Solving geoinformatics parametric polynomial systems using the improved Dixon resultant

Improvements in computational and observational technologies in geoinformatics, e.g., the use of laser scanners that produce huge point cloud data sets, or the proliferation of global navigation satellite systems (GNSS) and unmanned aircraft vehicles (UAVs), have brought with them the challenges of handling and processing this “big data”. These call for improvement or development of better processing algorithms. One way to do that is integration of symbolically presolved sub-algorithms to speed up computations. Using examples of interest from real geoinformatic problems, we will discuss the Dixon-EDF resultant as an improved resultant method for the symbolic solution of parametric polynomial systems. We will briefly describe the method itself, then discuss geoinformatics problems arising in minimum distance mapping (MDM), parameter transformations, and pose estimation essential for resection. Dixon-EDF is then compared to older notions of “Dixon resultant”, and to several respected implementations of Gröbner bases algorithms on several systems. The improved algorithm, Dixon-EDF, is found to be greatly superior, usually by orders of magnitude, in both CPU usage and RAM usage. It can solve geoinformatics problems on which the other methods fail, making symbolic solution of parametric systems feasible for many problems

Hybrid symbolic-numeric methods in geosciences

Modern computing systems, like Mathematica, are blending numeric and symbolic evaluation of many algorithms improving their efficiencies (time, accuracy). In this contribution three toy-examples of the application of hybrid symbolic-numeric computations in geosciences are presented in order to illustrate the features of this advanced technique, namely: ranging GNSS satellites, computing GNSS cycle ambiguities and employing symbolic regression for verifying Kepler’s third law

Numeric-Symbolic Solution for Satellite Trajectory Control by Pole Placement

Control design of satellites based on pole placement method results in determined or underdetermined multivariable polynomial systems. Since only the real solutions can be considered for hardware implementation, we are looking for exclusively these solutions. In this study we suggest a numeric-symbolic approach to compute only the real solutions directly. Employing computer algebra (Dixon or reduced Gröbner basis) a condition can be formulated for the free variables as parameters of the underdetermined system, which ensures only real solutions. Numerical example illustrates the procedure and the effectivity of the control law

Pareto optimality solution of the multi-objective photogrammetric resection-intersection problem

Reconstruction of architectural structures from photographs has recently experienced intensive efforts in computer vision research. This is achieved through the solution of nonlinear least squares (NLS) problems to obtain accurate structure and motion estimates. In Photogrammetry, NLS contribute to the determination of the 3-dimensional (3D) terrain models from the images taken from photographs. The traditional NLS approach for solving the resection-intersection problem based on implicit formulation on the one hand suffers from the lack of provision by which the involved variables can be weighted. On the other hand, incorporation of explicit formulation expresses the objectives to be minimized in different forms, thus resulting in different parametric values for the estimated parameters at non-zero residuals. Sometimes, these objectives may conflict in a Pareto sense, namely, a small change in the parameters results in the increase of one objective and a decrease of the other, as is often the case in multi-objective problems. Such is often the case with error-in-all-variable (EIV) models, e.g., in the resection-intersection problem where such change in the parameters could be caused by errors in both image and reference coordinates.This study proposes the Pareto optimal approach as a possible improvement to the solution of the resection-intersection problem, where it provides simultaneous estimation of the coordinates and orientation parameters of the cameras in a two or multistation camera system on the basis of a properly weighted multi-objective function. This objective represents the weighted sum of the square of the direct explicit differences of the measured and computed ground as well as the image coordinates. The effectiveness of the proposed method is demonstrated by two camera calibration problems, where the internal and external orientation parameters are estimated on the basis of the collinearity equations, employing the data of a Manhattan-type test field as well as the data of an outdoor, real case experiment. In addition, an architectural structural reconstruction of the Merton college court in Oxford (UK) via estimation of camera matrices is also presented. Although these two problems are different, where the first case considers the error reduction of the image and spatial coordinates, while the second case considers the precision of the space coordinates, the Pareto optimality can handle both problems in a general and flexible way

