Cohen, A., Daubechies, I., Jawerth, B. and Vial, P. (1993). Multiresolution analysis, wavelets and fast algorithms on an interval. C. R. Acad. Sci. Paris 316, 417--421.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....moments; wavelet shrinkage; AMS 2000 Subject Classification: Primary 62G08;42C40 Secondary 62G20;65T60 1. Introduction The simplest setting for much of the theory of nonparametric function estimation is the Gaussian white noise model Y (t) f(s)ds #W (t) 0 1, 1) in which f L 2 [0, 1] is unknown and W is standard Brownian motion. If # # is an orthonormal basis for L 2 [0, 1] then the model takes the sequence form y # = # # #z # (2) by setting y # = # # dY , # # = # # f and z # = # # dW . Here z are i.i.d. N(0, 1) and the Parseval relation f) # # ....

....62G20;65T60 1. Introduction The simplest setting for much of the theory of nonparametric function estimation is the Gaussian white noise model Y (t) f(s)ds #W (t) 0 1, 1) in which f L 2 [0, 1] is unknown and W is standard Brownian motion. If # # is an orthonormal basis for L 2 [0, 1], then the model takes the sequence form y # = # # #z # (2) by setting y # = # # dY , # # = # # f and z # = # # dW . Here z are i.i.d. N(0, 1) and the Parseval relation f) # # ) # Postal address: Department of Statistics, Evans Hall, U.C. Berkeley, Berkeley, CA ....

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Cohen, A., Daubechies, I., Jawerth, B. and Vial, P. (1993). Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris (A) 316, 417--421.

....34 73. few of the largest coefficients. Impressive lossy compression rates for images have been achieved using this type of approximation [DeVore et al. 1992] There are many constructions of wavelets for functions parametrized over the interval [Andersson et al. 1993; Chui and Quak 1992; Cohen et al. 1993; Meyer 1992] These have found use in signal processing [Mallat 1989] B spline modeling [Finkelstein and Salesin 1994] motion planning [Liu et al. 1994] and many other applications involving functions parametrized in one dimension. Two dimensional wavelets are important for a variety of ....

COHEN, A., DAUBECHIES, I., AND VIAL, P. 1993. Multiresolution analysis, wavelets and fast algorithms on an interval. Appl. Comput. Harmon. Anal. 1, 1, 100--115.

.... = Z f(x) j;k (x)dx: 6) For functions supported in [0; 1] we can select an index set = f(j; k) j = 0; 1; Delta Delta Delta ; k = 0; Delta Delta Delta ; 2 j Gamma 1g and modify some of j;k , j; k) 2 , such that ( j;k ) j;k)2 forms a complete orthonormal basis for L 2 [0; 1] (Cohen, Daubechies, Jawerth, Vial 1993; Daubechies 1994; Donoho and Johnstone 1996) Wavelets allow us to estimate the smoothness of f(x) in terms of decay estimates of its wavelet coefficients for large j. For example, f(x) satisfies the Holderian condition with exponent ff if, and only, if for some positive constant C, j j;k j C ....

Cohen, A., Daubechies, I., Jawerth, B. and Vial, P. (1993) Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris (A), 316, 417-421.

....of Section 3. Let the vector Y = Y (t 1 ) Y (t n ) 0 represent a sample of n measurements on Y (t) The discrete wavelet transformation of Y can be written in the form WY , where W is an n Theta n orthogonal matrix which depends on both the wavelet and the boundary adjustment (Cohen et al. 1993; Daubechies, 1994) Fast algorithms with complexity of order n are available for performing both the transformation and the inversion of the transformation which results in the reconstruction of the original data. To label the n = 2 J coefficients of the transformation, we index n Gamma 1 of ....

Cohen, A., Daubechies, I., Jawerth, B. & Vail, P. (1993). Multiresolution analysis, wavelets and fast algorithms on an interval. Comptes Rendus des S'eances de l'Acad'emie des Sciences, S'erie I, Math'ematique 316, 417--21.

.... as f(x) X (x) 6) with wavelet coefficients of f = Z f(x) x)dx: 7) For functions supported in a fixed unit interval, say [0; 1] we can select an index set ae ZZ and modify some of the , 2 , such that ( 2 forms a complete orthonormal basis for L 2 [0; 1] see Cohen, Daubechies, Jawerth, Vial (1993), Daubechies (1994) Donoho and Johnstone (1995a) Meyer (1992) 2.2 Wavelet Estimates Let y = Z (x)Y (dx) 8) be the empirical wavelet coefficients of the data Y . Wavelet estimates of f are defined to be f (t j ) X ffi t j (y ) 9) with ffi t j (y ) sign(y ) jy j ....

.... (DWT) is a discretized version of the continuous wavelet transformation and can be written as linear transformation involving a n by n orthogonal matrix W which depends on parameters M (number of vanishing moments) S (support width) j 0 (Low resolution cutoff) and boundary adjustments [see Cohen, Daubechies, Jawerth and Vial (1993), Daubechies (1994) Donoho and Johnstone (1994) The rows of W correspond to a discretized version of the wavelets . Indeed, if we denote by W j;k (i) the ith element of the (j; k) th row of W, then n Gamma1 X i=1 i W j;k (i) 0; 0; M; j j 0 ; k = 0; 2 j ....

