An Introduction to Wavelets and some Applications

Transcription

1 An Introduction to Wavelets and some Applications Milan, May 2003 Anestis Antoniadis Laboratoire IMAG-LMC University Joseph Fourier Grenoble, France An Introduction to Wavelets and some Applications p.1/54

2 Outline General facts on wavelet decompositions An Introduction to Wavelets and some Applications p.2/54

3 Outline General facts on wavelet decompositions Multiresolution analyses An Introduction to Wavelets and some Applications p.2/54

4 Outline General facts on wavelet decompositions Multiresolution analyses Computational algorithms An Introduction to Wavelets and some Applications p.2/54

5 Outline General facts on wavelet decompositions Multiresolution analyses Computational algorithms Approximation and Compression An Introduction to Wavelets and some Applications p.2/54

6 Outline General facts on wavelet decompositions Multiresolution analyses Computational algorithms Approximation and Compression Linear An Introduction to Wavelets and some Applications p.2/54

7 Outline General facts on wavelet decompositions Multiresolution analyses Computational algorithms Approximation and Compression Linear Nonlinear An Introduction to Wavelets and some Applications p.2/54

8 Outline General facts on wavelet decompositions Multiresolution analyses Computational algorithms Approximation and Compression Linear Nonlinear Estimation and Denoising An Introduction to Wavelets and some Applications p.2/54

9 Outline General facts on wavelet decompositions Multiresolution analyses Computational algorithms Approximation and Compression Linear Nonlinear Estimation and Denoising Regularization An Introduction to Wavelets and some Applications p.2/54

10 Outline General facts on wavelet decompositions Multiresolution analyses Computational algorithms Approximation and Compression Linear Nonlinear Estimation and Denoising Regularization Asymptotics and Applications An Introduction to Wavelets and some Applications p.2/54

11 Generalities Interest in representing an "observed signal as an appropriate superposition of "elementary functions in an adapted way. An Introduction to Wavelets and some Applications p.3/54

12 Generalities Interest in representing an "observed signal as an appropriate superposition of "elementary functions in an adapted way. To obtain such representations that can be useful in practice, one needs fast computational algorithms. An Introduction to Wavelets and some Applications p.3/54

13 Generalities Interest in representing an "observed signal as an appropriate superposition of "elementary functions in an adapted way. To obtain such representations that can be useful in practice, one needs fast computational algorithms. Once such representations are derived, one would like to simplify them in a efficient way by choosing appropriately only few elementary components. This may be seen as an approximation or compression task. An Introduction to Wavelets and some Applications p.3/54

14 The Haar basis The Haar basis is the simplest example of a wavelet basis. It allows us to introduce in a clear and simple way the wavelet idea, without going to far into mathematical details. Suppose we are given a discretized version of an integrable function f on [0, 1] on an equidistant grid of 8 values. An Introduction to Wavelets and some Applications p.4/54

15 The Haar basis The Haar basis is the simplest example of a wavelet basis. It allows us to introduce in a clear and simple way the wavelet idea, without going to far into mathematical details. Suppose we are given a discretized version of an integrable function f on [0, 1] on an equidistant grid of 8 values. Let the discrete signal be: [ ] A digital signal on [0, 1] An Introduction to Wavelets and some Applications p.4/54

16 We could represent the above digital signal in a different way in order to exploit a possible correlation between adjacent points. To this end, for every pair of neighbors we compute their averages to obtain: [ ] Averages of neighbor values To avoid any loss of information we also need to record some other values that represent the loss of information when going from the finer grid to the coarser. An Introduction to Wavelets and some Applications p.5/54

17 We choose [ ] Differences of Input and its approximation Indeed, 3+(-1)=2, 3-(-1)=4, 10+(-2)=8,... An Introduction to Wavelets and some Applications p.6/54

18 We choose [ ] Differences of Input and its approximation Indeed, 3+(-1)=2, 3-(-1)=4, 10+(-2)=8,... input signal = signal with lower resolution (4 values) and a quadruple of differences (the details). An Introduction to Wavelets and some Applications p.6/54

19 The Haar transform We can repeat the procedure over the averages again and again to obtain: Resolution Averages Details 8 [ ] 4 [ ] [ ] 2 [ ] [ ] 1 [ ] [ ] thus representing the input as: [ ] An Introduction to Wavelets and some Applications p.7/54

