CN113569393B - Robust absolute orientation algorithm - Google Patents
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Abstract
The invention relates to a robust absolute orientation algorithm, which comprises the following steps of S1, constructing an absolute orientation problem solving model based on a weighting coefficient; step S2, setting the weight of each coordinate point asStep S3, deducing an analytic solution algorithm of an absolute directional problem solving model, and calculating a rotation matrix R and a translation vector t corresponding to the weight in the step S2; step S4, calculating the superposition error e i ＝||Rr' A,i ‑r' B,i After updating the weight by using the S estimation method, entering a step S3, and calculating a rotation matrix and a translation vector corresponding to the current weight; step S5, continuously looping steps S4 and S3, and outputting a rotation matrix R and a translation vector t when the iteration number of the S estimation method exceeds the set maximum iteration number. According to the invention, the coarse difference point can be automatically identified through the construction of the absolute directional problem solving model and the application of the updating weight of the S estimation method, and then through the analytic algorithm of the absolute directional problem solving model, the robustness of the algorithm is improved, and the solving precision of the algorithm is ensured.
Description
Technical Field
The invention relates to the field of photogrammetry, in particular to a robust absolute orientation algorithm.
Background
From the three-dimensional coordinates of a given n spatial points in two coordinate systems A, B, the problem of the rotation matrix R and translation vector t of coordinate system a relative to coordinate system B is solved, referred to as the absolute orientation problem. Absolute orientation is an important and fundamental problem in computer vision, and has wide application in the fields of point cloud matching, photogrammetry and the like.
The existing solving algorithm of the absolute orientation problem mainly comprises two types of an analysis algorithm and an iterative optimization algorithm. The analysis algorithm mainly comprises an absolute orientation algorithm based on an orthogonal matrix, an absolute orientation algorithm based on a quaternion and the like, and the main ideas of the algorithms are to solve and obtain an optimal solution of the absolute orientation problem through a matrix optimization technology. The iterative optimization algorithm comprises nonlinear optimization methods such as a beam adjustment algorithm, and the main idea of the algorithm is to use numerical optimization solving algorithms such as a least square method, and the like, and the residual error is reduced by continuous iteration until the residual error is smaller than a threshold value, so that an approximate optimal solution of the absolute orientation problem is obtained.
In solving the absolute orientation problem using actual data, measurement of three-dimensional space points often has observation noise and gross errors. For example, in point cloud registration, when three-dimensional point cloud data obtained by scanning in different directions are registered, rough differences exist in corresponding coordinates due to the reasons of limited measurement precision, different point cloud densities, partial point cloud data missing, characteristic point matching errors and the like. The traditional two resolving algorithms are solved under ideal conditions, the influence of coarse difference is not considered, and non-steady statistic such as arithmetic mean value and the like are used in the solving process, so that the robustness of the two resolving methods is not strong. The absolute orientation algorithm based on the MF estimator builds an objective function capable of inhibiting the influence of errors by modeling the errors and the coarse factors, so that the influence of the coarse is reduced to a certain extent, but the algorithm does not give an analytical solution, and the solving efficiency is limited.
Disclosure of Invention
Aiming at the problems, the invention provides a robust absolute orientation algorithm which can automatically identify rough difference points and improve algorithm robustness.