From the global scale to the Mediterranean plate kinematics

Plate motions with respect to the mantle represent the most direct evidence to understand the origin of plate tectonic processes. The research here described has the aim to improve the knowledge on the global scale plate kinematics in “absolute” reference frames, or better, relative to the mantle, incorporating both geological–geophysical and space geodesy data. Geophysical and geological signatures of subduction and rift zones independently show a global polarity of current plate motions, suggesting a west-ward displacement of the whole lithosphere relative to the underlying mantle. We analytically modeled this tectonic pattern in a suitable selected hotspot framework, taking into account variable depths of the hotspot source, obtaining new plate angular velocities and their uncertainties, by least squares inversion. Then, we focused our attention on the Italian area estimating the velocity field from continuous GPS observations both relative to Eurasia, and relative to the mantle, applying the global model previously estimated. However, the presence of the Apennine subduction, having more or less the same extent of the investigated area, makes locally less reliable our global model. Consequently, we applied a simple kinematic model to estimate the rates and spatial pattern of the subduction along the Apennines. The variable rates inferred after our analysis, better reconcile if the subduction process is conceived as a passive rather than active feature. Then, the analysis came back again to the global scale and to the basic argument if plates are passively riding along on the top of a mantle convection cell, or whether the plates themselves are active drivers. On the other hand, if plate motion occurs as an ordered undulated west-directed flow, the net-rotation of the lithosphere emerges as a passive process active at global scale, and then, it can be driven only by external forces. Thus, the last part of this research has been dedicated to find the experimental evidences connecting the tectonic processes to the Earth's rotation and the tidal drag

Investigation into the Accuracy and Practicality of Methods for Transforming Coordinates between Geodetic Datums

This thesis is a study of methods of transforming coordinates between geodetic datums, the methods being generally known as datum transformations. Direct methods are described and categorised as conformal, near-conformal and non-conformal. New variations on all three types are included in the direct methods: SMITSWAM (which avoids changes of coordinate-type), generalisations of Standard & Abridged Molodensky, and normalised generalisations of multiple regression equations (5 types). Reverse transformations are extensively covered, as are methods of derivation. In both cases, new algorithms are included. Direct methods, with the exception of multiple regression equations, do not capture distortions in datum transformations. The thesis therefore includes a review of composite methods which extract a trend model and apply a surface-fitting technique (SFT) to the residuals. Sometimes the SFT is used as a gridding method, producing regularly-spaced data that can be interpolated as a final stage of the composite process. The SFTs selected for detailed study include new variations on inverse-distance-to-a-power weighting and nearest-neighbour interpolation. These are called HIPFEAD and LIVONN respectively. In both cases, the variations are shown to have advantages in terms of accuracy of fit. Least-squares collocation and radial basis functions are shown to produce reusable vectors - described here as “revamped signals” – that enable interpolation without gridding. Where the composite methods are used for gridding, it is shown that geodetic coordinates can be used, avoiding the need for projected grid coordinates. The interpolation options applied are piecewise-bilinear and piecewise-bicubic, the latter being an algorithm (believed to be new) that uses up to 12 “grid” points. Case studies were considered using 6 datasets, two for Great Britain, one each for Western Australia, Ghana, Sweden and Slovenia. These showed beneficial properties of the new methods, both in the direct and composite categories. They also enabled comparisons of transformation methods generally

(Schw)Ehre, wem (Schw)Ehre gebührt : Festschrift zur Verabschiedung von Prof. Dr.-Ing. Dr. h.c. Bernhard Heck

Die Festschrift zur Verabschiedung von Prof. Bernhard Heck enthält 41 Beiträge aus dem Freundeskreis, der Kollegenschaft sowie von ehemaligen Promovierenden. Der Schwerpunkt der Arbeiten liegt auf den Gebieten der Physikalischen und Satellitengeodäsie sowie der Geodynamik und spiegelt das vielfältige Wirken von Bernhard Heck wider. Abgerundet wird die Schrift durch Beiträge zur Ausbildung am Geodätischen Institut des KIT und zur Stellung der Geodäsie im gesellschaftlichen Kontext

Vernetzt und ausgeglichen : Festschrift zur Verabschiedung von Prof. Dr.-Ing. habil. Dr.-Ing. E.h. Günter Schmitt

Im Rahmen der Festschrift zur Verabschiedung von Prof. Schmitt wurden von Mitarbeitern des Geodätischen Instituts sowie von ehemaligen Doktoranden insgesamt 24 Beiträge zu aktuellen Arbeiten der Verfasser erstellt. Das Spektrum dieser Arbeiten liegt in den Bereichen Geodäsie, GIS, GNSS, Netzausgleichung, Geodynamik sowie Wertermittlung. Innovative Arbeiten zu aktuellen Fragestellungen stehen in direktem Zusammenhang mit früheren wissenschaftlichen Arbeiten von Prof. Schmitt