20 Input = digital signal Average Difference Average Difference Average Difference The Haar transform An Introduction to Wavelets and some Applications p.8/54

21 The digital input may be considered as a piecewise constant function on [0, 1] on the intervals I 3,k = [2 3 k, 2 3 (k + 1)[, k = 0,..., If φ(x) = I [0,1[ (x) and φ j,k (x) = φ(2 j x k), the function may be written as f(x) = 2φ 3,0 (x) + 4φ 3,1 (x) + 8φ 3,2 (x) + 12φ 3,3 (x) + 14φ 3,4 (x) + 0φ 3,5 (x) + 2φ 3,6 (x) + 1φ 3,7 (x). We then may re-write: f(x) = 3φ 2,0 (x) + 10φ 2,1 (x) + 7φ 2,2 (x) + 1.5φ 2,3 (x) + ( 1)ψ 2,0 (x) + ( 2)ψ 2,1 (x) + 7ψ 2,2 (x) + 0.5ψ 2,3 (x), where ψ(x) = I [0,1/2[ (x) I [1/2,1[ (x). An Introduction to Wavelets and some Applications p.9/54

22 V 0 subspace of L 2 ([0, 1[) spanned by the constant functions on [0, 1[. An Introduction to Wavelets and some Applications p.10/54

23 V 0 subspace of L 2 ([0, 1[) spanned by the constant functions on [0, 1[. V 1 subspace spanned by the piecewise constant functions on [0, 1/2[ and [1/2, 1[ An Introduction to Wavelets and some Applications p.10/54

24 V 0 subspace of L 2 ([0, 1[) spanned by the constant functions on [0, 1[. V 1 subspace spanned by the piecewise constant functions on [0, 1/2[ and [1/2, 1[ V j subspace spanned by the piecewise constant functions on the intervals I j,k, k = 0, 2 j 1. An Introduction to Wavelets and some Applications p.10/54

25 V 0 subspace of L 2 ([0, 1[) spanned by the constant functions on [0, 1[. V 1 subspace spanned by the piecewise constant functions on [0, 1/2[ and [1/2, 1[ V j subspace spanned by the piecewise constant functions on the intervals I j,k, k = 0, 2 j 1. They are nested. Moreover, for each of them, the families {φ j,k, k = 0,..., 2 j 1} form a basis. For the usual inner product f, g = 1 0 f(x)ḡ(x)dx, these families are orthogonal and {ψ j,k, k = 0,..., 2 j 1} is a basis of the vector space W j, orthogonal complement of V j in V j+1. An Introduction to Wavelets and some Applications p.10/54

26 Multiresolution analysis on the R A multiresolution analysis of L 2 (R) is a nested sequence of closed subspaces V j, j Z, of L 2 (R), V 2 V 1 V 0 V 1 V 2, such that j V j = {0}, j V j = L 2 (R), f(x) V j f(2x) V j+1, There exists a function φ V 0 such that { V 0 = f L 2 (R) : f(x) = k Z α k φ(x k) }, {φ k, k Z} is a "stable basis of V 0, i.e. 0 < m φ k M < and A f 2 k α 2 k B f 2. An Introduction to Wavelets and some Applications p.11/54

27 The function φ is called the scaling function of the MRA. Let φ j,k (x) = 2 j/2 φ(2 j x k). For an orthogonal MRA, an orthonormal basis of V j is {φ j,k : k Z} and P j f = k Z < f, φ j,k > φ j,k, is the approximation of f at resolution 2 j. If W j is the orthogonal complement of V j in V j+1, we obtain another sequence {W j : j Z} of closed orthogonal subspaces of L 2 (R), such that each W j is a refinement of W 0, and their direct sum is L 2 (R). An Introduction to Wavelets and some Applications p.12/54

28 One can show that there exists a function ψ such that W 0 is spanned by its integer translates. Then ψ is called the wavelet associated to ϕ. For each integer j,the family is an orthonormal basis of W j. If g L 2 (R) we have: g = k Z {ψ j,k, k Z} c j0,kϕ j0,k + j j 0 k Z d j,k ψ j,k where j 0 is a level of coarse approximation. An Introduction to Wavelets and some Applications p.13/54

29 The first part in the right hand side is the orthogonal projection P j0 g of g on V j0, and the second part represents the details. The coefficients are defined by c j,k = g, φ j,k and d j,k = g, ψ j,k. The c s are called the scaling coefficients while the d s are the wavelet coefficients. An Introduction to Wavelets and some Applications p.14/54

