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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, Hilbert-space formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shift-invariant” 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 (anti-aliasing) prefilters that are not necessarily ideal low-pass. 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—Band-limited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,

: A complete characterization is given of closed shift-invariant subspaces of L 2 (IR d ) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace. AMS (MOS) Subject Classifications: 41A25, 41A63; 41A30, 41A15, 42B99, 46E30 Key Words and phrases: approximation order, Strang-Fix conditions, shift-invariant spaces, radial basis functions, orthogonal projection. Authors' affiliation and address: 1 Center for Mathematical Sciences University of Wisconsin-Madison 610 Walnut St. Madison WI 53705 and 2 Department of Mathematics University of South Carolina Columbia SC 29208 This work...

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Abstract—Based on the theory of approximation, this paper presents a unified analysis of interpolation and resampling techniques. An important issue is the choice of adequate basis functions. We show that, contrary to the common belief, those that perform best are not interpolating. By opposition to traditional interpolation, we call their use generalized interpolation; they involve a prefiltering step when correctly applied. We explain why the approximation order inherent in any basis function is important to limit interpolation artifacts. The decomposition theorem states that any basis function endowed with approximation order can be expressed as the convolution of a B-spline of the same order with another function that has none. This motivates the use of splines and spline-based functions as a tunable way to keep artifacts in check without any significant cost penalty. We discuss implementation and performance issues, and we provide experimental evidence to support our claims. Index Terms—Approximation constant, approximation order, B-splines, Fourier error kernel, maximal order and minimal support (Moms), piecewise-polynomials. I.

: 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., shift-invariant 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 shift-invariant 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...

We propose a new method for the intermodal registration of images using a criterion known as mutual information. Our main contribution is an optimizer that we specifically designed for this criterion. We show that this new optimizer is well adapted to a multiresolution approach because it typically converges in fewer criterion evaluations than other optimizers. We have built a multiresolution image pyramid, along with an interpolation process, an optimizer, and the criterion itself, around the unifying concept of spline-processing. This ensures coherence in the way we model data and yields good performance. We have tested our approach in a variety of experimental conditions and report excellent results. We claim an accuracy of about a hundredth of a pixel under ideal conditions. We are also robust since the accuracy is still about a tenth of a pixel under very noisy conditions. In addition, a blind evaluation of our results compares very favorably to the work of several other researchers.

In this paper, we consider Lp{approximation byinteger translates of a finite set of functions ( =0�:::�r; 1) which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vector = ( ) r;1 =0 is refinable, necessary and sufficient conditions for the refinement mask are derived. In particular, if algebraic polynomials can be exactly reproduced by integer translates of, then a factorization of the refinement mask of can be given. This result is a natural generalization of the result for a single function, where the refinement mask

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 two-scale 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 least-squares approximation error. Another contribution is the application of these results to B-splines and Daubechies scaling functions, which yields explicit asymptotic developments and upper bounds. Thanks to these explicit expressions, we can quantify ...