The technical scheme adopted for solving the technical problems is as follows: a robust absolute orientation algorithm comprising the steps of:
s1, constructing an absolute orientation problem solving model based on a weighting coefficient,
let n spatial points have three-dimensional coordinates r in coordinate systems A and B A,i And r B,i (i=1, 2, …, n) solving the model by a weighted method using an absolute directional problem in weighted form
Is used as an estimate of the rotation matrix and translation vector, where ω i More than or equal to 0 represents the weight corresponding to the ith coordinate point andSO (3) represents a third-order orthogonal matrix set with a determinant of 1, and I represents a 2-norm;
s2, setting the weight of each coordinate pointI.e. < ->
S3, deducing an analytic solution algorithm of an absolute directional problem solving model to obtain an optimal solution of a formula (2), calculating a rotation matrix R and a translation vector t corresponding to the weight value in S2,
the weighted centroid expression in the A, B coordinate system is recorded asAnd->(2) In the model of the formula, the solution of the rotation matrix and the translation vector is coupled, and new coordinates of an ith space point are introduced for decoupling>And->Ideal case->Wherein R is true And t true True values of rotation matrix and translation vector respectively, and then
r' B,i ＝R true r' A,i ， (3)
And therefore, the optimization model is solved first
Obtaining a rotation matrix R corresponding to the current weight * Through again
Obtaining translation vector t in displacement parameter * Thereby realizing the aim of optimizing parameter decoupling,
due to
First two itemsIndependent of the rotation matrix, therefore, the optimization model of equation (4) is equivalent to
Wherein the method comprises the steps oftr () is the trace function of the matrix, and for solving the formula (7), singular value decomposition is performed on the matrix M to obtain +.>Wherein U and V are 3×3 orthogonal matrices, σ j 0 (j=1, 2, 3) is the singular value of matrix M, when
R * ＝V T U, (8)
(7) The maximum value is obtained and the maximum value is obtained,
to obtain R * Corresponding translation vector, deriving t from (5)The derivative is set to 0, and the translation direction corresponding to the weight value is obtainedThe amount is
S4, calculating the superposition error e i ＝||Rr' A,i -r' B,i After updating the weight by using the S estimation method, entering a step S3, calculating a rotation matrix corresponding to the current weight according to the formula (8), calculating a superposition error, and calculating a translation vector corresponding to the current weight according to the formula (9);
s5, continuously cycling the steps S4 and S3, and outputting a rotation matrix R and a translation vector t when the iteration number of the S estimation method exceeds the set maximum iteration number.
Preferably, the S estimation method in step S4 is specifically,
normalizing the error when the cycle number is 1Wherein->For the purpose of the scale estimation,represented |e i The median, 0.6754, is a constant in the S estimation, using
When the number of loops is greater than 1, the new weight is usedNormalization of the observation error>Using
As a new weight.
Compared with the prior art, the invention has the following beneficial effects:
1. according to the invention, the coarse difference points can be automatically identified through the construction of the absolute directional problem solving model and the application of the updating weight of the S estimation method, and then through the analytic algorithm of the absolute directional problem solving model, the influence caused by the coarse difference points can be effectively restrained, and under the condition of more coarse difference points, the solving precision can be ensured, and the robustness of the absolute directional algorithm is improved;
2. the invention provides an iterative weight calculation method based on an S estimation method, which can automatically identify rough difference points and automatically calculate proper weights of all coordinate points to realize the inhibition of the rough difference points.
Drawings
FIG. 1 is a graph showing the average solution error of a rotation matrix and a translation vector as a function of the percentage increase in the coarse difference;
wherein, the graph (a) corresponds to the rotation matrix error and the graph (b) corresponds to the translation vector error.
Detailed Description
The present invention will be described in detail with reference to fig. 1, wherein the exemplary embodiments and descriptions of the present invention are provided for explaining the present invention, but not limiting the present invention, and the absolute orientation algorithm of the present invention is the Proposed method algorithm.
