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The Lifting Scheme: A Construction Of Second Generation Wavelets
, 1997
"... . We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to ..."
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Cited by 541 (16 self)
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. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, inplace calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...
Using the refinement equation for the construction of prewavelets II
, 1991
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Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 149 (18 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
Cardinal spline filters: Stability and convergence to the ideal sinc interpolator
, 1992
"... In this paper, we provide an interpretation of polynomial spline interpolation as a continuous filtering process. We prove that the frequency responses of the cardinal spline filters converge to the ideal lowpass filter in all L fnorms with 1 ~<p < + ~ as the order of the spline tends to inf ..."
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Cited by 61 (28 self)
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In this paper, we provide an interpretation of polynomial spline interpolation as a continuous filtering process. We prove that the frequency responses of the cardinal spline filters converge to the ideal lowpass filter in all L fnorms with 1 ~<p < + ~ as the order of the spline tends to infinity. We provide estimates for the resolution errors and the interpolation errors of the various filters. We also derive an upper bound for the error associated with the reconstruction of bandlimited signals using polynomial splines.
Wavelet Families Of Increasing Order In Arbitrary Dimensions
, 1997
"... . We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its ..."
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Cited by 58 (0 self)
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. We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, inplace calculation, and integerto integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. Submitted to IEEE Transactions on Image Processing Over the last decade several constructions of compactly supported wavelets have originated both from signal processing and mathematical analysis. In signal processing, critically sampled wavelet transforms are known as filter banks or subband transforms [32, 43, 54, 56]. In mathematical analysis, wavelets are defined as translates and dilates of one fixed function and ar...
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. ..."
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Cited by 32 (9 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.
Construction of Bivariate Compactly Supported Biorthogonal Box Spline Wavelets with Arbitrarily High Regularities
 APPL. COMPUT. HARMON. ANAL
"... We give a simple formula for the duals of the filters associated with bivariate box spline functions. We show how to construct bivariate nonseparable compactly supported biorthogonal wavelets associated with box spline functions which have arbitrarily high regularities. ..."
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Cited by 24 (8 self)
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We give a simple formula for the duals of the filters associated with bivariate box spline functions. We show how to construct bivariate nonseparable compactly supported biorthogonal wavelets associated with box spline functions which have arbitrarily high regularities.
Examples of Bivariate Nonseparable Compactly Supported Orthonormal Continuous Wavelets
 Wavelet Applications in Signal and Image Processing IV, proceedings of SPIE, 3169
, 1997
"... We give several examples of bivariate nonseparable compactly supported orthonormal wavelets whose scaling functions are supported over [0,3]x[0,3]. The Holder continuity properties of these wavelets are studied. Keywords: Nonseparable, Compact support, Orthonormal, Continuous, Wavelet 1. ..."
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Cited by 15 (4 self)
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We give several examples of bivariate nonseparable compactly supported orthonormal wavelets whose scaling functions are supported over [0,3]x[0,3]. The Holder continuity properties of these wavelets are studied. Keywords: Nonseparable, Compact support, Orthonormal, Continuous, Wavelet 1.
Construction of Trivariate Compactly Supported Biorthogonal Box Spline Wavelets
 J. Approx. Theory
, 1998
"... this paper, we are interested in constructing the compactly supported biorthogonal wavelets associated with trivariate box spline functions. Let B l;m;n;p;q;r be the trivariate box spline function whose Fourier transform is ..."
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Cited by 5 (2 self)
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this paper, we are interested in constructing the compactly supported biorthogonal wavelets associated with trivariate box spline functions. Let B l;m;n;p;q;r be the trivariate box spline function whose Fourier transform is
Biorthogonal Refinable Spline Functions
"... We give a construction for refinable spline functions of degree n with compact support and simple knots in 1 4 ZZ which are biorthogonal to uniform Bsplines of degree n with simple knots at 1 3 ZZ. x1. ..."
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Cited by 3 (1 self)
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We give a construction for refinable spline functions of degree n with compact support and simple knots in 1 4 ZZ which are biorthogonal to uniform Bsplines of degree n with simple knots at 1 3 ZZ. x1.