30 Filter banks Since V 0 V 1, any function of V 0 has an expansion in terms of the basis {φ 1,k, k Z} of V 1. For φ(x) = φ 0 (x) = φ 0,0 (x) we have φ(x) = k Z a k φ 1,k (x) = 2 k Z a k φ(2x k) with a k = φ, φ 1,k l 2 (Z). An Introduction to Wavelets and some Applications p.15/54

31 When the scaling functions are compactly supported, there is only a finite number of non zero coefficients among the a k s and φ(x) = 2 D 1 k=0 a k φ(2x k) The coefficients a = {a k } define the filter corresponding to φ. The scaling function is compactly supported if and only if a has a finite number of non zero coefficients. An Introduction to Wavelets and some Applications p.16/54

32 By analogy, and since W 0 V 1 we also have ψ(x) = 2 D 1 k=0 b k φ(2x k) with b k = ψ, φ 1,k l 2 (Z). The filters {a k } and {b k } are conjugate mirror filter banks. An Introduction to Wavelets and some Applications p.17/54

33 A perfect reconstruction filter bank decomposes a signal by filtering and subsampling. It reconstructs it by inserting zeroes, filtering and summation. The filter bank is said to be a perfect reconstruction filter bank when a 2 = a 0. If, additionally, h = h 2 and g = g 2, the filters are called conjugate mirror filters. An Introduction to Wavelets and some Applications p.18/54

34 Numerical evaluation No analytic formulas for evaluating numerically φ and ψ. For compactly supported wavelets we have two algorithms : An iterative algorithm (Daubechies and Lagarias cascade algorithm). An iterative method for solving the dilatation equations An Introduction to Wavelets and some Applications p.19/54

35 The Cascade Algorithm Build φ using the filter a 0,..., a D 1 with D even and 4. Start with φ 0 (x) = I [0,1[ (x). Compute then recursively the successive approximations of φ(x) using a 0,..., a D 1, i.e. φ m (x) = D 1 k=0 a k φ m 1 (2x k). One can show that this algorithm converges towards φ, when the iterations m converge to. In practice, 8 iterations suffice for a good discretization of φ. An Introduction to Wavelets and some Applications p.20/54

36 Example: φ Daubechies D = Building φ by the cascade algorithm An Introduction to Wavelets and some Applications p.21/54

37 Periodic wavelets Until now the functions were defined on R. While this seems reasonable for some applications, in practice most functions are observed over bounded domains. There exist many ways to define a MRA adapted to a bounded domain. One of these, the most simple and direct way, is by using periodic wavelets. Let the scaling function φ L 2 (R) and the associated wavelet ψ L 2 (R) be given. An Introduction to Wavelets and some Applications p.22/54

38 For all j, l Z, we define the periodic scaling function of period 1, by φ j,l (x) = + n= φ j,l (x + n) and the associated periodic wavelet by ψ j,l (x) = + n= ψ j,l (x + n). An Introduction to Wavelets and some Applications p.23/54

39 Note that, for j 0 and l Z, φ j,l (x) = 2 j/2 n φ(2 j (x + n 2 j l)) = φ j,0 (x) Therefore j 0 φ j,l (x) = 2 j/2. Similarly we can show that j 1, ψ j,l = 0. An Introduction to Wavelets and some Applications p.24/54

40 For φ and ψ generated with a filter of length D: φ j,l (x) = 2 j/2 for all j 0 and all l Z. An Introduction to Wavelets and some Applications p.25/54

41 For φ and ψ generated with a filter of length D: φ j,l (x) = 2 j/2 for all j 0 and all l Z. ψ j,l (x) = 0 for all j 1 and all l Z. An Introduction to Wavelets and some Applications p.25/54

42 For φ and ψ generated with a filter of length D: φ j,l (x) = 2 j/2 for all j 0 and all l Z. ψ j,l (x) = 0 for all j 1 and all l Z. for all j > J 0 log 2 (D 1) and x [0, 1], φ j,l (x) = φ j,l (x)i Ij,l (x) + φ j,l (x + 1)I I c (x) j,l with an identical relation for ψ. An Introduction to Wavelets and some Applications p.25/54