COHEN, A., DAUBECHIES, I., JAWERTH, B. and VIAL, P. (1993). Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris (A). 316 417-421.

....= i=n, i = 1; n, or equivalently, we observe f from model (5) and have discrete data y 1 ; Delta Delta Delta ; y n . Consequently we need to perform discrete versions of wavelet and vaguelette transformations. Fast algorithms are available for computing discrete wavelet transformation (Cohen, Daubechies, Jawerth and Vial (1993), Meyer (1993) Nason and Silverman (1994) and discrete vaguelette transformation (Kolaczyk (1994) As in the discrete wavelet transformation case (Donoho and Johnstone (1994) Wang (1995) the n Gamma 1 elements of the discrete vaguelette transformation of (y 1 ; Delta Delta Delta ; y n ) ....

Cohen, A., Daubechies, I., Jawerth, B. and Vial, P. (1993). Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris (A).

....later, results for model (2.1) 2.3) will imply Theorems 1.1 and 1.2 by suitable identifications. Thus we will want ultimately to interpret [1] I ) as the empirical wavelet coefficients of (f(t i ) n Gamma1 i=0 ; 2] I ) as the empirical wavelet coefficients of an estimate f n [3] (2.2) as a norm equivalent to n Gamma1 P E( f (t i ) Gamma f(t i ) 2 ; and [4] 2.3) as a condition guaranteeing that f is smoother than f . We will explain such identifications further in sections 5 6 below. 3 Soft Thresholding and Optimal Recovery Before tackling (2.1) 2.3) we ....

Cohen, A., Daubechies, I., Jawerth, B., and Vial, P. (1992). Multiresolution analysis, wavelets, and fast algorithms on an interval. To appear, Comptes Rendus Acad. Sci. Paris (A).

....be performed. 6 Delta2. Discrete wavelet transformation The discrete wavelet transformation is a discretized version of the continuous wavelet transformation and can be written as linear transformation involving a n byn orthogonal matrix W which depends on the wavelet and the boundary adjustment (Cohen, Daubechies, Jawerth Vial, 1993; Daubechies, 1994) Let y = y 1 ; Delta Delta Delta ; y n ) The discrete wavelet transformation of the data y is given by w = Wy. Because W is orthogonal, the inverse discrete wavelet transformation is y = W T w. The n Gamma 1 elements of w are indexed dyadically: w j;k , k = 0; ....

COHEN, A., DAUBECHIES, I., JAWERTH, B. & VIAL, P. (1993). Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris (A). 316, 417-421.

....unit interval. In particular, it is no longer possible to have all wavelets be simple dilations and translations of a single mother wavelet as in (2) Wavelets on the boundary must be adapted to retain orthonormality. Some approaches to this problem include Meyer (1991) Chui and Quak (1992) and Cohen, Daubechies, Jawerth, and Vial (1993). Simpler approaches use reflection or symmetric handling of the boundary. This work will presuppose the choice of an appropriate orthonormal wavelet basis on [0; 1] denoted f j;k ; k = 0; 2 j Gamma 1; j = 0; 1; g. Given such a basis and with knowledge of the underlying function f ....

Cohen, A., Daubechies, I., Jawerth, B., & Vial, P. (1993). Multiresolution analysis, wavelets and fast algorithms on an interval. Comptes Rendus des Seances de l' Academie des Sciences, Serie I, 316, 417--421.

....for the Haar wavelets, since supp j;k = 2 Gammaj k; 2 Gammaj (k 1) but it is not as straightforward for smoother wavelets. One common method for adapting smooth wavelets to the interval is periodic boundary handling, in which the function f is considered to be periodic with period 1. Cohen, Daubechies, and Vial (1993) develop a method to modify the usual Daubechies wavelets to live on the interval the resulting wavelet functions are not all simply dilations and translations of a single mother wavelet however. When wavelets (and their corresponding scaling functions) are appropriately modified, they form a ....

....For this illustration, the change point actually occurred at = 0:72, so the values line up much more closely than for = 0:20. The analysis discussed thus far can be generalized by allowing for a smoother wavelet basis. In particular, the Daubechies wavelets adapted to the interval (see Cohen, Daubechies, and Vial (1993)) are used. These wavelets also provide an orthonormal basis for L 2 [0; 1] so the expression (4) for the conditional distribution of the empirical coefficient w j;k given , Delta, and oe holds, with q j;k ( defined to be q j;k ( Z 1 j;k (t) dt: In practice, it is not necessary ....

Cohen, A., Daubechies, I., and Vial, P. (1993). Multiresolution analysis, wavelets and fast algorithms on an interval. Applied Computational Harmonic Analysis 1: 54--81.

