Deterministic Sampling Scheme Research Articles

Sparsity-driven deterministic sampling strategy for coded aperture x-ray computed tomography

The sub-sampling strategies in x-ray computed tomography (CT) have gained importance due to their practical relevance. In this direction of research, also known as coded aperture x-ray computed tomography (CAXCT), both random and deterministic strategies have been proposed in the literature. Of the techniques available, those based on compressive sensing (CS) have recently gained more traction because CS-based ideas efficiently exploit inherent duplication present in the system. The quality of the reconstructed CT images nevertheless depends on the sparse signal recovery properties of the sub-sampled Radon matrices. In the present work, we optimize CAXCT deterministically in such a way that the corresponding sub-sampled Radon matrices remain close to the equiangular tight frames for better numerical behaviour. We show that this optimization, involving the Khatri–Rao product, leads to a non-negative sparse approximation. While comparing and contrasting our method with its existing counterparts, we show that the proposed algorithm is computationally less involved. Finally, we demonstrate efficacy of the proposed deterministic sub-sampling strategy in recovering CT images in both noiseless and noisy cases.

A Discrete Quaternion Particle Filter Based on Deterministic Sampling for IMU Attitude Estimation

A discrete quaternion filtering algorithm following high-density uncertainty matching-based deterministic sampling scheme is proposed for IMU attitude estimation. Existing quaternion filters rely on the specific distributions directly or the on-tangent-plane Gaussian distribution indirectly to model the underlying uncertainty. The key idea of this paper is to apply a Dirac mixture at few deterministic weighted samples for non-parametric interpreting the high-density uncertainty of unit quaternion manifold. In contrast to standard Monte Carlo techniques based on random sampling, the proposed approach is deterministic. Furthermore, the deterministic quaternion particles are obtained by projecting on-sphere uniform points to the unit quaternion manifold via an exponential map. The on-manifold quaternions are then exploited in a discrete Bayesian filtering procedure. We compare the new filter with the genetic algorithm embedded quaternion particle filter (GA-QPF) and the unscented quaternion estimator (USQUE) for accuracy and computation cost by using simulated and practical IMU data. The results illustrate that the proposed estimator possesses lower computational cost than the GA-QPF while providing a comparable accuracy, higher accuracy than the USQUE, as well as more effective against large initialization error.

Inversion and precision estimation of earthquake fault parameters based on scaled unscented transformation and hybrid PSO/Simplex algorithm with GPS measurement data

The Scaled Unscented Transformation (SUT) method based on deterministic sampling strategy is introduced to nonlinear inversion and precision estimation of earthquake fault parameters. Firstly, the SUT method is used to estimate the precision of the curve fitting parameters obtained by Particle Swarm Optimization (PSO) algorithm and compared with least square method and Monte Carlo method to verify the effectiveness of the proposed method. Considering the inevitable error in the observations, we preprocess the simulated GPS data to obtain the approximate mean of observations, then contrast and analyze the fault parameter estimatesand precision information obtained by SUT method and Monte Carlo method based on the hybrid PSO/Simplex algorithm (MPSO). The influence of the “adjusted” observations, positions and number of the observation points and the level of noise added to deformation observations are also considered. Finally, the methods in this paper are applied to the Lushan (China) earthquake. The experimental results show that both SUT method and Monte Carlo method can obtain better parameter estimates with “adjusted” observations; SUT method can obtain more accurate precision information of fault parameters; Monte Carlo method can only judge mean square errors of parameters from the order of magnitude and the correlations between parameters are not accurate; When using GPS measurement data to invert the fault parameters, the positions and number of GPS points have certain influence on some fault parameters; When the noise added to the GPS measurement data increases, the fitting of the fault parameter estimate become worse and the precision of fault parameter estimate decreases.

A Deterministic Far-Field Sampling Strategy for Array Diagnosis Using Sparse Recovery

A deterministic sampling strategy for failure identification in uniform linear arrays using compressed sensing/sparse recovery technique is proposed. Numerical results confirm the good performance of the technique described in this letter compared to the mean performance of the method based on random sampling strategy.

A Novel Particle Filter Based on Hybrid Deterministic and Random Sampling

Particle filters are vital tools to solve the nonlinear/non-Gaussian filtering problems. However, it suffers from the problems of sample degeneracy and impoverishment. This paper proposes a framework of particle filter with a hybrid sampling strategy, generates particles by means of deterministic and random sampling, and then, restores the fixed number particles by a resampling method. Compared with traditional sampling strategies in the particle filters, the hybrid sampling framework is more suitable for the characteristics of a time-varying system. In order to demonstrate the effectiveness of the proposed framework, a novel particle filter based on hybrid deterministic and random sampling (HDRSPF) is designed, where the deterministic sampling strategy is UT transform and the random sampling strategy is Mont Carlo sampling. Simulation results of single-dimensional and multi-dimensional scenarios show that the proposed filter HDRSPF has a higher filtering accuracy at the cost of lower computational complexity as compared with the existing methods.