43 For φ and ψ generated with a filter of length D: φ j,l (x) = 2 j/2 for all j 0 and all l Z. ψ j,l (x) = 0 for all j 1 and all l Z. for all j > J 0 log 2 (D 1) and x [0, 1], φ j,l (x) = φ j,l (x)i Ij,l (x) + φ j,l (x + 1)I I c (x) j,l with an identical relation for ψ. 1 0 f(x) φ j,l (x)dx = f(x)φ j,l (x)dx (similarly for ψ) where f(x) = f(x [x]), x R. An Introduction to Wavelets and some Applications p.25/54

44 The periodic scaling functions and corresponding periodic wavelets define an orthonormal MRA of L 2 ([0, 1]). The approximation and details spaces are given by: { } Ṽ j = span φ j,l, l = 0,..., 2 j 1 and { } W j = span ψ j,l, l = 0,..., 2 j 1 We obtain, for J 0 0 the following decomposition of L 2 ([0, 1]): L 2 ([0, 1]) = Ṽ J0 ( j J0 W j ). An Introduction to Wavelets and some Applications p.26/54

45 Fast discrete wavelet transform Orthogonality of the scaling functions and the associated wavelets lead to a fast computational algorithm for decomposing or synthetizing a function of V j. Let f L 2 (R). We have formulated as (P Vj f)(x) = (P Vj 1 f)(x) + (P Wj 1 f)(x), (P Vj f)(x) = l Z c j 1,l φ j 1,l (x) + l Z d j 1,l ψ j 1,l (x). Aim: find a relation between the sequence of coefficients c j,l and the sequences c j 1,l and d j 1,l. An Introduction to Wavelets and some Applications p.27/54

46 The key is φ j 1,l (x) = D 1 k=0 a k φ j,2l+k (x) and ψ j 1,l (x) = D 1 k=0 b k φ j,2l+k (x) We have c j 1,l = = D 1 k=0 f(x)φ j 1,l (x)dx = a k f(x) f(x)φ j,2l+k (x)dx = D 1 k=0 D 1 k=0 a k φ j,2l+k (x)dx a k c j,2l+k and d j 1,l = D 1 k=0 b kc j,2l+k. An Introduction to Wavelets and some Applications p.28/54

47 Conversely, noting that we obtain φ j,l (x) = k a l 2k φ j 1,k (x) + k b l 2k ψ j 1,k (x) c j,l = k a l 2k c j 1,k + k b l 2k d j 1,k (x) These two above equations lead to the discrete fast wavelet transform. An Introduction to Wavelets and some Applications p.29/54

48 An example As an example, let c = c 3,0 c 3,1 c 3,2 c 3,3 c 3,4 c 3,5 c 3,6. c 3,7 An Introduction to Wavelets and some Applications p.30/54

49 The decomposition steps are: c 3,0 c 3,1 c 3,2 c 3,3 c 3,4 c 3,5 c 3,6 c 2,0 c 2,1 c 2,2 c 2,3 d 2,0 d 2,1 d 2,3 c 1,0 c 1,1 d 1,0 d 1,1 d 2,0 d 2,1 d 2,3 c 0,0 d 0,0 d 1,0 d 1,1 d 2,0 d 2,1 d 2,2. c 3,7 d 2,4 d 2,4 d 2,4 The final vector is the results of the DWT of the initial values. An Introduction to Wavelets and some Applications p.31/54

50 Matrix Representation of the DWT Let and c j = (c j,0,..., c j,2 j 1 )T d j = (d j,0,..., d j,2 j 1 )T The DWT equations define linear maps from R 2j to R 2j 1 and may be written as: c j 1 d j 1 = A j c j = B j c j where A j and B j are 2 j 1 2 j matrices defined via the filters. An Introduction to Wavelets and some Applications p.32/54

51 Approximation properties A linear approximation of square integrable function f on an orthonormal basis B = {e n } n Z projects f on the space spanned by M vectors chosen a priori in B: f M = M 1 f, e n e n. n=0 The quality of the approximation f f M 2 = n M f, e n 2 depends on the properties of f with respect to B. An Introduction to Wavelets and some Applications p.33/54

52 Fourier analysis provides efficient approximations for global smooth functions by projecting them on the space spanned by sinusoidal waves of frequencies the first M low frequencies. In a wavelet basis the signal is projected on V M. Here too, the quality of approximation depends on the global regularity of the function l f. An Introduction to Wavelets and some Applications p.34/54

