Groebner Basis in Geodesy and Geoinformatics

NRLI-UAV: Non-rigid registration of sequential raw laser scans and images for low-cost UAV LiDAR point cloud quality improvement

Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10−8.

Abstract. Utilizing ground control points (GCPs) to georeference photogrammetry-based point cloud data is a common practice in unmanned aerial system (UAS) mapping. Direct georeferencing or integrated sensor orientation (ISO) can be used to obtain georeferenced point clouds from UAS without relying heavily on GCPs. However, the accuracy of the point cloud may be impacted by the accuracy of the trajectory solution obtained by GNSS. To improve point cloud accuracy, post-processing kinematic (PPK) solutions can be applied to the UAS trajectory, which may provide higher accuracy than low-accuracy trajectory solutions and minimize the reliance on GCPs. This study compares the accuracy and precision of two different point clouds generated using different methods. One point cloud was generated using traditional photogrammetric methods with low accuracy Global Navigation Satellite System (GNSS) observations from the UAS and GCPs that have an average accuracy of one to two centimeters, while the other was generated using PPK trajectory solution for the UAS’s trajectory with two software: open-source Emlid Studio and the widely used Inertial Explorer. The use of PPK techniques in UAS mapping may have several potential benefits over traditional methods. By correcting the errors in the UAS's trajectory, a user may only need to depend on fewer ground control points, which can reduce the time and cost associated with fieldwork. This is particularly useful in areas that are difficult to access or have limited ground control point options, such as in urban or forested areas. To evaluate performance, a GNSS receiver is used to obtain measurements on checkpoints, which are used to assess the accuracies of the point clouds. In our experiments, the accuracy of the point clouds generated using PPK trajectory solution with high accuracy GCPs was found to be higher than those generated with low accuracy GNSS observations while aided with high accuracy ground control points. While the use of PPK with GCPs is generally expected to provide more accurate and reliable data than low-accuracy GNSS observations even after adjusting with GCPs, the number and distribution of GCPs can still significantly impact overall accuracy. Therefore, careful consideration of the number of GCPs and their placement is essential to achieve the desired level of efficiency and effectiveness in UAS mapping.

Abstract. Kinematic laser scanning enables efficient and accurate acquisition of 3D data, specifically point clouds. Transforming the laser ranging measurements from the scanner coordinate system to a georeferenced coordinate system requires integration with auxiliary navigation systems, in this case using GNSS (global navigation satellite system) and an IMU (inertial measurement unit). GNSS and IMU data are commonly fused via Kalman filter, and the resulting trajectory is used for georeferencing. Errors in the trajectory propagate through the georeferencing and cause discrepancies in the point cloud. To mitigate this, we present a holistic integration method incorporating GNSS, IMU and LiDAR measurements in a single adjustment, which can be seamlessly adapted to a setup with two GNSS receivers and antennas. This tight coupling of LiDAR and IMU together with the dual-GNSS setup allows the use of an ultra low-cost IMU while still achieving high-quality point clouds. We demonstrate this methodology on two datsets, where we discuss in particular the boresight calibration of such a system, and the impact of the LiDAR measurements and the dual-GNSS set-up on the trajectory and the point cloud.

Currently, 3D point clouds are obtained via LiDAR (Light Detection and Ranging) sensors to compute vegetation parameters to enhance agricultural operations. However, such a point cloud is intrinsically dependent on the GNSS (global navigation satellite system) antenna used to have absolute positioning of the sensor within the grove. Therefore, the error associated with the GNSS receiver is propagated to the LiDAR readings and, thus, to the crown or orchard parameters. In this work, we first describe the error propagation of GNSS over the laser scan measurements. Second, we present our proposal to overcome this effect based only on the LiDAR readings. Such a proposal uses a scan matching approach to reduce the error associated with the GNSS receiver. To accomplish such purpose, we fuse the information from the scan matching estimations with the GNSS measurements. In the experiments, we statistically analyze the dependence of the grove parameters extracted from the 3D point cloud -specifically crown surface area, crown volume, and crown porosity- to the localization error. We carried out 150 trials with positioning errors ranging from 0.01 meters (ground truth) to 2 meters. When using only GNSS as a localization system, the results showed that errors associated with the estimation of vegetation parameters increased more than 100 % when positioning error was equal or bigger than 1 meter. On the other hand, when our proposal was used as a localization system, the results showed that for the same case of 1 meter, the estimation of orchard parameters improved in 20 % overall. However, in lower positioning errors of the GNSS, the estimation of orchard parameters were improved up to 50% overall. These results suggest that our work could lead to better decisions in agricultural operations, which are based on foliar parameter measurements, without the use of external hardware.

