Finite covariance functions

Because of the full covariance matrices and the computer storage limitations the number of measurements which can be handled by the collocation method simultaneously, is limited. This paper presents a method to compute covariance functions with a finite support yielding sparse covariance matrices. The theoretical background is pointed out and, for the one- and two-dimensional case, special functions are developed which can be combined with the usually used covariance functions to get a “finite covariance function”. Simulated examples to demonstrate the behaviour of different solution methods to solve these special, sparse covariance matrices supplement our investigations.     

An elliptical basis function network for classification of remote sensing images

An elliptical basis function (EBF) network is employed in this study for the classification of remotely sensed images. Though similar in structure, the EBF network differs from the well-known radial basis function (RBF) network by incorporating full covariance matrices and employing the expectation-maximization (EM) algorithm to estimate the basis functions. Since remotely sensed data often take on mixture-density distributions in the feature space, the network not only possesses the advantage of the RBF mechanism, but also utilizes the EM algorithm to compute the maximum likelihood estimates of the mean vectors and covariance matrices of a Gaussian mixture distribution in the training phase. Experimental results show that the EM-based EBF network is more effective in training and simpler in structure than an RBF network constructed for the same task.The research was supported by grant 40101021 from the Natural Science Foundation of China, and grant 2002AA135230 from Hi-Tech research and development program of China. The authors would like to thank the reviewers for their valuable comments.     

Multi-view dense matching supported by triangular meshes

We present a new procedure to compute dense 3D point clouds from a sequential set of images. This procedure is considered as a second step of a three-step algorithm for 3D reconstruction from image sequences, whose first step consists of image orientation and the last step is shape reconstruction. We assume that the camera matrices as well as a sparse set of 3D points are available and we strive for obtaining a dense and reliable 3D point cloud. Three novel ideas are presented: (1) for sparse tracking and triangulation, the search space for correspondences is reduced to a line segment by means of known camera matrices and disparity ranges are provided by triangular meshes from the already available points; (2) triangular meshes from extended sets of points are used for dense matching, because these meshes help to reconstruct points in weakly textured areas and present a natural way to obtain subpixel accuracy; (3) two non-local optimization methods, namely, 1D dynamic programming along horizontal lines and semi-global optimization were employed for refinement of local results obtained from an arbitrary number of images. All methods were extensively tested on a benchmark data set and an infrared video sequence. Both visual and quantitative results demonstrate the effectiveness of our algorithm.     

Approximating covariance matrices estimated in multivariate models by estimated auto- and cross-covariances

Quantities like tropospheric zenith delays or station coordinates are repeatedly measured at permanent VLBI or GPS stations so that time series for the quantities at each station are obtained. The covariances of these quantities can be estimated in a multivariate linear model. The covariances are needed for computing uncertainties of results derived from these quantities. The covariance matrix for many permanent stations becomes large, the need for simplifying it may therefore arise under the condition that the uncertainties of derived results still agree. This is accomplished by assuming that the different time series of a quantity like the station height for each permanent station can be combined to obtain one time series. The covariance matrix then follows from the estimates of the auto- and cross-covariance functions of the combined time series. A further approximation is found, if compactly supported covariance functions are fitted to an estimated autocovariance function in order to obtain a covariance matrix which is representative of different kinds of measurements. The simplification of a covariance matrix estimated in a multivariate model is investigated here for the coordinates of points of a grid measured repeatedly by a laserscanner. The approximations are checked by determining the uncertainty of the sum of distances to the points of the grid. To obtain a realistic value for this uncertainty, the covariances of the measured coordinates have to be considered. Three different setups of measurements are analyzed and a covariance matrix is found which is representative for all three setups. Covariance matrices for the measurements of laserscanners can therefore be determined in advance without estimating them for each application.     

Combination of spatial networks using an estimated covariance matrix

When combining satellite and terrestrial networks, covariance matrices are used which have been estimated from previous data. It can be shown that the least-squares estimator of the unknown parameters using such an estimated covariance matrix is not necessarily the best. There are a number of cases where a more efficient estimator can be obtained in a different way. The problem occurs frequently in geodesy, since in least-squares adjustment of correlated observations estimated covariance matrices are often used. If the general structure of the covariance matrix is known, results can often be improved by a method called covariance adjustment. The statistical model used in least-squares collocation leads to a type of covariance matrix which fits into this framework. It is shown in which way improvements can be made using a modified approach of principal component analysis. As a numerical example the combination of a satellite and a terrestrial network has been computed with varying assumptions on the covariance matrix. It is shown which types of matrices are critical and where the usual least-squares approach can be applied without hesitation. Finally, a simplified representation of covariances for spatial networks by means of a suitable covariance function is suggested. Paper presented at the International Symposium on Computational Methods in Geometrical Geodesy-Oxford, 2–8 September, 1973.     