53 Nonlinear approximation A linear approximation of f may be enhanced if the M vectors are chosen a posteriori, in a way that depends on f. For M fixed, the approximation error is minimized by taking the M vectors for which the coefficients f, e n are the largest. If B is a wavelet basis,the amplitude of the coefficients is related to the local regularity of f and a nonlinear approximation amounts in using an adaptative sampling whose resolution increases locally where the function is irregular. An Introduction to Wavelets and some Applications p.35/54

54 First application: Compression Consider a function f expanded in a basis as f(x) = m k=1 c k e k (x). The input data are the coefficients c 1,..., c m. The aim of compression is to define an approximation of f using much less coefficients (possible in a different basis) with a minimum loss of information. Given an upper bound for the error, say ɛ > 0, we seek f(x) = m k=1 c k ẽ k (x) with m < m and f f ɛ. An Introduction to Wavelets and some Applications p.36/54

55 For simplicity consider that the basis is fixed one for all and that it is orthonormal. Let σ be a permutation of {1, 2,..., m} and let f the approximation using the first m coefficients of the permutation σ : f(x) = m k=1 c σ(k) e σ(k) (x). The L 2 error of this approximation is f f 2 2 = m k= m+1 c σ(k) 2. To minimize it σ must rank the coefficients in a decreasing order of their absolute magnitude. An Introduction to Wavelets and some Applications p.37/54

56 Example Compression JumpSine Signal Top Fourier Coeffs Top Wavelet Coeffs error = Top Fourier Coeffs error = Compression example An Introduction to Wavelets and some Applications p.38/54

57 Second application: denoising Data : Y j = f(t j ) + ɛ j, j = 1,..., n Usually : t j equidistant, n = 2 m, and ɛ j i.i.d mean 0 and variance σ 2. In practice, nonequispaced design; An Introduction to Wavelets and some Applications p.39/54

58 Second application: denoising Data : Y j = f(t j ) + ɛ j, j = 1,..., n Usually : t j equidistant, n = 2 m, and ɛ j i.i.d mean 0 and variance σ 2. In practice, nonequispaced design; number of data points not a power of 2. An Introduction to Wavelets and some Applications p.39/54

59 Second application: denoising Data : Y j = f(t j ) + ɛ j, j = 1,..., n Usually : t j equidistant, n = 2 m, and ɛ j i.i.d mean 0 and variance σ 2. In practice, nonequispaced design; number of data points not a power of 2. design may be random; An Introduction to Wavelets and some Applications p.39/54

60 Second application: denoising Data : Y j = f(t j ) + ɛ j, j = 1,..., n Usually : t j equidistant, n = 2 m, and ɛ j i.i.d mean 0 and variance σ 2. In practice, nonequispaced design; number of data points not a power of 2. design may be random; the errors may not be of constant variance; An Introduction to Wavelets and some Applications p.39/54

61 Linear methods Extension of penalized least-squares methods An Introduction to Wavelets and some Applications p.40/54

62 Linear methods Extension of penalized least-squares methods choice of the smoothing parameter by GCV. An Introduction to Wavelets and some Applications p.40/54

63 Linear methods Extension of penalized least-squares methods choice of the smoothing parameter by GCV. An example. An Introduction to Wavelets and some Applications p.40/54

64 Simple idea: Approximate 2 m/2 f, φ m,k f(k/2 m ) and replace the raw data {Y i } by its interpolation in V m : ˆf m (t) = 2 m/2 k Z Y k φ m,k (t) Minimize ˆf m f 2 L 2 ([0,1]) + λj p spp(p VJ0 f) where J 0 is given et J spp is norm on B s pp([0, 1]). An Introduction to Wavelets and some Applications p.41/54

65 The norm of g B s pq([0, 1]) is equivalent to J spq (α, β) = α j0 p + ( ) 1/q (2 j(s+(1/2) (1/p)) β j p ) q. j=0 An Introduction to Wavelets and some Applications p.42/54

66 The solution f λ = 2 j 0 1 k=0 c j0,kϕ J0,k + m j=j 0 2 j 1 k=0 ˆβ j,k ψ j,k, c j0,k, DWT of ˆf m. k = 0,..., 2 j 0 1 are the scaling coefficients of the ˆβ j,k = d j,k 1+λ2 2sj, j j 0, k = 0,..., 2 j 1, with d j,k the wavelet coefficients of W ˆf m. An Introduction to Wavelets and some Applications p.43/54