Integrated TLS and GPR data can provide multisensor and multiscale spatial data for the comprehensive identification and analysis of surficial and subsurface information, but a reliable systematic methodology associated with data integration of TLS and GPR is still scarce. The aim of this research is to develop a methodology for the data integration of TLS and GPR for detailed, three-dimensional (3D) virtual reconstruction. GPR data and high-precision geographical coordinates at the centimeter level were simultaneously gathered using the GPR system and the Global Navigation Satellite System (GNSS) signal receiver. A time synchronization algorithm was proposed to combine each trace of the GPR data with its position information. In view of the improved propagation model of electromagnetic waves, the GPR data were transformed into dense point clouds in the geodetic coordinate system. Finally, the TLS-based and GPR-derived point clouds were merged into a single point cloud dataset using coordinate transformation. In addition, TLS and GPR (250 MHz and 500 MHz antenna) surveys were conducted in the Litang fault to assess the feasibility and overall accuracy of the proposed methodology. The 3D realistic surface and subsurface geometry of the fault scarp were displayed using the integration data of TLS and GPR. A total of 40 common points between the TLS-based and GPR-derived point clouds were implemented to assess the data fusion accuracy. The difference values in the x and y directions were relatively stable within 2 cm, while the difference values in the z direction had an abrupt fluctuation and the maximum values could be up to 5 cm. The standard deviations (STD) of the common points between the TLS-based and GPR-derived point clouds were 0.9 cm, 0.8 cm, and 2.9 cm. Based on the difference values and the STD in the x, y, and z directions, the field experimental results demonstrate that the GPR-derived point clouds exhibit good consistency with the TLS-based point clouds. Furthermore, this study offers a good future prospect for the integration method of TLS and GPR for comprehensive interpretation and analysis of the surficial and subsurface information in many fields, such as archaeology, urban infrastructure detection, geological investigation, and other fields.

Discrete isoperimetric variational problems which model single and double integral isoperimetric problems are formulated and some multiplier rules are derived. For quardratic functionals, the Euler-Lagrange equation is linear and can be analyzed bydeterminant methods. difference equations methods. or numerically bythealgebraic eigenvalue problem. A specific example is given wherethe eigenfuntions and eigenvalue of this discrete problem converge. as the step size goes to zero, to the eigenfunctionsand eigenvalues of the corresponding, continuous problem Recent developments in algebraic geometry offer the hopeof using Groebner (=Gröbner or Grobner) basis methods to numerically solve some systems of equationsgenerated by Lagrange multiplier methods. These methods mayapply to nonlinear systems of equations whenever they can be reduced to systems of polynomial equations. The Groebner basis algorithm of Buchberger is an extension of Gaussian elimination to polynomial systems.

The direct and inverse scattering problems in two-dimensional acoustics at a fixed frequency are considered. It is shown how the null-field approach can be modified so that the Q matrix (which in a straightforward manner gives the transition matrix) is obtained as a series instead of an integral. For an obstacle which is a perturbation of a circle this series form gives an approximate, very explicit, expression for the transition matrix. A few numerical examples are given to show the utility and limitations of this approximation. For the inverse problem, the series form of Q gives a system of nonlinear polynomial equations which are solved by the imbedding method. Some numerical examples show that quite accurate results are obtained by this method in cases where the system of equations can be kept small. The linear approximation of the system of polynomial equations yields a method that works surprisingly well and which is also promising for the more difficult three-dimensional and vector problems.

