Chapter 7 Wavelets and Multiresolution Processing. Subband coding Quadrature mirror filtering Pyramid image processing

Transcription

1 Chapter 7 Wavelets and Multiresolution Processing Wavelet transform vs Fourier transform Basis functions are small waves called wavelet with different frequency and limited duration Multiresolution theory: representation and analysis of signals/images at more than one resolutions. Subband coding Quadrature mirror filtering Pyramid image processing 2002 R. C. Gonzalez & R. E. Woods

2 7. Background Small objects or low contrast images are examined at high resolutions; whereas large object and high contrast images are examined in low resolution. Images are 2-D array of intensity with locally varying statistics

3 7. Background

4 7. Background Image Pyramids Base level J with size 2 J 2 J or N N where J=log 2 N Intermediate level j with size 2 j 2 j with 0 j J Most pyramids are truncated to P+ levels where j=j-p, J-,J The total number of pixel elements in a P+ level pyramid is N N P

5 7. Background Gaussian pyramid Laplacian Pyramid

6 7. Background

7 7. Background-Subband Coding In subband coding: an image is decomposed into a set of bandlimited components, called subbands. The subbands can be downsampled without loss of information. Reconstrunction of the original image is accomplished by upsampling, filtering, and summing the individual subbands

8 7. Background-Subband Coding

9 7. Background-Subband Coding The input is -D band-limited discrete time signals xn, n=0,,2, The output xˆ n is formed through the decomposition of xn into y 0 n and y n via analysis filter h 0 n and h n, and subsequent recombination via synthesis filters g 0 n and g n. n The Z-transform of xn is x down n=x2n X down z=/2[xz /2 +X-z /2 ] x up n=xn/2 for n=0,2,4,.. X up z=xz 2 X z = x n z

10 2002 R. C. Gonzalez & R. E. Woods 7. Background-Subband Coding is obtained from the downsampled and upsampled xn, therefore X-z is the modulated version of xn Z - [X-z]=- n xn The system output is ˆ n x [ ] 2 ˆ z X z X z X + = [ ] [ ] 2 z X z H z X z H z G ˆ z X z H z X z H z G z X + =

11 2002 R. C. Gonzalez & R. E. Woods 7. Background-Subband Coding Rearrange The second component contain the z dependence, represents the aliasing that is introduced by downsampling-upsampling process. For error-free reconstruction of the input, xn= The second component is zero. H 0 -zg 0 z+h -zg z=0 H 0 zg 0 z+h zg z=2 [ ] [ ] 2 2 ˆ z X z G z H z G z H z X z G z H z G z H z X = ˆ n x

12 2002 R. C. Gonzalez & R. E. Woods 7. Background-Subband Coding Reduce to matrix expression Where Assume that H m z is nonsingular then 7-2 The analysis and synthesis filtered are crossmodulated. [ ] [ ] = z z G z G m H = 0 0 z H z H z H z H z H m = det z H z H z z G z G H m

13 7. Background-Subband Coding For FIR filters, the determinate of the modulation matrix is a pure delay, i.e., det H m z =αz -2k+ Ignoring the delay and let α=2, by taking inverse Z transform we have g 0 n=- n h n and g n=- n+ h 0 n If α=-2 then g 0 n=- n+ h n and g n=- n h 0 n :

14 7. Background-Subband Coding The biorthogonality of the analysis and synthesis filters, let Pz be the product of he lowpass analysis and synthesis filter from 7.2 Pz=G 0 zh 0 z=2h 0 zh -z/deth m z also deth m z=-deth m -z and G zh z=-2h 0 -zh z/deth m z=p-z Thus G zh z =P-z=G 0 -zh 0 -z and G 0 -zh 0 -z+g 0 -zh 0 -z=2

15 7. Background-Subband Coding Inverse z-transform n g 0 k h0 n k + g0 k h0 n k = 2δ n k Odd index terms cancel, it is simplified as g0 k h0 2n k = g0 k, h0 2n k = δ n Express G 0 and H 0 as function of G and H g g k, k, h h 0 2n 2n k = δ n, k = 0 g 0 k, h 2n k = 0

