Results 1  10
of
22
Ideal spatial adaptation by wavelet shrinkage
 Biometrika
, 1994
"... With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, variable knot spline, or variable bandwidth kernel, to the unknown function. Estimation with the aid of an oracle o ers dramatic ad ..."
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Cited by 838 (4 self)
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With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, variable knot spline, or variable bandwidth kernel, to the unknown function. Estimation with the aid of an oracle o ers dramatic advantages over traditional linear estimation by nonadaptive kernels � however, it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone. We describe a new principle for spatiallyadaptive estimation: selective wavelet reconstruction. Weshowthatvariableknot spline ts and piecewisepolynomial ts, when equipped with an oracle to select the knots, are not dramatically more powerful than selective wavelet reconstruction with an oracle. We develop a practical spatially adaptive method, RiskShrink, which works by shrinkage of empirical wavelet coe cients. RiskShrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so. A new inequality inmultivariate normal decision theory which wecallthe oracle inequality shows that attained performance di ers from ideal performance by at most a factor 2logn, where n is the sample size. Moreover no estimator can give a better guarantee than this. Within the class of spatially adaptive procedures, RiskShrink is essentially optimal. Relying only on the data, it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle. In contrast, it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone.
Nonlinear solution of linear inverse problems by waveletvaguelette decomposition
, 1992
"... We describe the WaveletVaguelette Decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type { such asnumerical di erentiation, inversion of Abeltype ..."
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Cited by 182 (12 self)
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We describe the WaveletVaguelette Decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type { such asnumerical di erentiation, inversion of Abeltype transforms, certain convolution transforms, and the Radon Transform. We propose to solve illposed linear inverse problems by nonlinearly \shrinking" the WVD coe cients of the noisy, indirect data. Our approach o ers signi cant advantages over traditional SVD inversion in the case of recovering spatially inhomogeneous objects. We suppose that observations are contaminated by white noise and that the object is an unknown element of a Besov space. We prove that nonlinear WVD shrinkage can be tuned to attain the minimax rate of convergence, for L 2 loss, over the entire Besov scale. The important case of Besov spaces Bp;q, p <2, which model spatial inhomogeneity, is included. In comparison, linear procedures { SVD included { cannot attain optimal rates of convergence over such classes in the case p<2. For example, our methods achieve faster rates of convergence, for objects known to lie in the Bump Algebra or in Bounded Variation, than any linear procedure.
Framelets: MRABased Constructions of Wavelet Frames
, 2001
"... We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spl ..."
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Cited by 129 (50 self)
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We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudospline tight frames and symmetric biframes with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.
On the Construction of Multivariate (pre)wavelets
, 1992
"... : A new approach for the construction of wavelets and prewavelets on IR d from multiresolution is presented. The method uses only properties of shiftinvariant spaces and orthogonal projectors from L 2 (IR d ) onto these spaces, and requires neither decay nor stability of the scaling function. F ..."
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Cited by 78 (11 self)
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: A new approach for the construction of wavelets and prewavelets on IR d from multiresolution is presented. The method uses only properties of shiftinvariant spaces and orthogonal projectors from L 2 (IR d ) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multiresolution. AMS (MOS) Subject Classifications: primary: 41A63, 46C99; secondary: 41A30, 41A15, 42B99, 46E20. Key Words and phrases: wavelets, multiresolution, shiftinvariant spaces, box splines. Authors' affiliation and address: 1 Center for Mathematical Sciences University of WisconsinMadison 610 Walnut St. Madison WI 53705 and 2 Department of Mathematics University of South Carolina Columbia SC 29208 This work was carried out while t...
Multiresolution and wavelets
 Proc. Edinburgh Math. Soc
, 1994
"... Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general ..."
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Cited by 48 (24 self)
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Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skewsymmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.
Wavelet shrinkage for nonequispaced samples, The Annals of Statistics 26
, 1998
"... Standard wavelet shrinkage procedures for nonparametric regression are restricted to equispaced samples. There, data are transformed into empirical wavelet coefficients and threshold rules are applied to the coefficients. The estimators are obtained via the inverse transform of the denoised wavelet ..."
