Image Recovery from Sparse Samples, Discrete Sampling Theorem, and Sharply Bounded Band-Limited Discrete Signals

2011 ◽  
pp. 295-331 ◽  
Author(s):  
Leonid P. Yaroslavsky
Keyword(s):  
Sampling Theorem ◽  
Image Recovery ◽  
Discrete Sampling ◽  
Discrete Signals ◽  
Band Limited ◽  
Sparse Samples

scholarly journals Discrete sampling theorem to Shannon’s sampling theorem using the hyperreal numbers R

2021 ◽  
Vol 28 (2) ◽  
pp. 163-182
Author(s):  
José L. Simancas-García ◽  
Kemel George-González
Keyword(s):  
Computational Cost ◽  
Number System ◽  
Sampling Theorem ◽  
Discrete Sampling ◽  
Sampled Signal ◽  
Speed Up ◽  
Shannon’S Sampling Theorem ◽  
The Difference ◽  
Band Limited ◽  
Hyperreal Numbers

Shannon’s sampling theorem is one of the most important results of modern signal theory. It describes the reconstruction of any band-limited signal from a finite number of its samples. On the other hand, although less well known, there is the discrete sampling theorem, proved by Cooley while he was working on the development of an algorithm to speed up the calculations of the discrete Fourier transform. Cooley showed that a sampled signal can be resampled by selecting a smaller number of samples, which reduces computational cost. Then it is possible to reconstruct the original sampled signal using a reverse process. In principle, the two theorems are not related. However, in this paper we will show that in the context of Non Standard Mathematical Analysis (NSA) and Hyperreal Numerical System R, the two theorems are equivalent. The difference between them becomes a matter of scale. With the scale changes that the hyperreal number system allows, the discrete variables and functions become continuous, and Shannon’s sampling theorem emerges from the discrete sampling theorem.

scholarly journals Nonuniform sampling, image recovery from sparse data and the discrete sampling theorem

2009 ◽  
Vol 26 (3) ◽  
pp. 566 ◽  
Author(s):  
Leonid P. Yaroslavsky ◽  
Gil Shabat ◽  
Benjamin G. Salomon ◽  
Ianir A. Ideses ◽  
Barak Fishbain
Keyword(s):  
Sparse Data ◽  
Sampling Theorem ◽  
Nonuniform Sampling ◽  
Image Recovery ◽  
Discrete Sampling

scholarly journals A note on Whittaker's cardinal series in harmonic analysis

1986 ◽  
Vol 29 (3) ◽  
pp. 349-357 ◽  
Author(s):  
M. M. Dodson ◽  
A. M. Silva ◽  
V. Soucek
Keyword(s):  
Control Theory ◽  
Data Processing ◽  
Harmonic Analysis ◽  
Real Function ◽  
Sampling Theorem ◽  
Communication Engineering ◽  
Shannon Sampling Theorem ◽  
Band Limited ◽  
Image Position ◽  
Sampling Set

The sampling theorem, often referred to as the Shannon or Whittaker-Kotel'nikov- Shannon sampling theorem, is of considerable importance in many fields, including communication engineering, electronics, control theory and data processing, and has appeared frequently in various forms in engineering literature (a comprehensive account of its numerous extensions and applications is given in [3]). The result states that a band-limited signal, i.e. a real function f of the formwhere w>0, is under reasonable conditions on the even function F, determined by its values on the sampling set (l/2w)ℤ and can be reconstructed from the samples f(k/2w), k∈ℤ, by the series

On Derivative Sampling From Image Blur for Reconstruction of Band-Limited Signals

2014 ◽  
Author(s):  
Jacopo Tani ◽  
Sandipan Mishra ◽  
John T. Wen
Keyword(s):  
Numerical Study ◽  
Signal Reconstruction ◽  
Sampling Theorem ◽  
Image Sensors ◽  
Blurred Image ◽  
Derivative Sampling ◽  
Time Averages ◽  
Image Blur ◽  
Band Limited ◽  
Exposure Windows

Image sensors are typically characterized by slow sampling rates, which limit their efficacy in signal reconstruction applications. Their integrative nature though produces image blur when the exposure window is long enough to capture relative motion of the observed object relative to the sensor. Image blur contains more information on the observed dynamics than the typically used centroids, i.e., time averages of the motion within the exposure window. Parameters characterizing the observed motion, such as the signal derivatives at specified sampling instants, can be used for signal reconstruction through the derivative sampling extension of the known sampling theorem. Using slow image based sensors as derivative samplers allows for reconstruction of faster signals, overcoming Nyquist limitations. In this manuscript, we present an algorithm to extract values of a signal and its derivatives from blurred image measurements at specified sampling instants, i.e. the center of the exposure windows, show its application in two signal reconstruction numerical examples and provide a numerical study on the sensitivity of the extracted values to significant problem parameters.

