Methodology and use of tensor invariants for satellite gravity gradiometry

Although its use is widespread in several other scientific disciplines, the theory of tensor invariants is only marginally adopted in gravity field modeling. We aim to close this gap by developing and applying the invariants approach for geopotential recovery. Gravitational tensor invariants are deduced from products of second-order derivatives of the gravitational potential. The benefit of the method presented arises from its independence of the gradiometer instrument’s orientation in space. Thus, we refrain from the classical methods for satellite gravity gradiometry analysis, i.e., in terms of individual gravity gradients, in favor of the alternative invariants approach. The invariants approach requires a tailored processing strategy. Firstly, the non-linear functionals with regard to the potential series expansion in spherical harmonics necessitates the linearization and iterative solution of the resulting least-squares problem. From the computational point of view, efficient linearization by means of perturbation theory has been adopted. It only requires the computation of reference gravity gradients. Secondly, the deduced pseudo-observations are composed of all the gravitational tensor elements, all of which require a comparable level of accuracy. Additionally, implementation of the invariants method for large data sets is a challenging task. We show the fundamentals of tensor invariants theory adapted to satellite gradiometry. With regard to the GOCE (Gravity field and steady-state Ocean Circulation Explorer) satellite gradiometry mission, we demonstrate that the iterative parameter estimation process converges within only two iterations. Additionally, for the GOCE configuration, we show the invariants approach to be insensitive to the synthesis of unobserved gravity gradients.     

Three-dimensional outlier detection for GPS networks and their densification via the BLIMPBE approach

This article discusses outlier detection based on Baarda's theory, but applied to three-dimensional GPS baseline vectors, as compared to the traditional approach of investigating the vector components individually (one-dimensional). In addition, Schaffrin's recently proposed estimator of type BLIMPBE is discussed and contrasted with a minimum constrained least-squares adjustment, specifically Partial-MINOLESS. A more detailed discussion of these topics, which includes additional formulas and derivations, can be found in the first author's Master of Science thesis, hereinafter referred to as Report 465.     

Ground and building extraction from LiDAR data based on differential morphological profiles and locally fitted surfaces

This paper proposes a new framework for ground extraction and building detection in LiDAR data. The proposed approach constructs the connectivity of a grid over the LiDAR point-cloud in order to perform multi-scale data decomposition. This is realised by forming a top-hat scale-space using differential morphological profiles (DMPs) on points’ residuals from the approximated surface. The geometric attributes of the contained features are estimated by mapping characteristic values from DMPs. Ground definition is achieved by using features’ geometry, whilst their surface and regional attributes are additionally considered for building detection. A new algorithm for local fitting surfaces (LoFS) is proposed for extracting planar points. Finally, transitions between planar ground and non-ground regions are observed in order to separate regions of similar geometrical and surface properties but different contexts (i.e. bridges and buildings). The methods were evaluated using ISPRS benchmark datasets and show superior results in comparison to the current state-of-the-art.     

Nonlinear least-squares method via an isomorphic geometrical setup

The resolution of a nonlinear parametric adjustment model is addressed through an isomorphic geometrical setup with tensor structure and notation, represented by a u-dimensional “model surface” embedded in a flat n-dimensional “observational space”. Then observations correspond to the observational-space coordinates of the pointQ, theu initial parameters correspond to the model-surface coordinates of the “initial” pointP, and theu adjusted parameters correspond to the model-surface coordinates of the “least-squares” point . The least-squares criterion results in a minimum-distance property implying that the vector Q must be orthogonal to the model surface. The geometrical setup leads to the solution of modified normal equations, characterized by a positive-definite matrix. The latter contains second-order and, optionally, thirdorder partial derivatives of the observables with respect to the parameters. This approach significantly shortens the convergence process as compared to the standard (linearized) method.     

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.     

Some properties of decorrelation techniques in the ambiguity space

The most recent contributions to ambiguity resolution techniques have mainly focused on resolution in the ambiguity domain. Two techniques utilizing a decorrelation approach are compared. These techniques are the least-squares ambiguity decorrelation adjustment method and the lattice basis reduction. The latter is also known as the LLL method. The main focus in this article is on the decorrelation performance of these state-of-the-art techniques, which are aiming at ambiguity space decorrelation through unimodular transformations. The performances of the two-decorrelation techniques are compared through their ability in making the ambiguity space as orthogonal as possible.     