The security of many recently proposed cryptosystems is based on the difficulty of solving large systems of quadratic multivariate polynomial equations. This problem is NP-hard over any field. When the number of equations m is the same as the number of unknowns n the best known algorithms are exhaustive search for small fields, and a Gröbner base algorithm for large fields. Gröbner base algorithms have large exponential complexity and cannot solve in practice systems with n ≥ 15. Kipnis and Shamir [9] have recently introduced a new algorithm called “relinearization”. The exact complexity of this algorithm is not known, but for sufficiently overdefined systems it was expected to run in polynomial time.In this paper we analyze the theoretical and practical aspects of relinearization. We ran a large number of experiments for various values of n and m, and analysed which systems of equations were actually solvable. We show that many of the equations generated by relinearization are linearly dependent, and thus relinearization is less efficient that one could expect. We then develop an improved algorithm called XL which is both simpler and more powerful than relinearization. For all 0 < ε ≤ 1/2, and m ≥ εn 2, XL and relinearization are expected to run in polynomial time of approximately \( n^{\mathcal{O}(1/\sqrt \varepsilon )} \). Moreover, we provide strong evidence that relinearization and XL can solve randomly generated systems of polynomial equations in subexponential time when m exceeds n by a number that increases slowly with n.KeywordsQuadratic EquationExhaustive SearchAsymptotic ComplexityExact ComplexityMultivariate EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Uncrewed aircraft systems (UASs) and structure-from-motion/multi-view stereo (SfM/MVS) photogrammetry are efficient methods for mapping terrain at local geographic scales. Traditionally, indirect georeferencing using ground control points (GCPs) is used to georeference the UAS image locations before further processing in SfM software. However, this is a tedious practice and unsuitable for surveying remote or inaccessible areas. Direct georeferencing is a plausible alternative that requires no GCPs. It relies on global navigation satellite system (GNSS) technology to georeference the UAS image locations. This research combined field experiments and simulation to investigate GNSS-based post-processed kinematic (PPK) as a means to eliminate or reduce reliance on GCPs for shoreline mapping and charting. The study also conducted a brief comparison of real-time network (RTN) and precise point positioning (PPP) performances for the same purpose. Ancillary experiments evaluated the effects of PPK base station distance and GNSS sample rate on the accuracy of derived 3D point clouds and digital elevation models (DEMs). Vertical root mean square errors (RMSEz), scaled to the 95% confidence interval using an assumption of normally-distributed errors, were desired to be within 0.5 m to satisfy National Oceanic and Atmospheric Administration (NOAA) requirements for nautical charting. Simulations used a Monte Carlo approach and empirical tests to examine the influence of GNSS performance on the quality of derived 3D point clouds. RTN and PPK results consistently yielded RMSEz values within 10 cm, thus satisfying NOAA requirements for nautical charting. PPP did not meet the accuracy requirements but showed promising results that prompt further investigation. PPK experiments using higher GNSS sample rates did not always provide the best accuracies. GNSS performance and model accuracies were enhanced when using base stations located within 30 km of the survey site. Results without using GCPs observed a direct relationship between point cloud accuracy and GNSS performance, with R2 values reaching up to 0.97.

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p ∈ P is seen by at least one guard g∈ G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃ ℝ, which has been studied earlier by other communities. The class ∃ ℝ consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP ⊆ ∃ ℝ. We prove that the art gallery problem is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃ ℝ. As an illustration of our techniques, we can show that for every compact semi-algebraic set S⊂ [0,1]2 there exists a polygon with rational coordinates that enforces one of the guards to be at any position p∈ S, in any optimal guarding. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. In the ∃ ℝ-hardness proof for the art gallery problem we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it can be used to obtain ∃ ℝ-hardness proofs for other problems. In particular, ETR-INV has been very recently used to prove ∃ ℝ-hardness of other geometric problems.

Using Einstein’s notation [4]), the polynomial differential systems of finite dimension n and of degree at most k with coefficients in a field k of characteristic zero can be written as dx dt = a + ajα1x α1 + ajα1α2x 1x2 + ajα1···αrx α1 · · · xr (1) where j, α1, αr ∈ {1, . . . , n}, 1 ≤ r ≤ k, and for j = 1, . . . , n and for 2 ≤ r ≤ k, the tensor ajα1···αr (1 time contravariant and r times covariant) is symmetric with respect to the lower subscripts and we consider the action of the general linear group GL(n, k) on the phase space k. In this work, starting from a minimal system of generators of the ideal of algebraic invariants of polynomial differential systems (1) we will develop an algorithmic method to study the local stability of a given cubic differential system with the help of Grobner bases and using the GL(n, k)-invariants of this cubic system [2]. Our algorithmic method is motivated by the important role of the invariant theory in the survey of polynomial differential systems and the numerous works on Groebner basis one of the main practical tools for solving algebraic systems and computing algebraic varieties. Keywords Polynomial differential systems,linear transformations, cubic differential systems,singular points, invariant, covariant, Grobner bases.

