Exact recovery of higher order moments

This correspondence addresses the problem of exact recovery of higher order moments of unquantized signals from those of their quantized counterparts, in the context of nonsubtractive dithered quantization. It introduces a new statistical characterization of the dithered quantizer in the form of a pth-order moment-sense input/ouput function hp (x). A class of signals for which the solution to the exact moment recovery problem is guaranteed is defined, and some of its key properties are stated and proved. Two approaches to this problem are discussed and the practical gains accruing from the 1-bit implementation of the second approach are highlighted. Finally, a fruitful extension of this work to the exact recovery of cumulants is briefly pointed ou

On the exact recovery of higher-order moments of noisy signals

The importance of moments in science and engineering, as witnessed by the continuous and wide applicability of second-order moments (correlations) and the use of their higher-order brethren is clearly unquestionable. Due to the predominance of digital, rather than analogue, signal processing, it is of practical importance to investigate the impact of amplitude quantization on the exact recovery of unquantized moments from their quantized counterparts. We extend the results of Cheded (see IEEE ICASSP'95, p.1816-19, Detroit, USA) to the more general and interesting case where no a priori knowledge of the quantizer input's membership of the class Lp is available. We introduce a new moment-sense input/output function hp(x) that statistically characterizes the quantizer. Two new theorems are also stated that solve the exact moment recovery problem. Finally, two approaches to this problem are presented with some simulation results: based on a 1-bit quantizer, that substantiate very well the theor

On the exact recovery of higher-order moments of noisy signals

The importance of moments in science and engineering, as witnessed by the continuous and wide applicability of second-order moments (correlations) and the use of their higher-order brethren is clearly unquestionable. Due to the predominance of digital, rather than analogue, signal processing, it is of practical importance to investigate the impact of amplitude quantization on the exact recovery of unquantized moments from their quantized counterparts. We extend the results of Cheded (see IEEE ICASSP'95, p.1816-19, Detroit, USA) to the more general and interesting case where no a priori knowledge of the quantizer input's membership of the class Lp is available. We introduce a new moment-sense input/output function hp(x) that statistically characterizes the quantizer. Two new theorems are also stated that solve the exact moment recovery problem. Finally, two approaches to this problem are presented with some simulation results: based on a 1-bit quantizer, that substantiate very well the theor

On the exact recovery of cumulants

The importance of higher-order statistics (HOS) is clearly reflected in their increasingly wide applicability. However, because of the high dimensionality involved, the digital estimation of HOS is still plagued by heavy computational loads which tend to severely reduce the potential for real-time applications. This paper proposes a solution to this problem, based on a successful exact moments recovery (EMR) theory which makes it possible to exploit the attractive practical advantages of 1-bit quantization schemes while avoiding their associated large quantization errors. Two new theorems on the variance aspects of 3 unbiased linear sample moment estimators are also presented. Finally, some simulation results on the recovery of cumulants, which are in very good agreement with the theory, are include

Sampling and Reconstruction of Shapes with Algebraic Boundaries

We present a sampling theory for a class of binary images with finite rate of innovation (FRI). Every image in our model is the restriction of \mathds{1}_{\{p\leq0\}} to the image plane, where \mathds{1} denotes the indicator function and $p$ is some real bivariate polynomial. This particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial $p$. We show that the image parameters --i.e., the polynomial coefficients-- satisfy a set of linear annihilation equations with the coefficients being the image moments. The inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels. As a remedy to these problems, we replace conventional moments with more stable \emph{generalized moments} that are adjusted to the given sampling kernel. The benefits are threefold: (1) it relaxes the requirements on the sampling kernels, (2) produces annihilation equations that are robust at numerical precision, and (3) extends the results to images with unbounded boundaries. We further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points, and sequentially enforcing measurement consistency. We consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images, including low to moderate noise levels and a range of realistic sampling kernels.Comment: 12 pages, 14 figure

Quadrature-based Lattice Boltzmann Model for Relativistic Flows

A quadrature-based finite-difference lattice Boltzmann model is developed that is suitable for simulating relativistic flows of massless particles. We briefly review the relativistc Boltzmann equation and present our model. The quadrature is constructed such that the stress-energy tensor is obtained as a second order moment of the distribution function. The results obtained with our model are presented for a particular instance of the Riemann problem (the Sod shock tube). We show that the model is able to accurately capture the behavior across the whole domain of relaxation times, from the hydrodynamic to the ballistic regime. The property of the model of being extendable to arbitrarily high orders is shown to be paramount for the recovery of the analytical result in the ballistic regime.Comment: 6 pages, 2 figures, proceedings of TIM 15-16 conference (26-28 May 2016, Timisoara, Romania

Propagation of epistemic uncertainty in queueing models with unreliable server using chaos expansions

In this paper, we develop a numerical approach based on Chaos expansions to analyze the sensitivity and the propagation of epistemic uncertainty through a queueing systems with breakdowns. Here, the quantity of interest is the stationary distribution of the model, which is a function of uncertain parameters. Polynomial chaos provide an efficient alternative to more traditional Monte Carlo simulations for modelling the propagation of uncertainty arising from those parameters. Furthermore, Polynomial chaos expansion affords a natural framework for computing Sobol' indices. Such indices give reliable information on the relative importance of each uncertain entry parameters. Numerical results show the benefit of using Polynomial Chaos over standard Monte-Carlo simulations, when considering statistical moments and Sobol' indices as output quantities