高分五号高光谱影像的关联对齐域适应与分类

高分五号(GF-5)搭载的高光谱传感器兼顾宽覆盖和高分辨率的特性,但在实际应用中宽覆盖范围内各种地物类别的标注十分困难。当标记样本很少甚至没有标记样本时,遥感图像分类异常困难。此时,可以采用域适应方法,借助已标记的历史数据(源域)实现对未标记数据(目标域)的分类。本文提出了一种基于稀疏矩阵变换的关联对齐域适应分类算法。首先,利用稀疏矩阵变换估计源域和目标域的协方差矩阵;然后,运用协方差关联对齐方法估计源域到目标域的变换矩阵;接着,运用估计得到的变换矩阵将源域数据进行变换,使得其与目标域对齐;最后,在变换后的源域数据上建立分类器,实现对目标域数据的分类。本文的算法在两个真实的GF-5高光谱数据集上进行了验证。实验结果表明,本文算法要优于常用的子空间对齐算法和关联对齐算法。特别地,在黄河口GF-5数据上,本文算法比原始关联对齐方法的最近邻分类准确率提升了3.5%,支持向量机分类准确率提升了2.3%。     

Shaping filters and the upward continuation problem

An attempt is made to bridge the gap between closed-form harmonic upward continuation (HUC) of analytic covariance functions of the disturbing potential of the anomalous local gravity field and the numerical shaping filter construction when the local gravity vector is modelled in the framework of Kalman filtering. Some fundamental concepts of the local gravity field, interpreted as a stochastic process that is stationary in the plane and harmonic in the upper half space, are reviewed. The shaping-filter modelling technique for the local gravity vector is introduced. To determine the relation between the disturbing potential covariance function and the gravity vector covariance matrix, the role of the so-called admissible pair is established. It is shown that rescaling an admissible pair leads to an analogue rescaling of the shaping filter matrices derived hereof; no cumbersome numerical recalculations are necessary. The class of covariance functions whose corresponding shaping filters possess a closed-form HUC are identified as models whose HUC can be interpreted as a rescaling. Received: 17 December 1997 / Accepted: 7 September 1998     

A unified scheme for processing GPS dual-band phase observations

A unified computational scheme is presented for sequential least-squares processing ofGPS dual-band carrier-beat-phase observations in network-mode positioning with orbit relaxation, and in orbit determination applications. This scheme is applicable to any spatial and temporal distribution of stations and satellites during a particularGPS experiment. Full covariance matrices can be specified for carrier-beat-phases and for weighted constraints on the ionosphere in order to improve phase ambiguity resolution. Physically meaningful choices for these covariance matrices are developed.     

A general framework for error analysis in measurement-based GIS Part 3: Error analysis in intersections and overlays

This is the third of a four-part series on the development of a general framework for error analysis in measurement-based geographic information systems (MBGIS). In this paper, we study the characteristics of error structures in intersections and polygon overlays. When locations of the endpoints of two line segments are in error, we analyze errors of the intersection point and obtain its error covariance matrix through the propagation of the error covariance matrices of the endpoints. An approximate law of error propagation for the intersection point is formulated within the MBGIS framework. From simulation experiments, it appears that both the relative positioning of two line segments and the error characteristics of the endpoints can affect the error characteristics of the intersection. Nevertheless, the approximate law of error propagation captures nicely the error characteristics under various situations. Based on the derived results, error analysis in polygon-on-polygon overlay operation is also performed. The relationship between the error covariance matrices of the original polygons and the overlaid polygons is approximately established.This project was supported by the earmarked grant CUHK 4362/00H of the Hong Kong Research grants Council.     

An alternative method for non-negative estimation of variance components

A typical problem of estimation principles of variance and covariance components is that they do not produce positive variances in general. This caveat is due, in particular, to a variety of reasons: (1) a badly chosen set of initial variance components, namely initial value problem (IVP), (2) low redundancy in functional model, (3) an improper stochastic model, and (4) data’s possibility of containing outliers. Accordingly, a lot of effort has been made in order to design non-negative estimates of variance components. However, the desires on non-negative and unbiased estimation can seldom be met simultaneously. Likewise, in order to search for a practical non-negative estimator, one has to give up the condition on unbiasedness, which implies that the estimator will be biased. On the other hand, unlike the variance components, the covariance components can be negative, so the methods for obtaining non-negative estimates of variance components are not applicable. This study presents an alternative method to non-negative estimation of variance components such that non-negativity of the variance components is automatically supported. The idea is based upon the use of the functions whose range is the set of all positive real numbers, namely positive-valued functions (PVFs), for unknown variance components in stochastic model instead of using variance components themselves. Using the PVF could eliminate the effect of IVP on the estimation process. This concept is reparameterized on the restricted maximum likelihood with no effect on the unbiasedness of the scheme. The numerical results show the successful estimation of non-negativity estimation of variance components (as positive values) as well as covariance components (as negative or positive values).     