16 7. Background-Subband Coding More general expression: g j k, h 2n k = δ i j δ n i, j = i { 0, } Filter banks satisfying this condition are biorthogonal Quadrature mirror filterqmf Table 7. Conjugate quadrature filter CQF Table 7. Orthonormal filter Table 7. for fast wavelet transform, it requires that 7.22 g i n, g n + 2m = δ i j δ m i, j = i { 0, } 7.2

17 7. Background-Subband Coding

18 7. Background-Subband Coding

19 7. Background-Subband Coding They satisfy perfect reconstruction and 7.2 and 7.22

20 7. Background-Subband Coding

21 7. Background-Harr Transform The oldest and simplest known orthonormal wavelets The Harr transform is both separable and symmetric as T=HFH where F is N N image matrix and H is N N transform matrix and T is the resulting N N transform. For Harr transform, the transformation matrix H contains the Harr basis function h k z defined over the continuous closed interval [0,] for k=0,,.n- where N=2 n.

22 7. Background-Harr Transform To generate H, we define k=2 p +q- where 0 p n-, q=0 or for p=0, and q 2 p for p 0 The Harr basis functions are and h h 0 z=h 00 z=/n /2, z [0,] k z = h pq z = N p / 2 p / 2 q / 2 q p z 0.5 / 2 p < q z 0.5 / 2 < q / 2 otherwise, z [0,] p p

23 7. Background-Harr Transform The ith row of an N N Harr transformation matrix contains the elements of h i z, z=0/n, /N,.N-/N If N=4, k, q, and p are assumed as the following values: k p q

24 2002 R. C. Gonzalez & R. E. Woods 7. Background-Harr Transform The 4 4 transformation matrix is The 2 2 transformation matrix is The basis functions satisfy the QMF prototype filter in Table 7., the h 0 n and h n are the elements of the first and the second rows of H 2. = H 4 = 2 H 2

25 7. Background- Harr Transform

26 7.2 Multiresolution Expansions Multi-resolution analysis MRA Ascaling function is used to create a series of approximations of a function or image, each different by a factor of 2 from its neighboring approximation Additional functions, called wavelet, are used to encode the difference between two adjacent approximation.

27 7.2 Multiresolution Expansions Series expansion A signal fx can be represented as a linear combination of expansion functions as f x = α kϕk x k The expressible functions form a function space that is the closed span of the expansion set denoted as V = Span { ϕ x} fx V means that fx is in the closed span of {ϕ k x} k k

28 7.2 Multiresolution Expansions For any function space V and the corresponding expansion set {ϕ k x} there is a set of dual functions denoted as { ~ ϕ k x} that can be used to compute the α k as α = ~ ϕ = k x f x ~ ϕ * k x f x dx k

29 7.2 Multiresolution Expansions Case : The expansion functions form orthonormal basis for V that ϕ j x ϕk x = δ jk The basis and its dual are equivalent ϕ k x= ~ ϕ k x and then α k = ϕ k x f x Case 2: If the expansion functions are not orthonormal, but orthogonal basis for V then ϕ x ϕ x j k and the basis functions and their duals are called biorthogonal. The biorthogonal basis and their duals are ϕ x ~ ϕ x = δ j = 0 k if j k jk

30 7.2 Multiresolution Expansions Case 3: If the expansion set is not a basis for V, but support the expansion, then it is a spanning set in which there is more than one set of α k for any fx V The expansions and their duals are said to be overcomplete or redundant. They form a frame in which A f x ϕ x f x k A and B frame the normalized inner products of the expansion coefficients and the function. k B f x

31 7.2 Multiresolution Expansions If A=B the expansion is called a tight frame, and it can shown that f x = ϕk x f x ϕk x A k