The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation

The GPS double difference carrier phase measurements are ambiguous by an unknown integer number of cycles. High precision relative GPS positioning based on short observational timespan data, is possible, when reliable estimates of the integer double difference ambiguities can be determined in an efficient manner. In this contribution a new method is introduced that enables very fast integer least-squares estimation of the ambiguities. The method makes use of an ambiguity transformation that allows one to reformulate the original ambiguity estimation problem as a new problem that is much easier to solve. The transformation aims at decorrelating the least-squares ambiguities and is based on an integer approximation of the conditional least-squares transformation. This least-squares ambiguity decorrelation approach, flattens the typical discontinuity in the GPS-spectrum of ambiguity conditional variances and returns new ambiguities that show a dramatic improvement in correlation and precision. As a result, the search for the transformed integer least-squares ambiguities can be performed in a highly efficient manner.     

GPS double difference statistics: with and without using satellite geometry

In this contribution GPS statistics are presented for the case that the relative receiver-satellite geometry is included in the single baseline model and for the case that the relative receiver-satellite geometry is excluded. It is shown that the statistics are linked through a particular form of a phased adjustment. Based on the stepwise approach of a phased adjustment, the impact of using satellite geometry or dispensing with it, on the least-squares estimators, on the teststatistics and their associated reliability, and on the integer ambiguity estimation, is presented and analyzed. Received 1 March 1996; Accepted 22 July 1996     

Precision, volume and eigenspectra for GPS ambiguity estimation based on the time-averaged satellite geometry

In this contribution we consider the time-averaged GPS single-baseline model and study in a qualitative sense its relation with the geometry-free model and the geometry-based model. The least-squares estimators of the model are derived and their properties discussed. Special attention is given to the ambiguity search space, since it plays such a crucial role in the problem of integer ambiguity estimation and validation. Easy-to-evaluate, closed-form expressions are presented for the volumes of the ambiguity search spaces that belong to the geometry-free model, the single-epoch geometry-based model and the time-averaged model. By means of an eigenvalue analysis, the geometry of the ambiguity search spaces is revealed and its impact on the search for the integer least-squares ambiguities discussed. Received: 3 April 1996; Accepted: 6 January 1997     

磁梯度张量系统的非线性集成矢量校正

磁梯度张量系统测量精度受到单磁传感器系统误差与传感器阵列间非对准误差的严重影响。为了获得精确的张量测量输出,建立了单磁传感器零漂、标度因子与非正交角等系统误差和多传感器轴系间非对准误差的集成数学模型,提出了基于十字磁梯度张量系统最小二乘非线性集成校正方法。相比两步标量校正,利用建立的集成数学模型能够一次性估计出十字形张量系统的48个误差参数,以"人造"平台输出为参考实现低成本矢量校正,极大提高了校正效率和参数估计准确率。仿真和实测表明,张量系统误差参数仿真估计准确率高于99.75%,实验校正后总场输出均方根误差（root mean square error,RMSE）小于2 nT,张量分量RMSE小于50 nT/m,参数估计具有较高的鲁棒性。     

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.     

张量组稀疏表示的高光谱图像去噪算法

提出了一种基于张量组稀疏表示的高光谱遥感影像降噪。高光谱影像数据可视为三阶张量。首先,高光谱图像被划分为小的张量分块,然后,对相似的张量分块进行聚类,并对聚类分组进行稀疏表示。基于高光谱图像的空间非局部自相似性和光谱相关性,将张量组稀疏表示模型分解为一系列无约束低秩张量的近似问题,进而通过张量分解进行求解。对模拟和真实高光谱数据进行试验,验证了该算法的有效性。     

基于物方空间几何约束最小二乘匹配的建筑物半自动提取方法

提出了一种从数字航空立体像对半自动提取建筑物的方法。操作员通过人机界面选择房屋的种类并输入初始位置，然后经过边缘检测、直线段提取和据此房屋几何模型的线段自动编组等处理得到各房屋角点的初始位置，最后，为了获得房屋的精确定位和符合物方严格几何约束的结果，基于物方空间几何约束的最小二乘匹配平差模型用于求取房屋直线边缘和物方几何模型的最优匹配，试验表明，该方法能提高建筑物的提取精度并可为半自动建筑物提取提供一个灵活的框架。     

Solving three types of satellite gravity gradient boundary value problems by least-squares

The principle and method for solving three types of satellite gravity gradient boundary value problems by least-squares are discussed in detail. Also, kernel function expressions of the least-squares solution of three geodetic boundary value problems with the observations {Гzz}, {Гxz, Гyz} and {Гxx ? Гyy, 2Гxy} are presented. From the results of recovering gravity field using simulated gravity gradient tensor data, we can draw a conclusion that satellite gravity gradient integral formulas derived from least-squares are valid and rigorous for recovering the gravity field.     