Error-covariances of the estimates of spherical harmonic coefficients computed by LSC,using second-order radial derivative functionals associated with realistic GOCE orbits

Least-squares collocation may be used for the estimation of spherical harmonic coefficients and their error and error correlations from GOCE data. Due to the extremely large number of data, this requires the use of the so-called method of Fast Spherical Collocation (FSC) which requires that data is gridded equidistantly on each parallel and have the same uncorrelated noise on the parallel. A consequence of this is that error-covariances will be zero except between coefficients of the same signed order (i.e., the same order and the same coefficient type C–C or S–S). If the data distribution and the characteristics of the data noise are symmetric with respect to the equator, then, within a given order and coefficient type, the error-covariances amongst coefficients whose degrees are of different parity also vanish. The deviation from this “ideal” pattern has been studied using data-sets of second order radial derivatives of the anomalous potential. A total number of points below 17,000 were used having an equi-angular or an equal area distribution or being associated with points on a realistic GOCE orbit but close to the nodes of a grid. Also the data were considered having a correlated or an uncorrelated noise and three different signal covariance functions. Grids including data or not including data in the polar areas were used. Using the functionals associated with the data, error estimates of coefficients and error-correlations between coefficients were calculated up to a maximal degree and order equal to 90. As expected, for the data-distributions with no data in the polar areas the error-estimates were found to be larger than when the polar areas contained data. In all cases it was found that only the error-correlations between coefficients of the same order were significantly different from zero (up to 88%). Error-correlations were significantly larger when data had been regarded as having non-zero error-correlations. Also the error-correlations were largest when the covariance function with the largest signal covariance distance was used. The main finding of this study was that the correlated noise has more pronounced impact on gridded data than on data distributed on a realistic GOCE orbit. This is useful information for methods using gridded data, such as FSC.     

Non-stationary covariance function modelling in 2D least-squares collocation

Standard least-squares collocation (LSC) assumes 2D stationarity and 3D isotropy, and relies on a covariance function to account for spatial dependence in the observed data. However, the assumption that the spatial dependence is constant throughout the region of interest may sometimes be violated. Assuming a stationary covariance structure can result in over-smoothing of, e.g., the gravity field in mountains and under-smoothing in great plains. We introduce the kernel convolution method from spatial statistics for non-stationary covariance structures, and demonstrate its advantage for dealing with non-stationarity in geodetic data. We then compared stationary and non- stationary covariance functions in 2D LSC to the empirical example of gravity anomaly interpolation near the Darling Fault, Western Australia, where the field is anisotropic and non-stationary. The results with non-stationary covariance functions are better than standard LSC in terms of formal errors and cross-validation against data not used in the interpolation, demonstrating that the use of non-stationary covariance functions can improve upon standard (stationary) LSC.     

Code and carrier phase based short baseline GPS positioning: computational aspects

A recursive least squares algorithm is presented for short baseline GPS positioning using both carrier phase and code measurements. We take advantage of the structure of the problem to make the algorithm computationally efficient and use orthogonal transformations to ensure that the algorithm is numerically reliable. Details are given for computing position estimates and error covariance matrices with possible satellite rising and setting. Real data test results suggest our algorithm is effective.This research was supported by NSERC of Canada Grant RGPIN217191–99, FCAR of Quebec Grant 2001-NC-66487, and NSERCGEOIDE Network Project ENV#14 for Xiao-Wen Chang, and by NSERC of Canada Grant RGPIN9236–01 for Christopher C. Paige.The online version of the original article can be found at     

ISO2002: an analytical stochastic model of multi-difference GPS carrier-phase data

The differencing technique is useful in global positioning system (GPS) positioning when two or more GPS receivers collect simultaneous observables from common satellites at each epoch, and all carrier-phase observables have the same normal distribution. An analytical probability distribution of the single-, double-, triple- and multi-difference GPS observables is obtained. This analytical model, called ISO2002, has a good matrix structure, in which I indicates the number of receivers, S indicates the number of observed satellites, and O indicates the number of epochs. The variance–covariance matrix can be expressed as the Kronecker product of several small matrices, so its inverse is equal to the Kronecker product of the inverses of these sub-matrices. Moreover, these small matrices are circulant or symmetric diagonal Toeplitz matrices, so their inverses have analytical solutions. The analytical model ISO2002 proposed to compute the inverse variance–covariance matrix is shown to be very effective.     

