Explicit solution of the overdetermined three-dimensional resection problem
TL;DR: In this article, the Gauss-Jacobi combinatorial algorithm was proposed to solve the over-determined three-dimensional resection problem in a closed form without reverting to iterative or linearization procedures.
Abstract: Several procedures for solving in a closed form the three-dimensional resection problem have already been presented. In the present contribution, the overdetermined three-dimensional resection problem is solved in a closed form in two steps. In step one a combinatorial minimal subset of observations is constructed which is rigorously converted into station coordinates by means of the Groebner basis algorithm or the multipolynomial resultant algorithm. The combinatorial solution points in a polyhedron are then reduced to their barycentric in step two by means of their weighted mean. Such a weighted mean of the polyhedron points in ℝ3 is generated via the Error Propagation law/variance–covariance propagation. The Fast Nonlinear Adjustment Algorithm was proposed by C.F. Gauss, whose work was published posthumously, and C.G.I. Jacobi. The algorithm, here referred to as the Gauss–Jacobi Combinatorial algorithm, solves the overdetermined three-dimensional resection problem in a closed form without reverting to iterative or linearization procedures. Compared to the actual values, the obtained results are more accurate than those obtained from the closed-form solution of a minimano of three known stations.
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01 Jan 2007
41 citations
TL;DR: Various methods for calculating the scale factor are discussed and solutions based on quaternion with those that are based on rotation matrix making use of skew-symmetric matrix are compared.
Abstract: The present work deals with an important theoretical problem of geodesy: we are looking for a mathematical dependency between two spatial coordinate systems utilizing common pairs of points whose coordinates are given in both systems. In geodesy and photogrammetry the most often used procedure to move from one coordinate system to the other is the 3D, 7 parameter (Helmert) transformation. Up to recent times this task was solved either by iteration, or by applying the Bursa–Wolf model. Producers of GPS/GNSS receivers install these algorithms in their systems to achieve a quick processing of data. But nowadays algebraic methods of mathematics give closed form solutions of this problem, which require high level computer technology background. In everyday usage, the closed form solutions are much more simple and have a higher precision than earlier procedures and thus it can be predicted that these new solutions will find their place in the practice. The paper discusses various methods for calculating the scale factor and it also compares solutions based on quaternion with those that are based on rotation matrix making use of skew-symmetric matrix.
22 citations
TL;DR: In this article, a closed-form of the Newton method was proposed to solve over-determined pseudo-distance equations using the compacted Hessian matrix to save the computation and storage required by Newton method.
Abstract: The Newton method has been widely used for solving nonlinear least-squares problem. In geodetic adjustment, one would prefer to use the Gauss–Newton method because of the parallel with linear least-squares problem. However, it is proved in theory as well as in practice that the Gauss–Newton method has slow convergence rate and low success rate. In this paper, the over-determined pseudo-distance equations are solved by nonlinear methods. At first, the convergence of decent methods is discussed after introducing the conditional equation of nonlinear least squares. Then, a compacted form of the Hessian matrix from the second partial derivates of the pseudo-distance equations is given, and a closed-form of Newton method is presented using the compacted Hessian matrix to save the computation and storage required by Newton method. At last, some numerical examples to investigate the convergence and success rate of the proposed method are designed and performed. The performance of the closed-form of Newton method is compared with the Gauss–Newton method as well as the regularization method. The results show that the closed-form of Newton method has good performances even for dealing with ill-posed problems while a great amount of computation is saved.
21 citations
Cites background from "Explicit solution of the overdeterm..."
...Algebraic approaches to nonlinear global minimization problem have made great achievements in earth sciences (Awange 2010; Awange and Grafarend 2003)....
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TL;DR: In this article, the Groebner basis is used to solve the nonlinear Grunert distance equation in a single step once the equations have been converted into algebraic form.
Abstract: The three-dimensional (3-D) resection problem is usually solved by first obtaining the distances connecting the unknown point P{X,Y,Z} to the known points P
i
{X
i
,Y
i
,Z
i
}∣i=1,2,3 through the solution of the three nonlinear Grunert equations and then using the obtained distances to determine the position {X,Y,Z} and the 3-D orientation parameters {ΛΓ,ΦΓ, ΣΓ}. Starting from the work of the German J. A. Grunert (1841), the Grunert equations have been solved in several substitutional steps and the desire as evidenced by several publications has been to reduce these number of steps. Similarly, the 3-D ranging step for position determination which follows the distance determination step involves the solution of three nonlinear ranging (`Bogenschnitt') equations solved in several substitution steps. It is illustrated how the algebraic technique of Groebner basis solves explicitly the nonlinear Grunert distance equations and the nonlinear 3-D ranging (`Bogenschnitt') equations in a single step once the equations have been converted into algebraic (polynomial) form. In particular, the algebraic tool of the Groebner basis provides symbolic solutions to the problem of 3-D resection. The various forward and backward substitution steps inherent in the classical closed-form solutions of the problem are avoided. Similar to the Gauss elimination technique in linear systems of equations, the Groebner basis eliminates several variables in a multivariate system of nonlinear equations in such a manner that the end product normally consists of a univariate polynomial whose roots can be determined by existing programs e.g. by using the roots command in Matlab.
20 citations
Additional excerpts
...The overdetermined version of the problem is solved by the present authors in Awange and Grafarend (2003) ....
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TL;DR: In this article, the authors present direct procedures for solving distance nonlinear system of equations without linearization, iteration, forward and backward substitution, using algebraic softwares of Mathematica, Maple and Matlab.