32 7.2 Multiresolution Expansions Scaling Functions Consider a set of expansion functions composed of integer translations and binary scaling of the real square-integrable function ϕx, that is the set {ϕ j,k x} where ϕ j,k x= 2 j/2 ϕ2 j x-k For allj, k I and ϕx L 2 R j determines the position of ϕ j,k x, and k determines the width of ϕ j,k x ϕ j,k x is called the scaling function

33 7.2 Multiresolution Expansions If we restrict j to a specific value j=j 0 then {ϕ j0,k x} is a subset of {ϕ j,k x} and f x αϕ x V = Span ϕ x = { } k j,k j j,k k k Figure 7.9 fx= 0.5ϕ,0 x +ϕ, x 0.25ϕ,4 x Expansion function can be decomposed as ϕ 0,k x= /2 0.5 ϕ,2k x+ /2 0.5 ϕ,2k x

34 7.2 Multiresolution Expansions

35 7.2 Multiresolution Expansions Four requirements for MRA MRA requirement : The scaling function is orthogonal to its integer translates MRA requirement 2: The subspace spanned by the scaling function at low scales are nested within those spanned at higher scales. Subspaces containing high resolution function must also contain all lower resolution functions V -.. V - V 0 V V 2. V If fx V j then f2x V j+

36 7.2 Multiresolution Expansions

37 7.2 Multiresolution Expansions MRA requirement 3: The only function that is common to all V j is fx=0 MRA requirement 4: Any function can be represented with arbitrary precision. All measureable square-integrable functions can be represented in the limit as j as V ={L 2 r} The expansion function of subspace V j can be expressed as a weighted sum of expansion functions in V j+ space as ϕ j,k x= Σ n α n ϕ j+,n x

38 7.2 Multiresolution Expansions Change α n to h ϕ n ϕ j,k x= Σ n h ϕ n2 j+/2 ϕ2 j+/2 x-n Set j=k=0 then ϕ 0,0 x=ϕ x ϕx= Σ n h ϕ n2 /2 ϕ2x-n It is called the refinement equation, MRA equation, or the dilation equation. h ϕ n coefficients are called scaling function coefficients. h ϕ is referred as scaling vector The scaling function for Harr function are h ϕ 0=h ϕ =/2 /2 then ϕx= /2 /2 [2 /2 ϕ2x+ 2 /2 ϕ2x-] = ϕ2x+ϕ2x-

39 7.2 Multiresolution Expansions Wavelet functions We define the set {ψ j,k x} as ψ j,k x=2 j/2 ψ2 j x-k The W j space is W Span { ψ } j, k x If fx W j then fx=σ k α k ψ j,k x j = k

40 7.2 Multiresolution Expansions

41 7.2 Multiresolution Expansions The scaling and wavelet subspaces in Figure 7. are related as V j+ =V j W j where denotes the unions of spaces. The orthogonal complement of V j in V j+ is W j and all members of V j are orthogonal to the member of W j, Thus <ϕ j,k x ψ j,k x>=0 for all appropriate j, k, l Z We can express the space of all measurable squareintegrable functions as L 2 R=V 0 W 0 W or L 2 R=V W W 2.. Or L 2 R= W - W 0 W..

42 7.2 Multiresolution Expansions If fx is an element of V, but not V 0, an expansion of fx using V 0 scaling function; wavelet from W 0 would encode the difference between the approximation and the actual function. It can be generalized as L 2 R=V j0 W j0 W j0+. where j0 is an arbitrary starting scale Any wavelet function can be expressed as a weighted sum of shifted double-resolution scaling functions that is ψx= Σ n h ψ n2 /2 ϕ2x-n Where h ψ n is wavelet function coefficients and h ψ is the wavelet vector.

