A robust cylindrical fitting to point cloud data

Environmental, engineering and industrial modelling of natural features (e.g. trees) and man-made features (e.g. pipelines) requires some form of fitting of geometrical objects such as cylinders, which is commonly undertaken using a least-squares method that—in order to get optimal estimation—assumes normal Gaussian distribution. In the presence of outliers, however, this assumption is violated leading to a Gaussian mixture distribution. This study proposes a robust parameter estimation method, which is an improved and extended form of vector algebraic modelling. The proposed method employs expectation maximisation and maximum likelihood estimation (MLE) to find cylindrical parameters in case of Gaussian mixture distribution. MLE computes the model parameters assuming that the distribution of model errors is a Gaussian mixture corresponding to inlier and outlier points. The parameters of the Gaussian mixture distribution and the membership functions of the inliers and outliers are computed using an expectation maximisation algorithm from the histogram of the model error distribution, and the initial guess values for the model parameters are obtained using total least squares. The method, illustrated by a practical example from a terrestrial laser scanning point cloud, is novel in that it is algebraic (i.e. provides a non-iterative solution to the global maximisation problem of the likelihood function), is practically useful for any type of error distribution model and is capable of separating points of interest and outliers.     

Maximum Likelihood Estimation of Spatial Covariance Parameters

In this paper, the maximum likelihood method for inferring the parameters of spatial covariances is examined. The advantages of the maximum likelihood estimation are discussed and it is shown that this method, derived assuming a multivariate Gaussian distribution for the data, gives a sound criterion of fitting covariance models irrespective of the multivariate distribution of the data. However, this distribution is impossible to verify in practice when only one realization of the random function is available. Then, the maximum entropy method is the only sound criterion of assigning probabilities in absence of information. Because the multivariate Gaussian distribution has the maximum entropy property for a fixed vector of means and covariance matrix, the multinormal distribution is the most logical choice as a default distribution for the experimental data. Nevertheless, it should be clear that the assumption of a multivariate Gaussian distribution is maintained only for the inference of spatial covariance parameters and not necessarily for other operations such as spatial interpolation, simulation or estimation of spatial distributions. Various results from simulations are presented to support the claim that the simultaneous use of maximum likelihood method and the classical nonparametric method of moments can considerably improve results in the estimation of geostatistical parameters.     

Late Neogene Atlantic circulation model

A hydrodynamical-numerical model of the general circulation in the Paleo-Atlantic Ocean is applied to the climatological conditions of the period 3.6 to 2.4 m.a. b.p. For three different stages of this interval representing the transition from a warmer to a cooler climate the circulation and the temperature fields are calculated. Generally, a tendency towards a weakening of the currents is achieved.

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Zusammenfassung Ein hydrodynamisch-numerisches Modell der allgemeinen Zirkulation im Paläoatlantik wird mit den klimatologischen Randbedingungen der Periode 3.6 bis 2.4 m.a. zurück betrieben. Für drei verschiedene Zustände dieses Intervalls, die den Übergang von einer Warmzeit zu einer Kaltzeit darstellen, werden die Zirkulation und die Temperaturverteilung berechnet. Generell ergibt sich ein Trend zu einer Abschwächung des Strömungssystems.

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Résumé Un modèle numérique hydrodynamique de la circulation générale dans l'Océan Paléoatlantique est appliqué aux conditions climatiques de la période comprise entre 3.6 et 2.4 ma. Pour trois moments différents de cet intervalle, représentant la transition entre un climat plus chaud et un climat plus froid, on a calculé la distribution des circulations et des températures. En général, on observe que les courants ont tendance à atténuer.

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3,6 2,4 . , . .

     

A novel RANSAC approach to robustly solve the 3D similarity transformation problem

A novel RANSAC robust estimation technique is presented as an effiecient method for solving the seven-parameter datum transformation problem in the presence of outliers. RANSAC method, which is frequently employed in geodesy, has two sensitive features: (i) the user adjusts some parameters of the algorithm, making it subjective and a rather difficult procedure, and (ii) in its shell, a nonlinear system of equation should be solved repeatedly. In this contribution, we suggest an automatic adjustment strategy for the most important parameter, ‘the threshold value’, based on the ‘early stopping’ principle of the machine-learning technology. Instead of using iterative numerical methods, we propose the use of an algebraic polynomial system developed via a dual-quaternion technique and solved by a non-iterative homotophy method, thereby reducing the computation time considerably. The novelty of the proposed approach lies in three major contributions: (i) the provision for automatically finding the proper error limit parameter for RANSAC method, which has until now been a trial-and-error technique; (ii) employing the algebraic polynomial form of the dual-quaternion solution in the RANSAC shell, thereby accelerating the repeatedly requested solution process; and (iii) avoiding iterations via a heuristic approach of the scaling parameter. To illustrate the proposed method, the transformation parameters of the Western Australian Geodetic Datum (AGD 84) to Geocentric Datum Australia (GDA 94) are computed.     

On the condition number of covariance matrices in kriging, estimation, and simulation of random fields

The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from best to worst conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed.     

Comparison of structural and least-squares lines for estimating geologic relations

Two different goals in fitting straight lines to data are to estimate a true linear relation (physical law) and to predict values of the dependent variable with the smallest possible error. Regarding the first goal, a Monte Carlo study indicated that the structural-analysis (SA) method of fitting straight lines to data is superior to the ordinary least-squares (OLS) method for estimating true straight-line relations. Number of data points, slope and intercept of the true relation, and variances of the errors associated with the independent (X) and dependent (Y) variables influence the degree of agreement. For example, differences between the two line-fitting methods decrease as error in X becomes small relative to error in Y. Regarding the second goal—predicting the dependent variable—OLS is better than SA. Again, the difference diminishes as X takes on less error relative to Y. With respect to estimation of slope and intercept and prediction of Y, agreement between Monte Carlo results and large-sample theory was very good for sample sizes of 100, and fair to good for sample sizes of 20. The procedures and error measures are illustrated with two geologic examples.     

Some exact sampling distributions for variogram estimators

For equally spaced observations from a one-dimensional, stationary, Gaussian random function, the characteristic function of the usual variogram estimator for a fixed lag k is derived. Because the characteristic function and the probability density function form a Fourier integral pair, it is possible to tabulate the sampling distribution of a function of a using either analytic or numerical methods. An example of one such tabulation is given for an underlying model that is simple transitive.