Line Spectral Estimation Based on Compressed Sensing with Deterministic Sub-Nyquist Sampling

As an alternative to the traditional sampling theory, compressed sensing allows acquiring much smaller amount of data, still estimating the spectra of frequency-sparse signals accurately. However, the previous methods focus more on single measurement vector and default to using impractical random sampling. In this paper, we employ a deterministic and simple sampling scheme, that is, using three sub-Nyquist channels with pairwise coprime undersampling ratios. A multi-task model is utilized to unite the three-channel samples and address the multiple measurement vector problem. The model is solved by the proposed complex-valued multi-task algorithm based on variational Bayesian inference. Simulations show that this method is feasible and robust at quite low sampling rates.

Effectively Subsampled Quadratures for Least Squares Polynomial Approximations

This paper proposes a new deterministic sampling strategy for constructing polynomial chaos approximations for expensive physics simulation models. The proposed approach, effectively subsampled quadratures, involves sparsely subsampling an existing tensor grid using QR column pivoting. For polynomial interpolation using hyperbolic or total order sets, we then solve the following square least squares problem. For polynomial approximation, we use a column pruning heuristic that removes columns based on the highest total orders and then solves the tall least squares problem. While we provide bounds on the condition number of such tall submatrices, it is difficult to ascertain how column pruning affects solution accuracy as this is problem specific. We conclude with numerical experiments on an analytical function and a model piston problem that show the efficacy of our approach compared with randomized subsampling. We also show an example where this method fails.

Unscented von Mises–Fisher Filtering

We introduce the unscented von Mises–Fisher filter (UvMFF), a nonlinear filtering algorithm for dynamic state estimation on the $n$ -dimensional unit hypersphere. Estimation problems on the unit hypersphere occur in computer vision, e.g., when using omnidirectional cameras, as well as in signal processing. As approaches in literature are limited to very simple system and measurement models, we propose a deterministic sampling scheme on the unit hypersphere, which allows us to handle nonlinear system and measurement models. The proposed approach can be seen as a hyperspherical variant of the unscented Kalman filter (UKF). The advantages of the novel method are shown by means of simulations.

Unscented transform-based uncertainty analysis of rotating coil transducers for field mapping.

The uncertainty of a rotating coil transducer for magnetic field mapping is analyzed. Unscented transform and statistical design of experiments are combined to determine magnetic field expectation, standard uncertainty, and separate contributions of the uncertainty sources. For nonlinear measurement models, the unscented transform-based approach is more error-proof than the linearization underlying the "Guide to the expression of Uncertainty in Measurements" (GUMs), owing to the absence of model approximations and derivatives computation. When GUM assumptions are not met, the deterministic sampling strategy strongly reduces computational burden with respect to Monte Carlo-based methods proposed by the Supplement 1 of the GUM. Furthermore, the design of experiments and the associated statistical analysis allow the uncertainty sources domain to be explored efficiently, as well as their significance and single contributions to be assessed for an effective setup configuration. A straightforward experimental case study highlights that a one-order-of-magnitude reduction in the relative uncertainty of the coil area produces a decrease in uncertainty of the field mapping transducer by a factor of 25 with respect to the worst condition. Moreover, about 700 trials and the related processing achieve results corresponding to 5 × 10(6) brute-force Monte Carlo simulations.

Unscented Orientation Estimation Based on the Bingham Distribution

Orientation estimation for 3D objects is a common problem that is usually tackled with traditional nonlinear filtering techniques such as the extended Kalman filter (EKF) or the unscented Kalman filter (UKF). Most of these techniques assume Gaussian distributions to account for system noise and uncertain measurements. This distributional assumption does not consider the periodic nature of pose and orientation uncertainty. We propose a filter that considers the periodicity of the orientation estimation problem in its distributional assumption. This is achieved by making use of the Bingham distribution, which is defined on the hypersphere and thus inherently more suitable to periodic problems. Furthermore, handling of non-trivial system functions is done using deterministic sampling in an efficient way. A deterministic sampling scheme reminiscent of the UKF is proposed for the nonlinear manifold of orientations. It is the first deterministic sampling scheme that truly reflects the nonlinear manifold of the orientation.