43 7.2 Multiresolution Expansions h ψ n is related to h ϕ n as h ψ n=- n h ϕ -n The Harr scaling factor illustrated as h ϕ 0=h ϕ =/2 /2 h ψ 0=- 0 h ϕ -0=/2 /2 h ψ =- h ϕ -=-/2 /2 We get ψx=ϕ2x- ϕ2x- The Harr wavelet function is 0 x < 5 ψ x = 0.5 x < 0 elsewhere

44 7.2 Multiresolution Expansions

45 7.2 Multiresolution Expansions A function in V that is not subspace in V 0 can be expanded using V 0 and W 0 as fx=f a x+f d x where f a x = ϕ0,0 x ϕ0, 2 x 4 8 f d 2 2 x = ψ 0,0 x ψ 0, 2 x 4 8 f a x is an approximation of fx using V 0 scaling function, whereas f d x is the difference fx-f a x as a sum of W 0 wavelets

46 7.3 Wavelets transform in one dimensionthe wavelet series expansion Expand fx related to wavelet ψx and scaling function ϕx as k j= j0 k Where j0 is an arbitrary starting scale and c j0 k s and dk are relabeled α. The c j0 k s are normally called the approximation or scaling coefficients and the d j k s are referred as the detail or wavelet coefficients f x = ck 0 ϕ0,kx + d jk ψ j,kx ck fx ϕ x fx ϕ xdx = = 0 0,k 0,k d k fx ψ x fx ψ xdx = = j j,k j,k

47 7.3 Wavelets transform in one dimension

48 7.3 Wavelets transform in one dimension- the wavelet series expansion Wavelet series expansion y = ϕ x + ψ x + 4 ψ x ψ x V 0 W 0 W

49 7.3 Wavelets transform in one dimension- the discrete wavelet transform For discrete case, the series expansion becomes for j j0 f x = M W ϕ j0, k = f x ϕ j0, k x M x W ψ j, k = f x ψ j, k x M x Wϕ j0, k ϕ j, k x + M x 0 Wψ j, k ψ j0, k x j= j0 x Here fx, ϕ j0,k x, ψ j,k x are functions of the discrete variable x=0,,2, M- For example fx=fx 0 +xδx for some x 0, Δx, and x=0,,2, M-. Normally, we select M=2 J, j=0,,2, J-, k=0,,. 2 j -

50 7.3 Wavelets transform in one dimension- the discrete wavelet transform Example 7.8 f0=, f=4, f2=-3, f3=0, M=4, J=2, and with j0=0, the summation is performed over x=0,,2,3, j=0,, and k=0 for j=0 or k=0, for j=. Using Harr scaling and wavelet functions the rows of H 4 and assume that four samples of fx are distributed over the support of basis functions. Wϕ 0,0 = f x ϕ 0, 0 x = [ ] = 2 x 2 Wψ 0,0 = f x ψ 0, 0 x = [ ] = x

51 7.3 Wavelets transform in one dimension- the discrete wavelet transform W = ψ,0 f x ψ, 0 x = x W = ψ, f x ψ, x = f x = W 2 + W ϕ ψ x 0,0 ϕ,0 ψ 0,0,0 x + W x + W ψ ψ 0, ψ, ψ, 0, x x

52 7.3 Wavelets transform in one dimensionthe continuous wavelet transform CWT: transform a continuous function into a highly redundant function of two continuous variables - translation and scale where W = dx ψ s, τ f x ψ s,τ x x τ ψ s, τ x = ψ s s s and τ are called the scale and translation parameters.

53 7.3 Wavelets transform in one dimension- the continuous wavelet transform ICWT where f ψ s, τ x Wψ s, τ 0 2 x = dτds C s C ψ = ψ Ψ u u 2 du

54 7.3 Wavelets transform in one dimension- the continuous wavelet transform The Mexican hat wavelet ψx=2/3 /2 π -/4 -x 2 e -x2 /2 Figure 7.4a fx= ψ,0 x+ψ 6,80 x Figure 7.4c shows a portion s 0 and τ 00 of the CWT of Figure 7.4a Continuous translation τ Continuous scaling s The set of transformation coefficients {W ψ s, τ} and basis functions {ψ s, τ x}are infinite.