67 Cross validation One may choose J 0 and λ given the data and show that the resulting estimation has good properties (at least asymptotically). One popular method is cross validation. 16 voltage drop vs time A real data example with D8 compared to spline smoothing An Introduction to Wavelets and some Applications p.44/54

68 Nonlinear methods Decompose the signal into its wavelet coefficients An Introduction to Wavelets and some Applications p.45/54

69 Nonlinear methods Decompose the signal into its wavelet coefficients Extract the most significant coefficients by shrinkage or thresholding. An Introduction to Wavelets and some Applications p.45/54

70 Nonlinear methods Decompose the signal into its wavelet coefficients Extract the most significant coefficients by shrinkage or thresholding. Denoise by applying the inverse wavelet transform on the resulting coefficients. An Introduction to Wavelets and some Applications p.45/54

71 Almost all existing nonlinear methods are of the form: 2 1 n (z i θ i ) 2 + λ p( θ i ), i=1 i i 0 where z i is the ith row of z = WY n and p λ is an appropriate penalty function. An Introduction to Wavelets and some Applications p.46/54

72 Hard thresholding 1. Transform the data via DWT : Θ = W Y. 2. To separate the signal form its noise, threshold the coefficients: set ˆθ j,k = θ j,k if θ j,k is large, 0 otherwise. 3. Estimate the signal by ˆf = W 1 ˆΘ. An Introduction to Wavelets and some Applications p.47/54

73 Soft thresholding 1. Transform the data via DWT : Θ = W Y. 2. To separate the signal form its noise, threshold the coefficients: set ˆθ j,k = { θ j,k λ, si θ j,k > λ, 0, si θ j,k λ, 3. Estimate the signal by ˆf = W 1 ˆΘ. Many ways for choosing the thresholds.the initial resolution j 0 is usually set to (log 2 N)/2. An Introduction to Wavelets and some Applications p.48/54

74 Examples (a) L p penalty with p = 1 (soft (p=1)), p = 0.6 (short dash) and p = 0.2 (solid); (b) hard; (c) SCAD (robust); (d) transformed soft. 3 (a) L p penalties p=1, 0.6, (b) Hard thresholding penalty penalty θ (c) Smoothly clipped L 1 penalty θ penalty penalty θ (d) Transformed L 1 penalty An Introduction θ to Wavelets and some Applications p.49/54

75 (a) Hard thresholding and L 1 penalty (b) L p penalty (p=0.6) (c) Smoothly clipped L 1 penalty 5 (d)transformed L 1 penalty Corresponding estimates An Introduction to Wavelets and some Applications p.50/54

76 Universal thresholding (VisuShrink) When the noise is Gaussina, most of the coefficients (normalized as: Nd j,k /σ) are essentially white noise. This suggests to take λ = 2 log N, producing the principle VisuShrink implemented in the package wavethresh of R. Noisy Signal Wavelet Coeffs idwt d1 d2 d3 d4 d5 d6 s Reconstruction VisuShrink idwt d1 d2 d3 d4 d5 d6 s VisuShrink An Introduction to Wavelets and some Applications p.51/54

77 Properties When compared, it seems that hard thresholding produces estimates with larger variance while soft thresholding is more biased. Some authors propose a robust version: ˆθ λ 1,λ 2 j,k = 0 si θ j,k λ 1 sgn( θ j,k ) λ 2( θ j,k λ 1 ) λ 2 λ si λ 1 1 < θ j,k λ 2 θ j,k si θ j,k > λ 2, which groups the advantages of both thresholding methods An Introduction to Wavelets and some Applications p.52/54

78 References Newland D. E (1994) An Introduction to Random Vibrations, Spectral and Wavelet Analysis, Third Edition, (Wiley). Koornwinder, T H (1993) Wavelets: An Elementary Treatment in Theory and Applications, (World Scientific, Singapore). Strang, G. (1989) Wavelets and Dilation equations: A Brief Introduction, dans SIAM Review, 31(4) Daubechies, I. (1992) Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics (SIAM Publications, Philadelphia). Chui, C. K. (1992) An Introduction to Wavelets, (Academic Press). An Introduction to Wavelets and some Applications p.53/54

79 Software WaveLab for Matlab. Wavelet Toolbox for Matlab. Wavelab for Scilab. S+Wavelets for S-plus. Wavetresh 2.2 for S-plus and R. Many new denoising algorithms are implemented in MATLAB and are available at www-lmc.imag.fr/sms/software An Introduction to Wavelets and some Applications p.54/54