Abstract: In GPS atmospheric sounding, geodetic positioning, robotics and photogrammetric (perspective center and intersection) problems, distances (ranges) as observables play a key role in determining the unknown parameters. The measured distances (ranges) are however normally related to the desired parameters via nonlinear equations or nonlinear system of equations that require explicit or exact solutions. Procedures for solving such equations are either normally iterative, and thus require linearization or the existing analytical procedures require laborious forward and backward substitutions. We present in the present contribution direct procedures for solving distance nonlinear system of equations without linearization, iteration, forward and backward substitution. In particular, we exploit the advantage of faster computers with large storage capacities and the computer algebraic softwares of Mathematica, Maple and Matlab to test polynomial based approaches. These polynomial (algebraic based) approaches turn out to be the key to solving distance nonlinear system of equations. The algebraic techniques discussed here does not however solve all general types of nonlinear equations but only those nonlinear system of equations that can be converted into algebraic (polynomial) form.
18 citations
Cites methods from "Explicit solution of the overdeterm..."
...In case of more observations than the unknown, as is often the practise, the present algebraic techniques give way to the Gauss-Jacobi combinatorial approach where they are used as the computing engine (Awange and Grafarend, 2003)....
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References
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TL;DR: New results are derived on the minimum number of landmarks needed to obtain a solution, and algorithms are presented for computing these minimum-landmark solutions in closed form that provide the basis for an automatic system that can solve the Location Determination Problem under difficult viewing.
Abstract: A new paradigm, Random Sample Consensus (RANSAC), for fitting a model to experimental data is introduced. RANSAC is capable of interpreting/smoothing data containing a significant percentage of gross errors, and is thus ideally suited for applications in automated image analysis where interpretation is based on the data provided by error-prone feature detectors. A major portion of this paper describes the application of RANSAC to the Location Determination Problem (LDP): Given an image depicting a set of landmarks with known locations, determine that point in space from which the image was obtained. In response to a RANSAC requirement, new results are derived on the minimum number of landmarks needed to obtain a solution, and algorithms are presented for computing these minimum-landmark solutions in closed form. These results provide the basis for an automatic system that can solve the LDP under difficult viewing
23,396 citations
"Explicit solution of the overdeterm..." refers background in this paper
...…the problem by a substitution approach in three steps, the more recent desire has been to solve the distance equations in fewer steps, as exemplified in the works of Finsterwalder and Scheufele (1937), Merritt (1949), Fischler and Bolles (1981), Linnainmaa et al. (1988) and Grafarend et al. (1989)....
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TL;DR: The major direct solutions to the three point perspective pose estimation problems are reviewed from a unified perspective beginning with the first solution published in 1841 by a German mathematician and continuing through the solutions published in the German and then American photogrammetry literature, and most recently in the current computer vision literature.
Abstract: In this paper, the major direct solutions to the three point perspective pose estimation problems are reviewed from a unified perspective beginning with the first solution which was published in 1841 by a German mathematician, continuing through the solutions published in the German and then American photogrammetry literature, and most recently in the current computer vision literature. The numerical stability of these three point perspective solutions are also discussed. We show that even in case where the solution is not near the geometric unstable region, considerable care must be exercised in the calculation. Depending on the order of the substitutions utilized, the relative error can change over a thousand to one. This difference is due entirely to the way the calculations are performed and not due to any geometric structural instability of any problem instance. We present an analysis method which produces a numerically stable calculation.
574 citations
"Explicit solution of the overdeterm..." refers methods in this paper
...Extensive reviews of some of the above procedures are presented by Müller (1925) and Haralick et al. (1991, 1994)....
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01 Jan 1994
TL;DR: In this paper, the major direct solutions to the three point perspective pose estimation problems are reviewed from a unified perspective beginning with the first solution which was published in 1841 by a German mathematician, continuing through the solutions published in the German and then American photogrammetry literature, and most recently in the current computer vision literature.
Abstract: In this paper, the major direct solutions to the three point perspective pose estimation problems are reviewed from a unified perspective beginning with the first solution which was published in 1841 by a German mathematician, continuing through the solutions published in the German and then American photogrammetry literature, and most recently in the current computer vision literature. The numerical stability of these three point perspective solutions are also discussed. We show that even in case where the solution is not near the geometric unstable region, considerable care must be exercised in the calculation. Depending on the order of the substitutions utilized, the relative error can change over a thousand to one. This difference is due entirely to the way the calculations are performed and not due to any geometric structural instability of any problem instance. We present an analysis method which produces a numerically stable calculation.
546 citations
TL;DR: In this paper, a generalization of an earlier attempt by the author to obtain estimators of heteroscedastic variances in a regression model is presented, which is quite general, applicable to all experimental situations, and the computations are simple.
534 citations
01 Jul 1989-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: The authors propose an analytic solution for the perspective 4-point problem by replacing the four points with a pencil of three lines and by exploring the geometric constraints available with the perspective camera model.
Abstract: The perspective n-point (PnP) problem is the problem of finding the position and orientation of a camera with respect to a scene object from n correspondence points. The authors propose an analytic solution for the perspective 4-point problem. The solution is found by replacing the four points with a pencil of three lines and by exploring the geometric constraints available with the perspective camera model. The P4P problem is cast into the problem of solving a biquadratic polynomial equation in one unknown.
449 citations
"Explicit solution of the overdeterm..." refers background in this paper
...Other research done on the subject of resection includes the works of Müller (1925), Grafarend and Kunz (1965), Horaud et al. (1989), Lohse (1990), and Grafarend and Shan (1997a, b)....
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