CN111222250B - Method for improving parameter solving efficiency of geospatial coordinate transformation model - Google Patents
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Abstract
The invention discloses a method for improving the solving efficiency of parameters of a geospatial coordinate transformation model, and relates to the technical fields of surveying and mapping science, geographic information science and remote sensing image processing; the method comprises the steps of scaling an error item E in a coefficient matrix under a common coordinate conversion model to be a multiple of a conversion parameter, merging the multiple of the conversion parameter with an error of an observed value vector, and taking the whole of the multiple of the conversion parameter as a random error of the observed value vector by utilizing mutual offset among the errors; the iterative computation of the WTLS is realized by using a classical least squares method, the process is simple and clear, the implementation is easy, the parameter value close to the accurate solution of the WTLS can be obtained, and the computation efficiency is greatly improved; experiments prove that the method is reliable and effective, and is applicable to models comprising two-dimensional and three-dimensional four-parameter models, space seven-parameter models, polynomial fitting, affine transformation and the like.
Description
Technical Field
The invention relates to the technical fields of surveying and mapping science and technology, geographic information science and remote sensing image processing, in particular to a rapid solving method of geographic space coordinate transformation model parameters considering coincident point errors.
Background
With the continuous development of new technologies and new methods of mapping science, the shape and size of the earth are detected more and more accurately, and the countries around the world establish a country coordinate system which is more closely related to the earth and an independent coordinate system serving places.
Initially, the main technique for implementing coordinate transformation is to solve transformation parameters with the help of Least Square (LS) method by using the coordinates of coincident points under different coordinate systems. However, the LS method is theoretically better suited to the situation that the coefficient matrix has no error in coordinate transformation, in geospatial coordinate transformation, some elements of the coefficient matrix are also measured from observations with random errors, and the coefficient matrix also contains random errors like the observation vector, so that parameters obtained by using the classical LS method in geospatial coordinate transformation are approximate values and are not strict in theory. Meanwhile, a common calculation method taking random errors in the coefficient matrix and the observation vector into consideration is a whole least square method (total least squares, TLS), the common TLS method treats each element in the coefficient matrix according to equal weight, so that the method does not accord with the actual situation, the weighted total least square method (weighted total least squares, WTLS) method is an improvement of the common TLS method, the WTLS method respectively adds weights to the observation value vector and the coefficient matrix, and the defect of the common TLS method is overcome.
In general, there are mainly two ways to calculate TLS or WTLS, one based on singular value decomposition (singular value decomposition, SVD); another approach is an iterative algorithm based on lagrangian multipliers, which is a more commonly used algorithm at present. The WTLS iterative algorithm is the most rigorous algorithm for solving the problems of coordinate conversion and the like in the prior art, but the iterative method has the defects that the whole repeated iterative process is complex and time-consuming and the calculation efficiency is limited because the residual error of the coefficient matrix is required to be estimated in the iterative process. In the coordinate conversion, each coordinate point corresponds to two or three rows in the coefficient matrix, which results in two to three times increase in the data amount of the coefficient matrix when the number of coordinate points increases. Therefore, for the case of larger number of coordinate points, the problem of lower efficiency of the existing iterative algorithm is more serious.
In order to improve the calculation efficiency of WTLS, the WTLS iterative algorithm is optimized by improving the convergence speed, and the effect is not obvious, mainly because a large amount of matrix operations drag down the calculation efficiency.
Disclosure of Invention
The invention overcomes the defects of the prior art, provides a method for improving the solving efficiency of the parameters of the geospatial coordinate transformation model, and aims to obtain more accurate transformation parameter values and improve the calculating efficiency when transforming between geospatial coordinate systems.
The invention is realized by the following technical scheme.