Deterministic multidimensional nonuniform gap sampling

Born from empirical observations in nonuniformly sampled multidimensional NMR data relating to gaps between sampled points, the Poisson-gap sampling method has enjoyed widespread use in biomolecular NMR. While the majority of nonuniform sampling schemes are fully randomly drawn from probability densities that vary over a Nyquist grid, the Poisson-gap scheme employs constrained random deviates to minimize the gaps between sampled grid points. We describe a deterministic gap sampling method, based on the average behavior of Poisson-gap sampling, which performs comparably to its random counterpart with the additional benefit of completely deterministic behavior. We also introduce a general algorithm for multidimensional nonuniform sampling based on a gap equation, and apply it to yield a deterministic sampling scheme that combines burst-mode sampling features with those of Poisson-gap schemes. Finally, we derive a relationship between stochastic gap equations and the expectation value of their sampling probability densities.

Uncovering Representative Groups in Multidimensional Projections

AbstractMultidimensional projection‐based visualization methods typically rely on clustering and attribute selection mechanisms to enable visual analysis of multidimensional data. Clustering is often employed to group similar instances according to their distance in the visual space. However, considering only distances in the visual space may be misleading due to projection errors as well as the lack of guarantees to ensure that distinct clusters contain instances with different content. Identifying clusters made up of a few elements is also an issue for most clustering methods. In this work we propose a novel multidimensional projection‐based visualization technique that relies on representative instances to define clusters in the visual space. Representative instances are selected by a deterministic sampling scheme derived from matrix decomposition, which is sensitive to the variability of data while still been able to handle classes with a small number of instances. Moreover, the sampling mechanism can easily be adapted to select relevant attributes from each cluster. Therefore, our methodology unifies sampling, clustering, and feature selection in a simple framework. A comprehensive set of experiments validate our methodology, showing it outperforms most existing sampling and feature selection techniques. A case study shows the effectiveness of the proposed methodology as a visual data analysis tool.

On the Sampling of the Electromagnetic Field Radiated by Sparse Sources

The aim of this paper is to discuss the effectiveness of random sampling of the electromagnetic field in problems involving sparse radiating sources. The method adopted in this paper uses an intuitive geometrical approach based on the Johnson-Linderstrauss lemma applied to the $\epsilon$ -net of the set of the fields radiated by the sparse source. The paper shows that a proper Non Uniform Random Sampling (NURS) using some a-priori information on the problem allows a significant reduction in the cardinality of the set of the measured data compared to uniform random sampling and to random sampling from a $\lambda/2$ equispaced set of data. It is also shown that there is a limiting number of data after which random sampling is not convenient. In this case a deterministic sampling strategy, called Modified Non Uniform Sampling (MNUS), is proposed. Numerical results on linear sparse arrays confirm the effectiveness of NURS and MNUS.

A comparison of six sampling schemes for price index construction in a COICOP food group

This article compares the behaviour of sampling techniques for price indices using a scanner data set as a model population. Indices produced by two purposive deterministic cut-off designs and four probabilistic sampling schemes are compared with each other and with the ‘true’ population index from the whole data set. We found that the two deterministic cut-off sampling schemes show much different behaviour from the probabilistic sampling schemes. This is not unexpected, as the former schemes have a very restricted focus with respect to the variety of products. We also found that the probabilistic schemes are generally closer to each other and the ‘true’ value than the deterministic cut-off designs. The jackknife resampling technique is also explored as a means of estimating the SE of the index and compared with the actual results from repeated sampling.

An estimator for the quadratic covariation of asynchronously observed Itô processes with noise: Asymptotic distribution theory

The article is devoted to the nonparametric estimation of the quadratic covariation of non-synchronously observed Itô processes in an additive microstructure noise model. In a high-frequency setting, we aim at establishing an asymptotic distribution theory for a generalized multiscale estimator including a feasible central limit theorem with optimal convergence rate on convenient regularity assumptions. The inevitably remaining impact of asynchronous deterministic sampling schemes and noise corruption on the asymptotic distribution is precisely elucidated. A case study for various important examples, several generalizations of the model and an algorithm for the implementation warrant the utility of the estimation method in applications.