....kernel K is a C 2 function of compact support which is also a probability density, and if f = T V K;2 (y; ffi ) then f(t) 1 n n X i=1 y i K t Gamma t i ffi (t) OE ffi (t) 3) More refined versions of this formula would adjust K for boundary effects near t = 0 and t = 1. [5]. Variable Bandwidth High Order Kernels T V K;D (y; ffi ) D 2. Here ffi is again the local bandwidth, and the reconstruction formula is as in (3) only K( Delta) is a C D function integrating to 1, with vanishing intermediate moments Z t j K(t) dt = 0; j = 1; D Gamma 1: As D 2, ....

COHEN, A., DAUBECHIES, I., JAWERTH, B. & VIAL, P. (1993). Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris (A). 316., 417--421.

....see the WaveLab distribution, or the published articles [17, 19, 21, 23, 25] 5. 1 Wavelets WaveLab of course offers a full complement of wavelet transforms both the standard orthogonal periodized wavelet transforms FWT PO [14] standard boundary corrected wavelet transforms FWT CDJV [9], and the standard periodized biorthogonal wavelets FWT PBS [8] It also offers less standard wavelet transforms which have been developed as part of research at Stanford. Two examples include interpolating wavelet transforms based on interpolation schemes (FWT DD for what we call ....

Cohen, A., I. Daubechies, B. Jawerth and P. Vial (1992). Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris A 316 417-421.

....unit interval. In particular, it is no longer possible for all wavelets to be simple dilations and translations of a single mother wavelet as in (2) Wavelets on the boundary must be adapted to retain orthonormality. Some approaches to this problem include Meyer (1991) Chui and Quak (1992) and Cohen, Daubechies, Jawerth, and Vial (1993). Simpler approaches use reflection or symmetric handling of the boundary. This work will presuppose the choice of an appropriate orthonormal wavelet basis on [0; 1] denoted f j;k ; k = 0; 2 j Gamma 1; j = 0; 1; g. 1.2 The discrete wavelet transform If Y (no subscripts) represents ....

Cohen, A., Daubechies, I., Jawerth, B., & Vial, P. (1993). Multiresolution analysis, wavelets and fast algorithms on an interval. Comptes Rendus des Seances de l' Academie des Sciences, Serie I, 316, 417--421.

....2. The empirical wavelet transform of noisy data (d i ) n i=1 obeying (3) yields data y I = I ffl Delta z I ; I 2 I n ; 75) with ffl = oe= p n. This form of data is of the same general form as supposed in the sequence model (22) Detailed study of the Pyramid Filtering Algorithm of [10] reveals that all but O(log(n) of these coefficients are a standard Gaussian white noise with variance oe 2 =n; the other coefficients feel the boundaries , and have a slight covariance among themselves and a variance which is roughly, but not exactly, oe 2 =n. Nevertheless, the analog of ....

Cohen, A., Daubechies, I., Jawerth, B., and Vial, P. (1992). Multiresolution analysis, wavelets, and fast algorithms on an interval. To appear, Comptes Rendus Acad. Sci. Paris (A).

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Cohen, A., Daubechies, I., Jawerth, B. and Vial, P. (1993). Multiresolution analysis, wavelets and fast algorithms on an interval. C. R. Acad. Sci. Paris 316, 417--421.

No context found.

Cohen, A., Daubechies, I., Jawerth, B. and Vial, P. (1992) Multiresolution analysis, wavelets, and fast algorithms on an interval. C. R. Acad. Sci. Paris A, 316, 417--421.

No context found.

COHEN, A., DAUBECHIES, I., AND VIAL, P. 1993. Multiresolution analysis, wavelets and fast algorithms on an interval. Appl. Comput. Harmon. Anal. 1, 1, 100--115.

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Cohen, A., Daubechies, I., Jawerth, B. & Vial, P. (1993). Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris (A). 316, 417-421.

No context found.

Cohen, A., Daubechies, I., Jawerth, B. & Vial, P. (1993). Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris (A). 316, 417-421.

No context found.

Cohen, A., Daubechies, I., Jawerth, B. & Vial, P. (1993). Multiresolution analysis, wavelets and fast algorithms on an interval. Compt. Rend. Acad. Sci. Paris Ser. I, 316, 417-421.

No context found.

Cohen, A., Daubechies, I., Jawerth, B. and Vial, P. (1992) Multiresolution analysis, wavelets, and fast algorithms on an interval. Compt. Rend. Acad. Sci. Paris A, 316, 417--421.

No context found.

Cohen, A., Daubechies, I., Jawerth, B. & Vial, P. (1993). Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris (A). 316, 417-421.

No context found.

COHEN, A., DAUBECHIES, I., JAWERTH, B. and VIAL, P. (1993). Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris (A). 316 417-421.

No context found.

Cohen, A., Daubechies, I., Jawerth, B., and Vial, P. (1992). Multiresolution analysis, wavelets, and fast algorithms on an interval. To appear, Comptes Rendus Acad. Sci. Paris (A).

No context found.

Cohen, A., Daubechies, I., Jawerth, B. & Vial, P. (1993) Multiresolution analysis, wavelets and fast algorithms on an interval. Compt. Rend. Acad. Sci. Paris Ser. I, 316, 417--421.

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