Signal Sampling Theory Research Articles

Leveraging Multi-level Correlations for Imputing Monitoring Data in Water Supply Systems Using Graph Signal Sampling Theory

Leveraging Multi-level Correlations for Imputing Monitoring Data in Water Supply Systems Using Graph Signal Sampling Theory

Adaptive High-Resolution Imaging Method Based on Compressive Sensing.

Compressive sensing (CS) is a signal sampling theory that originated about 16 years ago. It replaces expensive and complex receiving devices with well-designed signal recovery algorithms, thus simplifying the imaging system. Based on the application of CS theory, a single-pixel camera with an array-detection imaging system is established for high-pixel detection. Each detector of the detector array is coupled with a bundle of fibers formed by fusion of four bundles of fibers of different lengths, so that the target area corresponding to one detector is split into four groups of target information arriving at different times. By comparing the total amount of information received by the detector with the threshold set in advance, it can be determined whether the four groups of information are calculated separately. The simulation results show that this new system can not only reduce the number of measurements required to reconstruct high quality images but can also handle situations wherever the target may appear in the field of view without necessitating an increase in the number of detectors.

Graph Signal Processing: Dualizing GSP Sampling in the Vertex and Spectral Domains

Vertex based and spectral based GSP sampling has been studied recently. The literature recognizes that methods in one domain do not have a counterpart in the other domain. This paper shows that in fact one can develop a unified graph signal sampling theory with analogous interpretations in both domains just like sampling in traditional DSP. To achieve it, we introduce a spectral shift $M$ acting in the spectral domain rather than shift $A$ that acts in the vertex domain. This leads to a GSP theory that starts from the spectral domain, for example, linear shift invariant (LSI) filtering in the spectral domain is with polynomials $P(M)$. We then develop GSP vertex and spectral domain dual versions for each of the four standard sampling steps of subsampling, decimation, upsampling, and interpolation. We show how GSP sampling reduces to DSP sampling when the graph is the directed time cycle graph. Simple examples illustrate the impact of choices that are available in GSP sampling.

Progressive coherence and spectral norm minimization scheme for measurement matrices in compressed sensing

Compressed sensing is a novel signal sampling theory based on the sparsity of signals. To recover a sparse signal from fewer measurements, it is desired that measurement matrices have low coherence and low spectral norm. For this purpose, a progressive coherence and spectral norm minimization (PCSNM) scheme is proposed by dividing the optimization problem into several ℓ∞-minimization problems with orthogonality constraints. Utilizing the smoothing technique and penalty function method, these non-smooth constrained problems are transformed into unconstrained approximate optimization problems and solved by a gradient-type based alternating minimization approach. Simulation results demonstrate that the proposed scheme outperforms some previous methods, especially in decreasing the spectral norm.

Graph Signal Sampling and Interpolation Based on Clusters and Averages

We consider a disjoint cover (partition) of an undirected weighted finite or infinite graph G by J connected subgraphs (clusters) $$\{S_{j}\}_{j\in J}$$ and select functions $$\psi _{j}$$ on each of the clusters. For a given signal f on G the set of its weighted average values samples is defined via inner products $$\{\langle f, \psi _{j}\rangle \}_{j\in J}$$ . The main results of the paper are based on Poincare-type inequalities that we introduce and prove. These inequalities provide an estimate of the norm of the signal f on the entire graph G from sets of samples of f and its local gradient on each of the subgraphs. This allows us to establish discrete Plancherel-Polya-type inequalities (or Marcinkiewicz-Zigmund-type or frame inequalities) for signals whose gradients satisfy a Bernstein-type inequality. These results enable the development of a sampling theory for signals on undirected weighted finite or infinite graphs. For reconstruction of the signals from their samples an interpolation theory by weighted average variational splines is developed. Here by a weighted average variational spline we understand a minimizer of a discrete Sobolev norm which takes on the prescribed weighted average values on a set of clusters (in particular, just values on a subset of vertices). Although our approach is applicable to general graphs it’s especially well suited for finite and infinite graphs with multiple clusters. Such graphs are known as community graphs and they find many important applications in materials science, engineering, computer science, economics, biology, and social studies.

Topological Signal Processing Over Simplicial Complexes

The goal of this paper is to establish the fundamental tools to analyze signals defined over a topological space, i.e. a set of points along with a set of neighborhood relations. This setup does not require the definition of a metric and then it is especially useful to deal with signals defined over non-metric spaces. We focus on signals defined over simplicial complexes. Graph Signal Processing (GSP) represents a special case of Topological Signal Processing (TSP), referring to the situation where the signals are associated only with the vertices of a graph. Even though the theory can be applied to signals of any order, we focus on signals defined over the edges of a graph and show how building a simplicial complex of order two, i.e. including triangles, yields benefits in the analysis of edge signals. After reviewing the basic principles of algebraic topology, we derive a sampling theory for signals of any order and emphasize the interplay between signals of different order. Then we propose a method to infer the topology of a simplicial complex from data. We conclude with applications to real edge signals and to the analysis of discrete vector fields to illustrate the benefits of the proposed methodologies.

GPS Signal Acquisition Based on Compressive Sensing and Modified Greedy Acquisition Algorithm

The first task of GPS receiver is to acquire the satellite signal, which tests the all likely 2-D hypotheses of the code phase and the Doppler frequency of the incoming GPS signal. Compressive sensing is a new signal sampling theory, for sparse signal, the signal can be sampled at a lower sampling rate than Nyquist sampling theory. In this paper, a new GPS signal acquisition method is proposed based on compressive sensing. We designed a GPS signal sparse representation dictionary according to the satellite; the position of sparse spike corresponds to the code phase and Doppler frequency. The modified greedy acquisition algorithm that is different from the compressive sensing signal reconstruction, which is used to acquire the GPS satellite. The numerical simulations illustrate that the new method can efficiently acquire the satellite with artificial or real GPS signal. The new method can compress the GPS signal greatly and reduce the transmission energy consumption accordingly, which is hoped to be applied to aircraft or animal monitoring that have little interference noise and need to record GPS data for a long time.