55 7.3 Wavelets transform in one dimension

56 7.4 Fast Wavelet Transform FWT FWT exploits a surprising but fortunate relationship between the coefficients of the DWT at adjacent scales known as Herringbone algorithm Consider the multi-resolution refinement equation: ϕx= Σ n h ϕ n2 /2 ϕ2x-n Scaling x by 2, translating by k, and letting m=2k+n then ϕ2 j x-k= Σ n h ϕ n2 /2 ϕ22 j x-k-n =Σ n h ϕ m-2k2 /2 ϕ2 j+ x-m h ϕ can be thought of as the weights used to expand ϕ2 j x-k as sum of the scale j+ scaling function.

57 2002 R. C. Gonzalez & R. E. Woods 7.4 Fast Wavelet Transform FWT Similarly, for ψ2 j x-k we have ψ2 j x-k = Σ n h ψ m-2k2 /2 ψ 2 j+ x-m For DWT, Replacing ψ2 j x-k, we have or = x j j k x x f M k j W 2 2, 2 / ψ ψ = + x m j j m x k m h x f M k j W , 2 / ϕ ψ ψ = + + m x j j m x x f M k m h k j W , 2 / ϕ ψ ψ

58 7.4 Fast Wavelet Transform FWT where the bracketed quantity is identical to the DWT transform pair with j0=j+, so W = hψ m 2k W j +, m ψ j, k ϕ m The DWT detail coefficients at scale j are a function of the DWT approximation coefficients at scale j+. Similarly, we have W = hϕ m 2k W j +, m ϕ j, k ψ m

59 2002 R. C. Gonzalez & R. E. Woods 7.4 Fast Wavelet Transform 7.4 Fast Wavelet Transform It is identical to the analysis portion of the two-band subband system of Figure 7.4 with h 0 n= h ϕ -n and h n= h ψ -n 0, 2,, = + = k k n n j W n h k j W ψ ϕ ϕ 0, 2,, = + = k k n n j W n h k j W ψ ψ ψ

60 7.4 Fast Wavelet Transform Figure 7.6 shows a two-stage filterbank for generating the coefficients at two highest scales of the transform. The highest level is W ϕ J, n=fn, where J is the highest scale. W ϕ J-, n is the low-pass approximation component, and W ψ J-, n is the high-pass detail component The second filter bank split the spectrum and the subspace V J-, the lower half-band, into quarter-band subspaces W J-2, and V J-2, with corresponding DWT coefficients: W ϕ J-2, n and W ψ J-2, n.

61 7.4 Fast Wavelet Transform

62 7.4 Fast Wavelet Transform Consider discrete function fn={, 4, -3, 0} The corresponding scaling and wavelet vectors {, 4, -3, 0} * {- ½ 0.5, ½ 0.5 }={-½ 0.5, -3½ 0.5, 7½ 0.5, -3½ 0.5, 0 } W ψ,k={-3½ 0.5, -3½ 0.5 } Or W ψ,k=h ψ -nw ϕ 2,n n=2k, k 0 =h ψ nfn n=2k, k 0 =Σ l h ψ 2k-lxl k=0, = ½ 0.5 x2k- ½ 0.5 x2k R. C. Gonzalez & R. E. Woods 2 n = 0, hϕ n = hψ n = 2 n = 0 otherwise 0 2 n = 0 otherwise

63 7.4 Fast Wavelet Transform

64 7.4 Fast Wavelet Transform- Inverse FWT W ϕ j+,k=h ϕ k W ϕ up j, k k 0 +h ψ k W ψ up j, k k 0

65 7.4 Fast Wavelet Transform- Inverse FWT

66 7.4 Fast Wavelet Transform- Inverse FWT

67 7.4 Fast Wavelet Transform- Inverse FWT

68 7.5 Wavelet transform in Two Dimension 2-D scaling function: ϕx, y=ϕxϕy 2-D wavelets: ψ H x, y=ψx ϕy ψ V x, y=ϕx ψy ψ D x, y=ψx ψy Translated and scaled basis functions: ϕ j,m,n x, y=2 j/2 ϕ2 j x-m, 2 j y-n ψ i j,m,n x, y=2j/2 ψ 2 j x-m, 2 j y-n, i={h, V, D}

