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 addresses the problem of recovering a super-resolved image from a set of warped blurred and decimated versions thereof. Several algorithms have already been proposed for the solution of this general problem. In this paper, we concentrate on a special case where the warps are pure translations, the blur is space invariant and the same for all the images, and the noise is white. We exploit previous results to develop a new highly efficient super-resolution reconstruction algorithm for this case, which separates the treatment into de-blurring and measurements fusion. The fusion part is shown to be a very simple noniterative algorithm, preserving the optimality of the entire reconstruction process, in the maximum-likelihood sense. Simulations demonstrate the capabilities of the proposed algorithm.

Recent years have seen growing interest in the problem of super-resolution restoration of video sequences. Whereas in the traditional single image restoration problem only a single input image is available for processing, the task of reconstructing super-resolution images from multiple undersampled and degraded images can take advantage of the additional spatiotemporal data available in the image sequence. In particular, camera and scene motion lead to frames in the source video sequence containing similar, but not identical information. The additional information available in these frames make possible reconstruction of visually superior frames at higher resolution than that of the original data. In this paper we review the current state of the art and identify promising directions for future research.

Super-resolution algorithms reconstruct a high-resolution image from a set of low-resolution images of a scene. Precise alignment of the input images is an essential part of such algorithms. If the low-resolution images are undersampled and have aliasing artifacts, the performance of standard registration algorithms decreases. We propose a frequency domain technique to precisely register a set of aliased images, based on their low-frequency, aliasing-free part. A high-resolution image is then reconstructed using cubic interpolation. Our algorithm is compared to other algorithms in simulations and practical experiments using real aliased images. Both show very good visual results and prove the attractivity of our approach in the case of aliased input images. A possible application is to digital cameras where a set of rapidly acquired images can be used to recover a higher-resolution final image. Copyright © 2006 Patrick Vandewalle et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A method is proposed for super-resolving multi-channel data with applications to PREDATOR video sequences. Using a generalization of Papoulis' sampling theorem, a closed-form solution has been obtained leading to a high-speed algorithm which can be realistically applied to large data sets such as video sequences. In existing multiframe methods it is a common practice to assume that the channel transfer functions are known and invariant from one frame to another, using empirical models such as Gaussian, sinc, etc. We have assumed that the transfer functions are unknown and may vary even when the same sensor is employed, and hence use the observed data to derive the Point Spread Function (PSF) for each frame. The estimated PSFs are used in the super-resolution algorithm. Results on PREDATOR video images are then given. Keywords: Super-resolution, video data processing, multi-channel sampling, datadriven estimation This work was partially supported by ONR Grant N00014-95-1-0521. 1 Intr...

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We examine the problem of periodic nonuniform sampling of a multiband signal and its reconstruction from the samples. This sampling scheme, which has been studied previously, has an interesting optimality property that uniform sampling lacks: one can sample and reconstruct the class ( ) of multiband signals with spectral support , at rates arbitrarily close to the Landau minimum rate equal to the Lebesgue measure of , even when does not tile under translation. Using the conditions for exact reconstruction, we derive an explicit reconstruction formula. We compute bounds on the peak value and the energy of the aliasing error in the event that the input signal is band-limited to the "span of " (the smallest interval containing ) which is a bigger class than the valid signals ( ), band-limited to . We also examine the performance of the reconstruction system when the input contains additive sample noise. Index Terms---Error bounds, Landau--Nyquist rate, multiband, nonuniform periodic samp...

any reduction in scan time offers a number of potential benefits ranging from high-temporal-rate observation of physiological processes to improvements in patient comfort. Following recent developments in Compressive Sensing (CS) theory, several authors have demonstrated that certain classes of MR images which possess sparse representations in some transform domain can be accurately reconstructed from very highly undersampled K-space data by solving a convex ℓ1-minimization problem. Although ℓ1-based techniques are extremely powerful, they inherently require a degree of over-sampling above the theoretical minimum sampling rate to guarantee that exact reconstruction can be achieved. In this paper, we propose a generalization of the Compressive Sensing paradigm based on homotopic approximation of the ℓ0 quasi-norm and show how MR image reconstruction can be pushed even further below the Nyquist limit and significantly closer to the theoretical bound. Following a brief review of standard Compressive Sensing methods and the developed theoretical extensions, several example MRI reconstructions from highly undersampled K-space data are presented.

Offset mismatch, gain mismatch, and sample-time error between time-interleaved channels limit the performance of time-interleaved analog-to-digital converters (ADCs). This paper focuses on the sample-time error. Techniques for correcting and detecting sample-time error in a two-channel ADC are described, and simulation results are presented.

In many applications, the sampling frequency is limited by the physical characteristics of the components: the pixel pitch, the rate of the analog-to-digital (A/D) converter, etc. A lowpass filter is usually applied before the sampling operation to avoid aliasing. However, when multiple copies are available, it is possible to use the information that is inherently present in the aliasing to reconstruct a higher resolution signal. If the different copies have unknown relative offsets, this is a nonlinear problem in the offsets and the signal coefficients. They are not easily separable in the set of equations describing the super-resolution problem. Thus, we perform joint registration and reconstruction from multiple unregistered sets of samples. We give a mathematical formulation for the problem when there are sets of samples of a signal that is described by expansion coefficients. We prove that the solution of the registration and reconstruction problem is generically unique