Gibbs sampler for computing and propagating large covariance matrices

The use of sampling-based Monte Carlo methods for the computation and propagation of large covariance matrices in geodetic applications is investigated. In particular, the so-called Gibbs sampler, and its use in deriving covariance matrices by Monte Carlo integration, and in linear and nonlinear error propagation studies, is discussed. Modifications of this technique are given which improve in efficiency in situations where estimated parameters are highly correlated and normal matrices appear as ill-conditioned. This is a situation frequently encountered in satellite gravity field modelling. A synthetic experiment, where covariance matrices for spherical harmonic coefficients are estimated and propagated to geoid height covariance matrices, is described. In this case, the generated samples correspond to random realizations of errors of a gravity field model. AcknowledgementsThe authors are indebted to Pieter Visser and Pavel Ditmar for providing simulation output that was used in the GOCE error generation experiments. Furthermore, the NASA/NIMA/OSU team is acknowledged for providing public ftp access to the EGM96 error covariance matrix. The two anonymous reviewers are thanked for their valuable comments.     

Code and carrier phase based short baseline GPS positioning: computational aspects

A recursive least squares algorithm is presented for short baseline GPS positioning using both carrier phase and code measurements. We take advantage of the structure of the problem to make the algorithm computationally efficient and use orthogonal transformations to ensure that the algorithm is numerically reliable. Details are given for computing position estimates and error covariance matrices with possible satellite rising and setting. Real data test results suggest our algorithm is effective.This research was supported by NSERC of Canada Grant RGPIN217191–99, FCAR of Quebec Grant 2001-NC-66487, and NSERCGEOIDE Network Project ENV#14 for Xiao-Wen Chang, and by NSERC of Canada Grant RGPIN9236–01 for Christopher C. Paige.An erratum to this article can be found at     

On weighted total least-squares adjustment for linear regression

The weighted total least-squares solution (WTLSS) is presented for an errors-in-variables model with fairly general variance–covariance matrices. In particular, the observations can be heteroscedastic and correlated, but the variance–covariance matrix of the dependent variables needs to have a certain block structure. An algorithm for the computation of the WTLSS is presented and applied to a straight-line fit problem where the data have been observed with different precision, and to a multiple regression problem from recently published climate change research.     

The combination of GNSS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise

We propose a methodology for the combination of a gravimetric (quasi-) geoid with GNSS-levelling data in the presence of noise with correlations and/or spatially varying noise variances. It comprises two steps: first, a gravimetric (quasi-) geoid is computed using the available gravity data, which, in a second step, is improved using ellipsoidal heights at benchmarks provided by GNSS once they have become available. The methodology is an alternative to the integrated processing of all available data using least-squares techniques or least-squares collocation. Unlike the corrector-surface approach, the pursued approach guarantees that the corrections applied to the gravimetric (quasi-) geoid are consistent with the gravity anomaly data set. The methodology is applied to a data set comprising 109 gravimetric quasi-geoid heights, ellipsoidal heights and normal heights at benchmarks in Switzerland. Each data set is complemented by a full noise covariance matrix. We show that when neglecting noise correlations and/or spatially varying noise variances, errors up to 10% of the differences between geometric and gravimetric quasi-geoid heights are introduced. This suggests that if high-quality ellipsoidal heights at benchmarks are available and are used to compute an improved (quasi-) geoid, noise covariance matrices referring to the same datum should be used in the data processing whenever they are available. We compare the methodology with the corrector-surface approach using various corrector surface models. We show that the commonly used corrector surfaces fail to model the more complicated spatial patterns of differences between geometric and gravimetric quasi-geoid heights present in the data set. More flexible parametric models such as radial basis function approximations or minimum-curvature harmonic splines perform better. We also compare the proposed method with generalized least-squares collocation, which comprises a deterministic trend model, a random signal component and a random correlated noise component. Trend model parameters and signal covariance function parameters are estimated iteratively from the data using non-linear least-squares techniques. We show that the performance of generalized least-squares collocation is better than the performance of corrector surfaces, but the differences with respect to the proposed method are still significant.     

基于密集匹配点云的DEM生产可行性研究

无人机倾斜摄影直接生产的成果通常包括三维模型、TDOM、DSM等,然而规划设计通常不能直接利用倾斜数据输出的DEM,需要辅以人工编辑。作为倾斜摄影影像处理的过程成果,密集匹配点云未得到充分利用。其与激光雷达点云具备相似的结构,且点云密度可自由选择,在不考虑数据量的情况下,密集匹配点云的点密度可数倍于激光雷达点云。此外,密集匹配点云无需单独赋色,即具有纹理信息, 对人工目视编辑自动分类后的地面点具有一定的辅助作用。本文对比分析了同一测区的密集匹配点云与激光雷达点云,验证了密集匹配点云用于房屋建筑区及稀疏林区地面点滤波并生产DEM的可行性。