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,

This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolution-based interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.

Abstract—Interpolation is required in a variety of medical image processing applications. Although many interpolation techniques are known from the literature, evaluations of these techniques for the specific task of applying geometrical transformations to medical images are still lacking. In this paper we present such an evaluation. We consider convolution-based interpolation methods and rigid transformations (rotations and translations). A large number of sincapproximating kernels are evaluated, including piecewise polynomial kernels and a large number of windowed sinc kernels, with spatial supports ranging from two to ten grid intervals. In the evaluation we use images from a wide variety of medical image modalities. The results show that spline interpolation is to be preferred over all other methods, both for its accuracy and its relatively low computational cost. Keywords—Convolution-based interpolation, spline interpolation, piecewise polynomial kernels, windowed sinc kernels, geometrical transformation, medical images, quantitative evaluation. 1

Nearest-neighbour, linear, and various cubic interpolation functions are frequently used in image resampling. Quadratic functions have been disregarded, largely because they have been thought to introduce phase distortions. This is shown not to be the case, and a family of quadratic functions is derived. The interpolating member of this family has visual quality close to that of the Catmull-Rom cubic, yet requires only sixty percent of the computation time.

Abstract—We examine the question of reconstruction of signals from periodic nonuniform samples. This involves discarding samples from a uniformly sampled signal in some periodic fashion. We give a characterization of the signals that can be reconstructed at exactly the minimum rate once a nonuniform sampling pattern has been fixed. We give an implicit characterization of the reconstruction system, and a design method by which the ideal reconstruction filters may be approximated. We demonstrate that for certain spectral supports the minimum rate can be approached or achieved using reconstruction schemes of much lower complexity than those arrived at by using spectral slicing, as in earlier work. Previous work on multiband signals have typically been those for which restrictive assumptions on the sizes and positions of the bands have been made, or where the minimum rate was approached asymptotically. We show that the class of multiband signals which can be reconstructed exactly is shown to be far larger than previously considered. When approaching the minimum rate, this freedom allows us, in certain cases to have a far less complex reconstruction system. Index Terms — Multiband, nonuniform, reconstruction, sampling. I.

Abstract—The reconstruction of images is an important problem in many applications. From sampling theory it is well known that the sinc function is the ideal interpolation kernel which, however, cannot be used in practice. In order to be able to obtain acceptable reconstructions, both in terms of computational speed and mathematical precision, it is required to design a kernel that is of finite extent and resembles the sinc function as much as possible. In this paper, the applicability of a particular class of sinc-approximating symmetrical piecewise nth-order polynomial kernels is investigated in satisfying these requirements. After the presentation of the general concept, kernels of first, third, fifth and seventh order are derived. An objective, quantitative evaluation of the reconstruction capabilities of these kernels is obtained by analyzing the spatial and spectral behavior using different measures and by using them to translate, rotate and magnify a number of real-life test images. From the experiments it is concluded that while the improvement of cubic convolution over linear interpolation is significant, the use of higher-order polynomials yields only marginal improvement. Keywords—Interpolation, image reconstruction, image resampling, piecewise polynomial kernels, cubic convolution, quintic convolution, septic convolution. I

We consider Shannon sampling theory for sampling sets which are unions of shifted lattices. These sets are not necessarily periodic. A function f can be reconstructed from its samples provided the sampling set and the support of the Fourier transform of f satisfy certain compatibility conditions. While explicit reconstruction formulas are possible, it is most convenient to use a recursive algorithm. The analysis is presented in the general framework of locally compact abelian groups, but several specific examples are given, including a numerical example implemented in MATLAB. 2000 Mathematics Subject Classification: 94A20, 94A12, 43A25, 42B99 Key words: Shannon sampling, multidimensional sampling, nonuniform sampling, periodic sampling, nonperiodic sampling, irregular sampling, locally compact abelian groups. # Mathematics Department, Western Oregon University, Monmouth, Oregon 97361 + Department of Mathematics, Oregon State University, Corvallis, OR 97331. This work was supported by ...

In this paper we study a simple oversampled analog-to-digital (A/D) conversion in shift invariant spaces. The Beurling-Landau type theorem for bandlimited signal spaces is extended to shift invariant spaces, and then a non-uniform sampling theorem for shift invariant spaces is established, which says, a uniformly discrete set is a stable sampling set for a shift invariant space if its Beurling lower density is larger than a fixed density determined by the generator of the shift invariant space. Consequently, an oversampling theorem for shift invariant spaces is attained. These sampling theorems together with a theorem concerning the stability of stable sampling in shift invariant spaces shown by us, are used to build a simple oversampled A/D conversion scheme in shift invariant spaces. In such a scheme, the quantization error e is found to behave as �e � 2 = O(τ 2) with respect to the sampling interval τ, which is the same as that for bandlimited signal spaces derived very recently. Moreover, we demonstrate that the bitrate required to encode the converted digital signal only increases as logarithm of sampling ratio. Keywords: sampling, oversampling, A/D conversion, shift invariant space, generator, Beurling-Landau theorem, quantization error, bit-rate.

space whose dimension is measured A measure F A field V A vector space U Open sets H s Hausdor# measure Appendix B # A Gaussian probability density function # # A Gaussian distribution tangent to a manifold ## A Gaussian distribution normal to a manifold # Sample covariance matrix E, E Error cost functions x x Center of a#ne subspace x Sample mean X A shifted data matrix Appendix C No special symbols Appendix D # Set of all images I An image I(x) Pixel brightness of image I at x P() A morph between two images Z(, , ) A general morph between images # A control line # Unit vector along the control line # # Vector perpendicular to the control line # # 1 ,# 0 The destination and source endpoints of # # The perpendicular proportion of a point to a control line # The signed perpendicular distance of a point to a control line d The Euclidean distance of a point to a control line #(#, , x) The point with the same relation to the control line # as ...

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