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Cited by 39 (3 self)
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Standard wavelet shrinkage procedures for nonparametric regression are restricted to equispaced samples. There, data are transformed into empirical wavelet coefficients and threshold rules are applied to the coefficients. The estimators are obtained via the inverse transform of the denoised wavelet coefficients. In many applications, however, the samples are nonequispaced. It can be shown that these procedures would produce suboptimal estimators if they were applied directly to nonequispaced samples. We propose a wavelet shrinkage procedure for nonequispaced samples. We show that the estimate is adaptive and near optimal. For global estimation, the estimate is within a logarithmic factor of the minimax risk over a wide range of piecewise Hölder classes, indeed with a number of discontinuities that grows polynomially fast with the sample size. For estimating a target function at a point, the estimate is optimally adaptive to unknown degree of smoothness within a constant. In addition, the estimate enjoys a smoothness property: if the target function is the zero function, then with probability tending to 1 the estimate is also the zero function. (1.1) 1. Introduction. Suppose
Affine systems in L_2(R^d): the analysis of the analysis operator
 J. Functional Anal
, 1996
"... Discrete affine systems are obtained by applying dilations to a given shiftinvariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are "global" in nature: ..."
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Cited by 25 (5 self)
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Discrete affine systems are obtained by applying dilations to a given shiftinvariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are "global" in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the affine system with the aid of two new notions: the affine product, and a quasiaffine system. This leads to a characterization of affine frames; the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the affine product can be factored during a multiresolution analysis con...
Construction of compactly supported biorthogonal wavelets II
, 1997
"... This paper deals with constructions of compactly supported biorthogonal wavelets from a pair of dual refinable functions in L 2 (R s ). In particular, an algorithmic method to construct wavelet systems and the corresponding dual systems from a given pair of dual refinable functions is given. Keyw ..."
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Cited by 24 (8 self)
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This paper deals with constructions of compactly supported biorthogonal wavelets from a pair of dual refinable functions in L 2 (R s ). In particular, an algorithmic method to construct wavelet systems and the corresponding dual systems from a given pair of dual refinable functions is given. Keywords: multivariate biorthogonal wavelets, multivariate wavelets, box splines, matrix extension 1. INTRODUCTION This paper deals with constructions of compactly supported biorthogonal wavelets, whose dilations and shifts form a Riesz basis for L 2 (R s ) and the dual basis is an affine system generated by compactly supported functions with required order of the smoothness, from a given pair of dual refinable functions. Constructions of compactly supported refinable dual pairs can be found in Ref. 6 and Ref. 3. With a pair of compactly supported refinable functions constructed, the key step to construct biorthogonal wavelets from a given pair of multiresolutions can be reduced to the follow...
Affine system in L2(IR d ): the analysis of the analysis operator
 CMSTSR # 96–02, University of Wisconsin
, 1995
"... Discrete affine systems are obtained by applying dilations to a given shiftinvariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are “global ” in nature: ..."
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Cited by 17 (13 self)
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Discrete affine systems are obtained by applying dilations to a given shiftinvariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are “global ” in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the affine system with the aid of two new notions: the affine product, and a quasiaffine system. This leads to a characterization of affine frames; the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the affine product can be factored during a multiresolution analysis construction, and this leads to a complete characterization of all tight frames that can be constructed by such methods. Moreover, this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution. Of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions, and that the freedom offered by redundancy can be fully exploited in these constructions.
Wavelet Estimation For Samples With Random Uniform Design
 Statist. Probab. Lett
"... We show that for nonparametric regression if the samples have random uniform design, the wavelet method with universal thresholding can be applied directly to the samples as if they were equispaced. The resulting estimator achieves within a logarithmic factor from the minimax rate of convergence ove ..."
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Cited by 14 (0 self)
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We show that for nonparametric regression if the samples have random uniform design, the wavelet method with universal thresholding can be applied directly to the samples as if they were equispaced. The resulting estimator achieves within a logarithmic factor from the minimax rate of convergence over a family of Holder classes. Simulation result is also discussed. Keywords: wavelets, nonparametric regression, minimax, adaptivity, Holder class. AMS 1991 Subject Classification: Primary 62G07, Secondary 62G20. 1 Introduction Wavelet shrinkage methods have been very successful in nonparametric regression. But so far most of the wavelet regression methods have been focused on equispaced samples. There, data are transformed into empirical wavelet coefficients and threshold rules are applied to the coefficients. The estimators are obtained via the inverse transform of the denoised wavelet coefficients. The most widely used wavelet shrinkage method for equispaced samples is the DonohoJoh...