Interest in small unmanned aircraft systems (sUAS) for topographic mapping has significantly grown in recent years, driven in part by technological advancements that have made it possible to survey small- to medium-sized areas quickly and at low cost using sUAS aerial photography and digital photogrammetry. Although this approach can produce dense point clouds of topographic measurements, they have not been tested extensively to provide insights on accuracy levels for topographic mapping. This case study examines the accuracy of a sUAS-derived point cloud of a parking lot located at the Citizens Bank Arena (CBA) in Ontario, California, by comparing it to ground control points (GCPs) measured using global navigation satellite system (GNSS) data corrected with real-time kinematic (RTK) and to data from a terrestrial laser scanning (TLS) survey. We intentionally chose a flat surface due to the prevalence of flat scenes in sUAS mapping and the challenges they pose for accurately deriving vertical measurements. When the GNSS-RTK survey was compared to the sUAS point cloud, the residuals were found to be on average 18 mm and −20 mm for the horizontal and vertical components. Furthermore, when the sUAS point cloud was compared to the TLS point cloud, the average difference observed in the vertical component was 2 mm with a standard deviation of 31 mm. These results indicate that sUAS imagery can produce point clouds comparable to traditional topographic mapping methods and support other studies showing that sUAS photogrammetry provides a cost-effective, safe, efficient, and accurate solution for topographic mapping.

We relate different approaches for proving the unsatisfiability of a system of real polynomial equations over Boolean variables. On the one hand, there are the static proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, we consider polynomial calculus, which is a dynamic algebraic proof system that models Gröbner basis computations. Our first result is that sum-of-squares simulates polynomial calculus: any polynomial calculus refutation of degree d can be transformed into a sum-of-squares refutation of degree 2d and only polynomial increase in size. In contrast, our second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree and exponential size. A corollary of our first result is that the proof systems Positivstellensatz and Positivstellensatz Calculus, which have been separated over non-Boolean polynomials, simulate each other in the presence of Boolean axioms.

The recent developments in spatial data collection, management and software require the availability of proper geodetic infrastructures for integrating different types and sources of coordinates without causing effective changes in positions. Nowadays, positions are mostly collected by GNSS data collectors based on WGS84/ITRF reference systems. The data are then subjected to transformations and projections to a locally used system. Another possibility is direct data collection based on the local coordinate system by classical surveys using land surveying, photogrammetry, laser scanning, etc. The spatial data management is commonly operated using Geographic Information Systems (GIS) software for mapping, analysis, planning, and other services. The conversions between different coordinate systems should be well defined to guarantee the consistency of the coordinates on all systems and tools. In Palestine, the classical and local surveys are all based on the local coordinate system Pal1923Grid for engineering, cadastral and planning applications. The different GNSS RTK-service providers use different definitions and transformation methods between WGS84 or the International Terrestrial Reference Frames (ITRF) and the local Palestine1923Grid, whereas the Land authority has adopted a group of parameters to be implemented on the Global Navigation Satellite Systems (GNSS) data collectors, which do not fit with Palestine1923Grid properties. Additionally, different transformation methods are used in GIS applications for converting the coordinates between the different systems using WGS84 as an intermediate system. Here, the coordinates of a group of the geodetic network in the West Bank of Palestine are used to assess the accuracy of the different transformations and systems by comparing the transformed coordinates using the GNSS system and the originally registered coordinates. Furthermore, a grid of points covering the coordinate system extents is used to describe the differences between the transformations and systems. It was found that the parameters provided by GNSS service providers have results that are consistent with each other and the geodetic network in the West Bank of Palestine compared to GIS-software parameters. By contrast, all systems have extremely deteriorated coordinates in the Gaza strip and the further parts of the Pal1923Grid extents.