Sampling and Reconstruction of Multiband Signals in Multiresolution Subspaces Associated with the Fractional Wavelet Transform

The fractional wavelet transform (FrWT), which generalizes the ordinary wavelet transform, is a very promising tool for signal processing and analysis. A sampling theory for signals in FrWT-based multiresolution subspaces has been proposed recently. However, the signals to be sampled are regarded as lowpass signals indiscriminately. When dealing with bandpass or multiband signals, the sampling rate could be unnecessarily high. This letter investigates the sampling theory of multiband signals in multiresolution subspaces associated with the FrWT. The nonideal sampling and reconstruction scheme we proposed can guarantee lower sampling rate, fewer sampling points but subequal reconstruction performances. Numerical experiments are presented to validate the theoretical derivations.

Discrete Signal Processing on Graphs: Sampling Theory

We propose a sampling theory for signals that are supported on either directed or undirected graphs. The theory follows the same paradigm as classical sampling theory. We show that perfect recovery is possible for graph signals bandlimited under the graph Fourier transform. The sampled signal coefficients form a new graph signal, whose corresponding graph structure preserves the first-order difference of the original graph signal. For general graphs, an optimal sampling operator based on experimentally designed sampling is proposed to guarantee perfect recovery and robustness to noise; for graphs whose graph Fourier transforms are frames with maximal robustness to erasures as well as for Erd\H{o}s-R\'enyi graphs, random sampling leads to perfect recovery with high probability. We further establish the connection to the sampling theory of finite discrete-time signal processing and previous work on signal recovery on graphs. To handle full-band graph signals, we propose a graph filter bank based on sampling theory on graphs. Finally, we apply the proposed sampling theory to semi-supervised classification on online blogs and digit images, where we achieve similar or better performance with fewer labeled samples compared to previous work.

An adaptive transpose measurement matrix algorithm for signal reconstruction in compressed sensing

Compressed sensing is a new signal sampling theory put forward in recent years. It can obtain a signal's discrete sample in the condition that the sampling rate of signal is far smaller than the Nyquist sampling rate, and then make the original signal perfectly reconstructed with nonlinear reconstruction algorithm. In this paper, three typical greedy reconstruction algorithms, i.e., orthogonal matching pursuit, regularised orthogonal matching pursuit, and subspace tracking algorithm are compared in terms of reconstruction accuracy, error rate and time of reconstruction. Furthermore, a new transpose measurement matrix reconstruction algorithm is proposed and tested through simulation. The experiments on both simulation and calculation showed that this new algorithm can effectively improve the optimisation of the signal transmission.

Study on Compressed Sensing Recovery Algorithms

Abastruct. Compressive sensing is a novel signal sampling theory under the condition that the signalis sparse or compressible.In this case,the small amount of signal values can be reconstructed when signal is sparse or compressible.This paper has reviewed the idea of OMP,GBP and SP,given algorithms and analyzed the experiment results,suggested some improvements.

Mathematics Approaches in Compressed Sensing

Mathematical approaches refer to make quantitative descriptions, deductions and calculations through the use of mathematics concepts, approaches and techniques, then draw some new conclusions and foresee by mathematical analysis and judgment. In recent years, Compressed Sensing theory (CS) provides solutions in alleviating the huge amount of information demand in the pressure of signal sampling, transmission and storage pressure. It is a novel signal sampling theory under the condition that the signal is compressible or sparse.In this case, the signal can be reconstructed accurately from the small amount of signal values if the signal is sparse or compressible. This paper introduces the CS theory framework and key technical issues, and focus on the analysis of the application of mathematical approaches in three aspects of the signal sparse representation, signal sparse transformation and reconstruction.In the end, Some mathematics problems of compressed sensing to solve are given and further development is pointed out. DOI: http://dx.doi.org/10.11591/telkomnika.v11i9.3303

Exact Feature Extraction Using Finite Rate of Innovation Principles With an Application to Image Super-Resolution

The accurate registration of multiview images is of central importance in many advanced image processing applications. Image super-resolution, for example, is a typical application where the quality of the super-resolved image is degrading as registration errors increase. Popular registration methods are often based on features extracted from the acquired images. The accuracy of the registration is in this case directly related to the number of extracted features and to the precision at which the features are located: images are best registered when many features are found with a good precision. However, in low-resolution images, only a few features can be extracted and often with a poor precision. By taking a sampling perspective, we propose in this paper new methods for extracting features in low-resolution images in order to develop efficient registration techniques. We consider, in particular, the sampling theory of signals with finite rate of innovation and show that some features of interest for registration can be retrieved perfectly in this framework, thus allowing an exact registration. We also demonstrate through simulations that the sampling model which enables the use of finite rate of innovation principles is well suited for modeling the acquisition of images by a camera. Simulations of image registration and image super-resolution of artificially sampled images are first presented, analyzed and compared to traditional techniques. We finally present favorable experimental results of super-resolution of real images acquired by a digital camera available on the market.

Application of basic splines for sampling multidimensional signals of finite duration

We investigate limitations of sampling theory for signals with finite spectra, which are particularly prone to appearance during measurements in multidimensional spaces. We present a method for sampling with several arguments, using basic splines. Fourier transforms, and the Parseval equality. We present an example of adaptive optimization of the sampling grid for a magnetic field that is associated with a spectrum of multidimensional signals having fractal similarity.