A robust absolute orientation algorithm comprising the steps of:
s1, constructing an absolute orientation problem solving model based on a weighting coefficient,
let n spatial points have three-dimensional coordinates r in coordinate systems A and B A,i And r B,i (i=1, 2, …, n) solving the model by a weighted method using an absolute directional problem in weighted form
Is used as an estimate of the rotation matrix and translation vector, where ω i More than or equal to 0 represents the weight corresponding to the ith coordinate point andSO (3) represents a third-order orthogonal matrix set with determinant 1, and I represents 2 norms, if the weight is selected to satisfy the i coordinate point, omega is greater in the degree of the rough difference i The principle that the difference point is close to 0 is that the influence of the rough difference point on the optimization objective function of the formula (2) can be effectively reduced, so that the correct rotation matrix and translation vector can be obtained by solving the optimization model of the formula (2);
s2, setting the weight of each coordinate pointI.e. < ->
S3, deducing an analytic solution algorithm of an absolute directional problem solving model to obtain an optimal solution of a formula (2), calculating a rotation matrix R and a translation vector t corresponding to the weight value in S2,
the weighted centroid expression in the A, B coordinate system is recorded asAnd->(2) In the model of the formula, the solution of the rotation matrix and the translation vector is coupled, and new coordinates of an ith space point are introduced for decoupling>And->Ideal case->Wherein R is true And t true True values of rotation matrix and translation vector respectively, and then
r' B,i ＝R true r' A,i ， (3)
And therefore, the optimization model is solved first
Obtaining a rotation matrix R corresponding to the current weight * Through again
Obtaining translation vector t in displacement parameter * Thereby realizing the aim of optimizing parameter decoupling,
due to
First two itemsIndependent of the rotation matrix, therefore, the optimization model of equation (4) is equivalent to
Wherein the method comprises the steps oftr () is the trace function of the matrix, and for solving the formula (7), singular value decomposition is performed on the matrix M to obtain +.>Wherein U and V are 3×3 orthogonal matrices, σ j 0 (j=1, 2, 3) is the singular value of matrix M, when
R * ＝V T U, (8)
(7) The maximum value is obtained and the maximum value is obtained,
to obtain R * Corresponding translation vector, deriving t from (5)Let the derivative be 0 to obtain a translation vector corresponding to the weight value as
S4, calculating the superposition error e i ＝||Rr' A,i -r' B,i After updating the weight by using the S estimation method, the method enters a step S3, calculates a rotation matrix corresponding to the current weight according to the formula (8), calculates the superposition error, calculates a translation vector corresponding to the current weight according to the formula (9),
normalizing the error when the cycle number is 1Wherein->For the purpose of the scale estimation,represented |e i The median, 0.6754, is a constant in the S estimation, using
When the number of loops is greater than 1, the new weight is usedNormalization of the observation error>Using
As a new weight;
s5, continuously cycling the steps S4 and S3, and outputting a rotation matrix R and a translation vector t when the iteration number of the S estimation method exceeds the set maximum iteration number.
In order to verify the effectiveness of the robust absolute orientation problem solving algorithm provided by the invention, the robustness is compared with a classical AbsPose algorithm based on an orthogonal matrix.
In the following simulation experiments, the spatial point coordinates and translation vector coordinates were in mm. The rotation matrix in the simulation test is a randomly generated SO (3) matrix, and three components of the translation vector t are in [0,1000 ]]The random number selected in the group. The coordinates of 20 ideal space points in the A coordinate system are uniformly distributed in [ -30,30]×[-30,30]×[-30,30]And (3) rotating and translating the ideal space point coordinate in the A coordinate system to obtain the ideal space point coordinate in the B coordinate system. To test robustness, the ideal spatial point coordinates in the A, B coordinate system are added to a mean value of 0, gaussian noise with variance of 0.2, then m (m=1,..10.) points were randomly selected in the B coordinate system, and 0,50 were added, respectively]Is used to simulate the coarse difference. The error of the rotation matrix is defined asWherein r is ij,true And r ij Elements of the ith row and jth column representing the rotation matrix true and resolved values, respectively, the translation vector error is defined as +.>Wherein t is true And t represent translation vector true and resolved values, respectively. For each m, independent experiments were performed 1000 times.