A haplotype inference algorithm for trios based on deterministic sampling

BackgroundIn genome-wide association studies, thousands of individuals are genotyped in hundreds of thousands of single nucleotide polymorphisms (SNPs). Statistical power can be increased when haplotypes, rather than three-valued genotypes, are used in analysis, so the problem of haplotype phase inference (phasing) is particularly relevant. Several phasing algorithms have been developed for data from unrelated individuals, based on different models, some of which have been extended to father-mother-child "trio" data.ResultsWe introduce a technique for phasing trio datasets using a tree-based deterministic sampling scheme. We have compared our method with publicly available algorithms PHASE v2.1, BEAGLE v3.0.2 and 2SNP v1.7 on datasets of varying number of markers and trios. We have found that the computational complexity of PHASE makes it prohibitive for routine use; on the other hand 2SNP, though the fastest method for small datasets, was significantly inaccurate. We have shown that our method outperforms BEAGLE in terms of speed and accuracy for small to intermediate dataset sizes in terms of number of trios for all marker sizes examined. Our method is implemented in the "Tree-Based Deterministic Sampling" (TDS) package, available for download at http://www.ee.columbia.edu/~anastas/tdsConclusionsUsing a Tree-Based Deterministic sampling technique, we present an intuitive and conceptually simple phasing algorithm for trio data. The trade off between speed and accuracy achieved by our algorithm makes it a strong candidate for routine use on trio datasets.

On numerical properties of the ensemble Kalman filter for data assimilation

Ensemble Kalman filter (EnKF) has been widely used as a sequential data assimilation method, primarily due to its ease of implementation resulting from replacing the covariance evolution in the traditional Kalman filter (KF) by an approximate Monte Carlo ensemble sampling. In this paper rigorous analysis on the numerical errors of the EnKF is conducted in a general setting. Error bounds are provided and convergence of the EnKF to the exact Kalman filter is established. The analysis reveals that the ensemble errors induced by the Monte Carlo sampling can be dominant, compared to other errors such as the numerical integration error of the underlying model equations. Methods to reduce sampling errors are discussed. In particular, we present a deterministic sampling strategy based on cubature rules (qEnKF) which offers much improved accuracy. The analysis also suggests a less obvious fact — more frequent data assimilation may lead to larger numerical errors of the EnKF. Numerical examples are provided to verify the theoretical findings and to demonstrate the improved performance of the qEnKF.

Asymptotic distributions of M-estimators in a spatial regression model under some fixed and stochastic spatial sampling designs

In this paper, we consider M-estimators of the regression parameter in a spatial multiple linear regression model. We establish consistency and asymptotic normality of the M-estimators when the data-sites are generated by a class of deterministic as well as a class of stochastic spatial sampling schemes. Under the deterministic sampling schemes, the data-sites are located on a regular grid but may have aninfill component. On the other hand, under the stochastic sampling schemes, locations of the data-sites are given by the realizations of a collection of independent random vectors and thus, are irregularly spaced. It is shown that scaling constants of different orders are needed for asymptotic normality under different spatial sampling schemes considered here. Further, in the stochastic case, the asymptotic covariance matrix is shown to depend on the spatial sampling density associated with the stochastic design. Results are established for M-estimators corresponding to certain non-smooth score functions including Huber’s ψ-function and the sign functions (corresponding to the sample quantiles).

Quasi-random algorithms for real-time spacecraft motion planning and coordination

In this paper we address the problem of planning in real time the motion of one or more spacecraft, in the presence of a variety of constraints, including collision avoidance with static obstacles or other maneuvering spacecraft, and plume impingement avoidance. Such problems are very challenging from the computational point of view, and push the limits of currently-available planning algorithms, based on randomization, which provide computational tractability by relaxing the completeness requirement to probabilistic completeness. In this paper we apply to the spacecraft maneuvering problem a set of recently-developed quasi-random algorithms, which, combiding optimal orbital maneuvers and deterministic sampling strategies, are able to provide extremely fast and efficient planners. Moreover, the proposed algorithms are able to guarantee the safety of the space system, that is, the satisfaction of collision and plume impingement avoidance constraints, even in the face of finite computation times (i.e., when the planner has to be pre-empted). Examples of orbital maneuvering close to large structures will illustrate the effectiveness of the methods, and a discussion of the results will conclude the paper.

Fast parallel algorithm for distance transform

We present an O((log log N)/sup 2/) -time algorithm for computing the distance transform of an N /spl times/ N binary image. Our algorithm is designed for the common concurrent read concurrent write parallel random access machine (CRCW PRAM) and requires O(N/sup 2+/spl epsi///log log N) processors, for any /spl epsi/ such that 0 < /spl epsi/ < 1. Our algorithm is based on a novel deterministic sampling scheme and can be used for computing distance transforms for a very general class of distance functions. We also present a scalable version of our algorithm when the number of processors is available p/sup 2+/spl epsi///log log p for some p < N. In this case, our algorithm runs in O((N/sup 2//p/sup 2/)+(N/p) log log p + (log log p)/sup 2/) time. This scalable algorithm is more practical since usually the number of available processors is much less than the size of the image.