一种基于多维正交判别子空间投影的人脸识别方法

人脸识别中,传统数据降维方法将人脸图像重排列成向量后进行处理,丢失了数据本身的结构特性,导致识别精度不高。本文发展了一种基于张量的数据降维方法———多维正交判别子空间投影。该算法直接用张量描述人脸,并通过张量到矢量投影(tensortovectorprojection,TVP)将张量数据投影到向量判别子空间。此方法寻找相互正交的投影向量集,使得判别子空间中数据类间离散度最大,同时类内离散度最小;进而利用TVP投影将高维张量数据映射成低维向量数据,在合适的约束条件下,这些降维后的向量特征数据是整个人脸数据中最具代表性的特征数据;最后,使用k最近邻(KNN)分类器将这些特征数据分类。利用经典人脸数据库ORL进行实验,验证了本文方法的有效性。     

Geometric methods for estimating representative sidewalk widths applied to Vienna’s streetscape surfaces database

Space, and in particular public space for movement and leisure, is a valuable and scarce resource, especially in today’s growing urban centres. The distribution and absolute amount of urban space—especially the provision of sufficient pedestrian areas, such as sidewalks—is considered crucial for shaping living and mobility options as well as transport choices. Ubiquitous urban data collection and today’s IT capabilities offer new possibilities for providing a relation-preserving overview and for keeping track of infrastructure changes. This paper presents three novel methods for estimating representative sidewalk widths and applies them to the official Viennese streetscape surface database. The first two methods use individual pedestrian area polygons and their geometrical representations of minimum circumscribing and maximum inscribing circles to derive a representative width of these individual surfaces. The third method utilizes aggregated pedestrian areas within the buffered street axis and results in a representative width for the corresponding road axis segment. Results are displayed as city-wide means in a 500 by 500 m grid and spatial autocorrelation based on Moran’s I is studied. We also compare the results between methods as well as to previous research, existing databases and guideline requirements on sidewalk widths. Finally, we discuss possible applications of these methods for monitoring and regression analysis and suggest future methodological improvements for increased accuracy.     

基于欧氏范数的Ⅱ类病态性诊断

测绘实践中的病态性,特别是Ⅱ类病态性问题已严重制约着测绘成果的质量。如何诊断病态性和削弱病态性的影响,是当前测绘工作者面临的一个重要课题。本文从观测空间的几何意义出发,基于欧氏范数,提出Ⅱ类病态性诊断方法。该方法几何意义明确,便于掌握。     

Parallel Cholesky-based reduction for the weighted integer least squares problem

The LLL reduction of lattice vectors and its variants have been widely used to solve the weighted integer least squares (ILS) problem, or equivalently, the weighted closest point problem. Instead of reducing lattice vectors, we propose a parallel Cholesky-based reduction method for positive definite quadratic forms. The new reduction method directly works on the positive definite matrix associated with the weighted ILS problem and is shown to satisfy part of the inequalities required by Minkowski’s reduction of positive definite quadratic forms. The complexity of the algorithm can be fixed a priori by limiting the number of iterations. The simulations have clearly shown that the parallel Cholesky-based reduction method is significantly better than the LLL algorithm to reduce the condition number of the positive definite matrix, and as a result, can significantly reduce the searching space for the global optimal, weighted ILS or maximum likelihood estimate.     

Vector-based approach for combining ascending and descending persistent scatterers interferometric point measurements

The combination of ascending and descending persistent scatterers interferometric (PSI) data by means of resampling and/or spatial interpolation, separately for each synthetic aperture radar geometry, is a commonly followed procedure, limited though by the reduced spatial coverage and the introduced uncertainties from multiple rasterization steps. Herein, an alternative approach is proposed for combining different PSI line-of-sight observables in the vector domain, based on the geographic proximity of PS point targets. In the proposed nearest neighbour vector approach all necessary analysis steps are performed by means of attributes transfer and calculations between features geodatabases, prior to any rasterization. By increasing the number of input point vectors during subsequent interpolation, the overall error budget coming from spatial interpolation is being reduced. The advantages of the proposed vector-based approach compared to the commonly used grid-based procedure are being demonstrated using real data.