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117
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 207 (22 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
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.
The Structure of Finitely Generated ShiftInvariant Spaces in ...
, 1992
"... : A simple characterization is given of finitely generated subspaces of L 2 (IR d ) which are invariant under translation by any (multi)integer, and used to give conditions under which such a space has a particularly nice generating set, namely a basis, and, more than that, a basis with desirable ..."
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Cited by 101 (21 self)
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: A simple characterization is given of finitely generated subspaces of L 2 (IR d ) which are invariant under translation by any (multi)integer, and used to give conditions under which such a space has a particularly nice generating set, namely a basis, and, more than that, a basis with desirable properties, such as stability, orthogonality, or linear independence. The last property makes sense only for `local' spaces, i.e., shiftinvariant spaces generated by finitely many compactly supported functions, and special attention is paid to such spaces. As an application, we prove that the approximation order provided by a given local space is already provided by the shiftinvariant space generated by just one function, with this function constructible as a finite linear combination of the finite generating set for the whole space, hence compactly supported. This settles a question of some 20 years' standing. AMS (MOS) Subject Classifications: primary: 41A25, 41A63, 46C99; secondary: 4...
Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets StrangFix
 IEEE Trans. on Signal Processing
, 2007
"... Abstract—Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, nonuniform splines or piecewise polynomials, and call the number of degrees of freedom per unit of time the rate of innovation. Cl ..."
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Cited by 92 (27 self)
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Abstract—Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, nonuniform splines or piecewise polynomials, and call the number of degrees of freedom per unit of time the rate of innovation. Classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited. Recently, it was shown that, by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation, it is possible to reconstruct such signals uniquely [34]. These sampling schemes, however, use kernels with infinite support, and this leads to complex and potentially unstable reconstruction algorithms. In this paper, we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm. The class of kernels that we can use is very rich and includes functions satisfying Strang–Fix conditions, exponential splines and functions with rational Fourier transform. This last class of kernels is quite general and includes, for instance, any linear electric circuit. We, thus, show with an example how to estimate a signal of finite rate of innovation at the output of an circuit. The case of noisy measurements is also analyzed, and we present a novel algorithm that reduces the effect of noise by oversampling. Index Terms—Analogtodigital conversion, annihilating filter method, multiresolution approximations, sampling methods, splines, wavelets. I.
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...
Frames and Stable Bases for ShiftInvariant Subspaces of . . .
, 1994
"... Let X be a countable fundamental set in a Hilbert space H, and let T be the operator T : ` 2 (X) ! H : c 7! X x2X c(x)x: Whenever T is welldefined and bounded, X is said to be a Bessel sequence. If, in addition, ran T is closed, then X is a frame. Finally, a frame whose corresponding T is inje ..."
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Cited by 75 (22 self)
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Let X be a countable fundamental set in a Hilbert space H, and let T be the operator T : ` 2 (X) ! H : c 7! X x2X c(x)x: Whenever T is welldefined and bounded, X is said to be a Bessel sequence. If, in addition, ran T is closed, then X is a frame. Finally, a frame whose corresponding T is injective is a stable basis (also known as a Riesz basis). This paper considers the above three properties for subspaces H of L 2 (IR d ), and for sets X of the form X = fOE(\Delta \Gamma ff) : OE 2 \Phi; ff 2 ZZ d g; with \Phi either a singleton, a finite set, or, more generally, a countable set. The analysis is performed on the Fourier domain, where the two operators TT and T T are decomposed into a collection of simpler "fiber" operators. The main theme of the entire analysis is the characterization of each of the above three properties in terms of the analogous property of these simpler operators. AMS (MOS) Subject Classifications: 42C15 Key Words: Riesz bases, stable bases, shif...
Approximation By Translates Of Refinable Functions
, 1996
"... . The functions f 1 (x); : : : ; fr (x) are refinable if they are combinations of the rescaled and translated functions f i (2x \Gamma k). This is very common in scientific computing on a regular mesh. The space V 0 of approximating functions with meshwidth h = 1 is a subspace of V 1 with meshwidth ..."
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Cited by 69 (14 self)
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. The functions f 1 (x); : : : ; fr (x) are refinable if they are combinations of the rescaled and translated functions f i (2x \Gamma k). This is very common in scientific computing on a regular mesh. The space V 0 of approximating functions with meshwidth h = 1 is a subspace of V 1 with meshwidth h = 1=2. These refinable spaces have refinable basis functions. The accuracy of the computations depends on p, the order of approximation, which is determined by the degree of polynomials 1; x; : : : ; x p\Gamma1 that lie in V 0 . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions f i (x) are known only through the coefficients c k in the refinement equationscalars in the traditional case, r \Theta r matrices for multiwavelets. The scalar "sum rules" that determine p are well known. We find the conditions on the matrices c k that yield approximation of order p from V 0 . These are equivalent to the StrangFix condition...
Approximation properties of multivariate wavelets
 Math. Comp
, 1998
"... Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of ..."
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Cited by 66 (10 self)
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Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in W k−1 1 (Rs) provides approximation order k. 1.
Quantitative Fourier Analysis of Approximation Techniques: Part II  Wavelets
 IEEE Trans. Signal Processing
, 1999
"... In a previous paper, we proposed a general Fourier method that provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the usual twoscale relation encountered in dyadic ..."
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Cited by 63 (28 self)
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In a previous paper, we proposed a general Fourier method that provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the usual twoscale relation encountered in dyadic multiresolution analysis. As a consequence of this additional constraint, the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter. This is, in particular, true for the asymptotic expansion of the approximation error for biorthonormal wavelets as the scale tends to zero. One of the contributions of this paper is the computation of sharp, asymptotically optimal upper bounds for the leastsquares approximation error. Another contribution is the application of these results to Bsplines and Daubechies scaling functions, which yields explicit asymptotic developments and upper bounds. Thanks to these explicit expressions, we can quantify ...