scholarly journals On the uncertainty inequality as applied to discrete signals

2006 ◽  
Vol 2006 ◽  
pp. 1-22 ◽  
Author(s):  
Y. V. Venkatesh ◽  
S. Kumar Raja ◽  
G. Vidyasagar
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Sampling Theorem ◽  
Discrete Signals ◽  
Bandlimited Signal ◽  
Time Signals ◽  
Shannon Sampling Theorem ◽  
Heisenberg Uncertainty ◽  
Comparison Of The Results ◽  
Uncertainty Inequality

Given a continuous-time bandlimited signal, the Shannon sampling theorem provides an interpolation scheme forexactly reconstructingit from its discrete samples. We analyze the relationship between concentration (orcompactness) in thetemporal/spectral domainsof the (i) continuous-time and (ii) discrete-time signals. The former is governed by the Heisenberg uncertainty inequality which prescribes a lower bound on the product ofeffectivetemporal and spectral spreads of the signal. On the other hand, the discrete-time counterpart seems to exhibit some strange properties, and this provides motivation for the present paper. We consider the following problem:for a bandlimited signal, can the uncertainty inequality be expressed in terms of the samples, using thestandard definitions of the temporal and spectral spreads of the signal?In contrast with the results of the literature, we present a new approach to solve this problem. We also present a comparison of the results obtained using the proposed definitions with those available in the literature.

AN EXTENDED FORM OF SUB-BAND INTERPOLATION OF MULTI-DIMENSIONAL DISCRETE SIGNALS AND ITS APPLICATIONS

1991 ◽  
Vol 01 (03) ◽  
pp. 273-302
Author(s):  
TAKURO KIDA
Keyword(s):  
Convergence Condition ◽  
Approximation Formula ◽  
Digital Television ◽  
Rectangular Lattice ◽  
Reciprocal Relation ◽  
Extended Form ◽  
Discrete Signals ◽  
Sample Points ◽  
Band Limited ◽  
Interpolation Functions

In this paper, we establish an extended form of the optimum sub-band interpolation for a family of n-dimensional discrete signals. We assume that the Fourier spectrums of these discrete signals have weighted L2 norms smaller than a given positive number. It is assumed that the sample points of these discrete signals are identical with the whole vertices of an n-dimensional rectangular lattice in Rn. Among these sample points, certain subsets are used for the interpolation. Selecting appropriate subsets of the sample points, we can realize a wide variety of periodic arrangements of sample points for interpolation such as hexagonal and octagonal lattices or a set of sample points used in interlaces scanning of digital television. The proposed method minimizes the measure of error which is equal to the envelope of the approximation errors with respect to the discrete signals. In the following discussion, we assume initially that the corresponding approximation formula uses an infinite number of interpolation functions having limited supports and functional forms different from each other. However, it should be noted that the resultant optimum interpolation functions are expressed as the parallel shifts of the impulse responses of the finite number of n-dimensional FIR filters. Equivalent analog approximation formula corresponding to the proposed discrete approximation, is derived and interesting reciprocal relation in the approximation, is also discussed. A necessary and sufficient condition for the convergence of the corresponding analog approximation formula to the original band limited signal, is presented. An equivalent expression of the analog approximation formula in the frequency domain, is derived in relation to the convergence condition.

Sampling analysis in the complex reproducing kernel Hilbert space

2014 ◽  
Vol 26 (1) ◽  
pp. 109-120 ◽  
Author(s):  
BING-ZHAO LI ◽  
QING-HUA JI
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sampling Theorem ◽  
Optimal Approximation ◽  
Sampling Points ◽  
Limited Function ◽  
Band Limited ◽  
Band Limited Function ◽  
Necessary And Sufficient

We consider and analyse sampling theories in the reproducing kernel Hilbert space (RKHS) in this paper. The reconstruction of a function in an RKHS from a given set of sampling points and the reproducing kernel of the RKHS is discussed. Firstly, we analyse and give the optimal approximation of any function belonging to the RKHS in detail. Then, a necessary and sufficient condition to perfectly reconstruct the function in the corresponding RKHS of complex-valued functions is investigated. Based on the derived results, another proof of the sampling theorem in the linear canonical transform (LCT) domain is given. Finally, the optimal approximation of any band-limited function in the LCT domain from infinite sampling points is also analysed and discussed.

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