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 Reconstruction of Periodic Band Limited Signals from Non-Uniform Samples with Sub-Nyquist Sampling Rate

Sensors ◽  
2020 ◽  
Vol 20 (21) ◽  
pp. 6246
Author(s):  
Dongxiao Wang ◽  
Xiaoqin Liu ◽  
Xing Wu ◽  
Zhihai Wang
Keyword(s):  
Signal Reconstruction ◽  
Sampling Rate ◽  
Sampling Theorem ◽  
Drive System ◽  
Reconstruction Method ◽  
Robot Arm ◽  
Reconstructed Signal ◽  
Band Limited ◽  
Simulation Signal ◽  
Important State

Important state parameters, such as torque and angle obtained from the servo control and drive system, can reflect the operating condition of the equipment. However, there are two problems with the information obtained through the network from the control and drive system: the low sampling rate, which does not meet the sampling theorem and the nonuniformity of the sampling points. By combing equivalent sampling and nonuniform signal reconstruction theory, this paper proposes a reconstruction method for signal obtained from servo system in periodic reciprocating motion. Equivalent sampling combines the low rate and nonuniform samples from multiple periods into one single period, so that the equivalent sampling rate is far increased. Then, the nonuniform samples with high density are further resampled to meet the reconstruction conditions. This step can avoid the amplitude error in the reconstructed signal and increase the possibility of successful reconstruction. Finally, the reconstruction formula derived from basis theory is applied to recover the signal. This method has been successfully verified by the simulation signal of the robot swing process and the actual current signal collected on the robot arm testbed.

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 Experimental and Numerical Study on Characterization and Evaluation of Surface Cracks with Wireless Ultrasonic Sensor

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Shuo Chen ◽  
Jing Li ◽  
Donghui Huang ◽  
Yuzhi Chen ◽  
Haitao Zhao
Keyword(s):  
Numerical Study ◽  
Experimental Testing ◽  
Quantitative Estimation ◽  
Signal Reconstruction ◽  
Characteristic Parameter ◽  
Crack Size ◽  
Waveform Analysis ◽  
Fe Model ◽  
Surface Cracks ◽  
Artificial Surface

Adopting both wireless ultrasonic sensing and numerical simulation techniques, this research investigates the interaction between Rayleigh wave and artificial surface cracks of varying depths. When analyzing experimental ultrasonic data collected by a wireless sensing node, the signals are enhanced through a two-step procedure including signal reconstruction and envelope extraction. The waveforms are interpreted in detail by analyzing wave components through time-of-flight technique. A finite element (FE) model is devised to properly simulate the experimental testing. The simulated waveforms are consistent with experimental results and corroborate the analysis and explanations of experimental waveforms. Based on both experimental and numerical waveform analysis, a relationship between ultrasonic characteristic parameter and crack size is established for the quantitative estimation purpose. The proposed model shows a good agreement with data from both test and literatures.

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

scholarly journals Ultrasonic Signal Reconstruction Using Compressed Sensing

2016 ◽  
Vol 855 ◽  
pp. 165-170
Author(s):  
Ren Jean Liou
Keyword(s):  
Compressed Sensing ◽  
Spline Interpolation ◽  
Signal To Noise Ratio ◽  
Signal Reconstruction ◽  
Sampling Theorem ◽  
Ultrasonic Signal ◽  
Ultrasonic Signals ◽  
Data Rates ◽  
Nyquist Rate ◽  
Simulation Results

Ultrasonic signal reconstruction for Structural Health Monitoring is a topic that has been discussed extensively. In this paper, we will apply the techniques of compressed sensing to reconstruct ultrasonic signals that are seriously damaged. To reconstruct the data, the application of conventional interpolation techniques is restricted under the criteria of Nyquist sampling theorem. The newly developed technique - compressed sensing breaks the limitations of Nyquist rate and provides effective results based upon sparse signal reconstruction. Sparse representation is constructed using Fourier transform basis. An l1-norm optimization is then applied for reconstruction. Signals with temperature characteristics were synthetically created. We seriously corrupted these signals and tested the efficacy of our approach under two different scenarios. Firstly, the signal is randomly sampled at very low rates. Secondly, selected intervals were completely blank out. Simulation results show that the signals are effectively reconstructed. It outperforms conventional Spline interpolation in signal-to-noise ratio (SNR) with low variation, especially under very low data rates. This research demonstrates very promising results of using compressed sensing for ultrasonic signal reconstruction.

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.

scholarly journals Approximation of Eigenvalues of Sturm-Liouville Problems by Using Hermite Interpolation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
M. M. Tharwat ◽  
S. M. Al-Harbi
Keyword(s):  
Boundary Conditions ◽  
Error Bounds ◽  
Eigenvalue Problems ◽  
Sampling Theorem ◽  
Eigenvalue Parameter ◽  
Derivative Sampling ◽  
Sturm Liouville ◽  
Truncation And Amplitude Errors ◽  
Computable Error Bounds ◽  
Hermite Interpolations

Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. In this paper, we use the derivative sampling theorem “Hermite interpolations” to compute approximate values of the eigenvalues of Sturm-Liouville problems with eigenvalue parameter in one or two boundary conditions. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Also, using computable error bounds, we obtain eigenvalue enclosures. Also numerical examples, which are given at the end of the paper, give comparisons with the classical sinc method and explain that the Hermite interpolations method gives remarkably better results.

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