69 7.5 Wavelet transform in Two Dimension The DWT of function fx,y of size M N is W W ϕ ψ 2002 R. C. Gonzalez & R. E. Woods j M N = 0, m, n f x, y ϕ j0, m, n x, y MN x= 0 y= 0 M N = i 0, m, n f x, y ψ x, y 0, MN j m n x= 0 y= 0 where i={h, V, D} Normally we let j0=0 and M=N=2 J, j=0,,2, J- Inverse DWT j, f x, y = MN + MN m n W ϕ i= H, V, D m n j0, m, n ϕ W ϕ j0, m, n i j0, m, n ψ j 0, m, n

70 7.5 Wavelet transform in Two Dimension Figure 7.22 the 2-D fast wavelet transform a the analysis filter bank; b the results of decomposition; c the synthesis filter bank

71 7.5 Wavelet transform in Two Dimension

72 7.5 Wavelet transform in Two Dimension

73 7.5 Wavelet Transform in Two Dimension Fig Con t

74 7.5 Wavelet Transform in Two Dimension DWT for edge detection a Emphasize the reconstructed image edge b Isolate the vertical edge

75 7.5 Wavelet Transform in Two Dimension The general wavelet-based for de-noising the image. Step : choose a wavelet and number of levels or scales, p, for decomposition, then compute FWT. Step 2: Thresholding the detail coefficients. Select and apply a threshold to the detail coefficients from scales J- to J-P. a Hard thresholding; b Soft thresholding, first hard-thresholding and then followed by scaling the nonzero coefficients toward zero to eliminate the discontinuity at the threshold. Step 3: Perform a wavelet reconstruction based on the original approximation coefficients at level J-P and the modified detail coefficients for level from J- to J-P R. C. Gonzalez & R. E. Woods

76 7.5 Wavelet Transform in Two Dimension

77 7.6 Wavelet packet DWT: the low frequencies are group into narrow bands, while the high frequencies are grouped into wider bands - constant-q filters. Wavelet packet a Generalized DWT, in which the details are teratively filtered and separated. Increased computation complexity raised from OM to OMlogM

78 7.6 Wavelet packet

79 7.6 Wavelet packet

80 7.6 Wavelet packet Double subscripts are introduced for Wavelet packet The first identifies the scale of the FWT parent node, and the second is a variable length string of A s and D s. A indicates the approximation filter. D indicates the detail filter. This creates a fixed logarithmic relationship between the frequency bands. In general, P scale, -D wavelet packet transform support DP+=[DP] 2 + unique decompositions, where D=.

81 7.6 Wavelet packet V J =V J-3 W J-3 W J-2,A W J-2, D W J-,AA W J-,AD W J-, DA W J-,DD

82 7.6 Wavelet packet

83 7.6 Wavelet packet V J =V J- W J-,D W J-, AA W J-,AD

84 7.6 Wavelet packet The spectrum resulting from th first iteration j+=j is shown in Figure 7.32 It divides the frequency plane into four equal areas. Figure 7.33, a three-scale full wavelet packet decomposition. In general, A P-scale, 2-D wavelet packet transform support DP+=[DP] 4 + unique decompositions, where D= Three-scale tree offer possible decompositions

85 7.6 Wavelet packet

86 7.6 Wavelet packet

87 7.6 Wavelet packet

88 7.6 Wavelet packet To select a optimal decompositions for the compression of the image is considering the cost function Ef=ΣΣ fx,y, which measures the entropy or information content of 2-D function f. Minimal entropy leaf nodes should be favored because they have more near-zero values that lead to greater compression. The cost function Ef can be used as a local measurement for the node under consideration.

89 7.6 Wavelet packet For each node of the analysis tree compute both the entropy of the node E P parent entropy and the entropy of the four offsprings, E A, E H, E V, E D. 2 Compare E p with E A + E H + E V + E D, If the combined entropy is greater than the entrop of the parent than prune the offspring, keep only the parent.

90 7.6 Wavelet packet

91 7.6 Wavelet packet

92 7.6 Wavelet packet