Fig. 1 shows the variation of the average solution error of the rotation matrix and the translation vector with the increase of the percentage of the coarse difference point, wherein the graph (a) corresponds to the rotation matrix error, the graph (b) corresponds to the translation vector error, the dots are the solution result of the absPose algorithm, and the square dots are the solution result of the algorithm provided by the invention. As can be seen from the graph, as the number of coarse difference points increases, both the rotation matrix error and the translation vector error of the abswise algorithm show a growing trend, and the growing speed is far greater than that of the robust absolute orientation problem solving algorithm provided by the invention. Under the condition that the percentages of the rough difference points are the same, the algorithm provided by the invention can greatly improve the measurement accuracy of the rotation matrix and the translation vector. When the rough difference point percentage is 50%, the average translation vector relative error obtained by the AbsPose algorithm is as high as 74.0%, and the translation vector relative error of the algorithm is only 2.3%, so that the solution accuracy of one order of magnitude is improved compared with the AbsPose. Therefore, the algorithm provided by the invention can effectively inhibit the influence caused by the rough difference points, can still ensure the solving precision under the condition of more rough difference points, and has good robustness.
The foregoing has described in detail the technical solutions provided by the embodiments of the present invention, and specific examples have been applied to illustrate the principles and implementations of the embodiments of the present invention, where the above description of the embodiments is only suitable for helping to understand the principles of the embodiments of the present invention; meanwhile, as for those skilled in the art, according to the embodiments of the present invention, there are variations in the specific embodiments and the application scope, and the present description should not be construed as limiting the present invention.
Claims (2)
1. A robust absolute orientation algorithm, characterized by: the method comprises the following steps:
s1, constructing an absolute orientation problem solving model based on a weighting coefficient,
let n spatial points have three-dimensional coordinates r in coordinate systems A and B A,i And r B,i (i=1, 2, …, n) solving the model by a weighted method using an absolute directional problem in weighted form
Is used as an estimate of the rotation matrix and translation vector, where ω i More than or equal to 0 represents the weight corresponding to the ith coordinate point andrepresenting a set of third-order orthogonal matrices of determinant 1, representation of 2 norms;
s2, setting the weight of each coordinate pointI.e. < ->
S3, deducing an analytic solution algorithm of an absolute directional problem solving model to obtain an optimal solution of a formula (2), calculating a rotation matrix R and a translation vector t corresponding to the weight value in S2,
the weighted centroid expression in the A, B coordinate system is recorded asAnd->(2) In the model of the formula, the solution of the rotation matrix and the translation vector is coupled, and new coordinates of an ith space point are introduced for decoupling>Andideal case->Wherein R is true And t true True values of rotation matrix and translation vector respectively, and then
r' B,i ＝R true r' A,i ， (3)
And therefore, the optimization model is solved first
Obtaining a rotation matrix R corresponding to the current weight * Through again
Obtaining translation vector t in displacement parameter * Thereby realizing the aim of optimizing parameter decoupling,
due to
First two itemsIndependent of the rotation matrix, therefore, the optimization model of equation (4) is equivalent to
Wherein the method comprises the steps oftr () is the trace function of the matrix, and for solving the formula (7), singular value decomposition is performed on the matrix M to obtain +.>Wherein U and V are 3×3 orthogonal matrices, σ j 0 (j=1, 2, 3) is the singular value of matrix M, when
R * ＝V T U, (8)
(7) The maximum value is obtained and the maximum value is obtained,
to obtain R * Corresponding translation vector, deriving t from (5)Let the derivative be 0 to obtain a translation vector corresponding to the weight value as
S4, calculating the superposition error e i ＝||Rr' A,i -r' B,i After updating the weight by using the S estimation method, entering a step S3, calculating a rotation matrix corresponding to the current weight according to the formula (8), calculating a superposition error, and calculating a translation vector corresponding to the current weight according to the formula (9);
s5, continuously cycling the steps S4 and S3, and outputting a rotation matrix R and a translation vector t when the iteration number of the S estimation method exceeds the set maximum iteration number.
2. A robust absolute orientation algorithm according to claim 1, characterized in that: the S estimation method in step S4 specifically includes,
normalizing the error when the cycle number is 1Wherein->For the scale estimator +.>Represented |e i Median, 0.6754 is a constant in the S estimation, using
When the number of loops is greater than 1, the new weight is usedNormalization of the observation error>Using
As a new weight.
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