 and all bases are ON: f j 2 = f j w j 1 2 Means: f j 1 f j Means: the more levels l p we have, the smaller is f 0 for the same f p Lecture 2H, Klas Nordberg, LiU 52

53 2D DWT For 2D images Apply the 1D DWT first on each column, and then on each row Or vice versa, gives the same result (why?) Lecture 2H, Klas Nordberg, LiU 53

54 2D DWT Horizontal Vertical A-AA D-A A-D D-DD Repeat the analysis on Repeat the analysis on the approximation part the approximation part Lecture 2H, Klas Nordberg, LiU 54

55 2D DWT, Example 2D DWT 2D DWT Lecture 2H, Klas Nordberg, LiU 55

56 What you should know include The 2 channel filter bank Conditions FB1 and FB2 for perfect reconstruction The CMF bank: condition on H 1 and the other three filters defined from H 1 Alternating flip Orthogonality of the CMF bank Definition of φ, as a function of continuous time t Definition of discrete sequence h[k] Construction of sequence g from h (alternating flip) Construction of ψ from φ (and g) Relation between MRA and CMF Construction of the sequence of spaces V p V p 1... V 1 V 0 Construction of the difference spaces W p 1,..., W 0, they are mutually orthogonal Decomposition of f p V p as f p = f 0 + w w p 1 1, f 0 V 0 and w j W j DWT: coordinates in W j = samples of CWT The corresponding sampling pattern Implementation of 2D DWT Lecture 2H, Klas Nordberg, LiU 56