Abstract
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.
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Communicated by: H. A. Babaie
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Appendix
Appendix
The Maple-FGb commands for the pose example:
Maple 2015 (X86 64 LINUX)
Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2015 > with(FGb): p := 0; v1 := [ x2, x3, x4 ]; v2 := [ x1, b1,b2,b3,b4,c12,c23,c34,c41 ]; sys := [x1^2 + x2^2 - c12*x1*x2 - b1, x2^2 + x3^2 - c23*x2*x3 - b2, x3^2 + x4^2 - c34*x3*x4 - b3, x4^2 + x1^2 - c41*x4*x1 - b4]; > ll1:=fgb_gbasis_elim(sys, p,v1,v2, {"step"=8,"verb"=3,"index"=40000000});
Magma commands for the pose example:
Magma V2.21-8 Thu Dec 10 2015 13:26:28 on ace-math01 [Seed = 2343837211] Type ? for help. Type <Ctrl>-D to quit. Q:=RationalField(); F<b1,b2,b3,b4,c12,c23,c34,c41> := FunctionField(Q,8); R<x1,x2,x3,x4> := PolynomialRing(F,4, "elim", [2,3,4]); I := Ideal ([x1^2 + x2^2 - c12*x1*x2 - b1, x2^2 + x3^2 - c23*x2*x3 - b2, x3^2 + x4^2 - c34*x3*x4 - b3, x4^2 + x1^2 - c41*x4*x1 - b4]); time G := GroebnerBasis(I);
Mathematica command for the general 3D conic problem, solving for x (the ei are the four equations from above):
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Lewis, R.H., Paláncz, B. & Awange, J. Solving geoinformatics parametric polynomial systems using the improved Dixon resultant. Earth Sci Inform 12, 229–239 (2019). https://doi.org/10.1007/s12145-018-0366-2
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DOI: https://doi.org/10.1007/s12145-018-0366-2