A method for improving the solving rate of geospatial coordinate transformation model parameters specifically comprises the following steps:
1) According to the geospatial coordinate transformation model and coincident point data for parameter acquisition, a coefficient matrix A, an observation vector L and a corresponding co-factor matrix Q are formed A And Q L ；
2) Setting a tiny quantity epsilon, and estimating an initial value theta of a parameter by using a least square method 0 ；
3) Will be a parameter theta 0 To make direct product change Representing the direct product of the two matrices; further calculate the co-factor matrix m=q L +X T Q A X。
4) Constraint criterion delta using LS T M -1 The delta = min estimated conversion parameters are:
5) Updating parameters using formula (IV): θ= [ A ] T M -1 A] -1 A T M -1 L；
6) Calculate delta= |θ - θ 0 I, if delta>Epsilon, then θ is substituted for θ 0 Repeating steps 3 to 5 until delta<Ending the iterative calculation; the parameter θ obtained by the last calculation is the final conversion parameter.
The geospatial coordinate transformation model is represented by a functional model and a stochastic model representation, wherein the functional model is represented by formula (I a):
L＝(A-E)θ+e (Ιa)
the random model is shown as formula (I b):
wherein A (n x t) is a coefficient matrix, L (n x 1) is an observation value vector, E (n x t) and delta (n x 1) are random errors in the coefficient matrix and the observation vector respectively, and theta (t x 1) is a parameter value to be solved; q (Q) A (nt×nt) and Q L (n x n) is coefficient matrix and observed value vector L-form coefficient matrix, sigma respectively 2 As unit weight variance, vec (·) represents the transform to the matrix.
Further, the coefficient matrix is subjected to error-free change in the formula (i a), and the transformation is expressed as:
L＝(A-E)θ+e＝Aθ+Δ (Ⅱa)
Δ＝e-Eθ (Ⅱb)
it can be seen that the error term E in the coefficient matrix is scaled by a factor θ and then transferred into E to form Δ.
Further, applying the co-factor propagation law to equation (IIb) to obtain a co-factor matrix M of delta;
M＝Q L +X T Q A X (Ⅲ)
in the formula (III), representing the direct product of the two matrices.
Further, constraint criterion Δ of LS is utilized in step 4 T M -1 Delta=min the conversion parameters of the estimation equation (iia) yields formula (iv).
Equation (iv) is the iterative calculation of the present invention, called the iterative weighted least squares method (iteration weighted least squares, IWLS).
The estimation formula of the unit weight variance of the formula (IV) is:
the geographic space coordinate conversion model is any one of a plane similar four-parameter coordinate conversion model, an affine transformation conversion model and a seven-parameter coordinate conversion model. The method comprises the following steps:
(1) Plane similarity four-parameter coordinate conversion model:
in which x is 0 And y 0 For the translation parameters, the auxiliary parameters u=kcosa, w=ksina. Where k is the scale parameter and a is the rotation parameter. T, S the target coordinate system and the original coordinate system, respectively.
(2) Affine transformation (two-dimensional six parameters) transformation model:
parameter c in 1 And c 2 The translation parameters along the x-axis and the y-axis, respectively. Other parameters a 1 ，a 2 ，b 1 And b 2 Associated with the four physical parameters of the 2D linear transformation, which include two dimensions along the x-axis and the y-axis, one rotation and one non-perpendicularity parameter. T, S the target coordinate system and the original coordinate system, respectively.
(3) Seven-parameter coordinate conversion model:
in which x is 0 、y 0 And z 0 For translation parameters, k is a scale parameter, a x 、a y And a z Is a rotation parameter. T, S represent the target coordinate system and the original coordinate system, respectively.
The coordinate observation value inevitably contains random errors under the influence of external environment, instruments and other factors, so that the coordinate observation value is contained in the coefficient matrix and the observation vector of the three models.
Compared with the prior art, the invention has the following beneficial effects.
The algorithm of the invention can obtain the parameter value close to the accurate solution of WTLS, and the calculation efficiency is greatly improved, and the effectiveness and the advancement of the algorithm can be proved from the data of two geospatial coordinate transformation experiments.
1) The invention combines the error term E in the coefficient matrix under the common coordinate conversion model after being scaled into the multiple of the conversion parameter with the error of the observed value vector, and utilizes the mutual offset among the errors to treat the whole as the random error of the observed value vector, thereby simplifying the complexity of the random model.
2) The method directly applies the least square criterion to solve the parameters of the simplified conversion model, simultaneously gives consideration to random errors in the original coordinate system and the target coordinate system, has simple calculation and high precision, has fewer matrix operations, and effectively improves the calculation efficiency of the geospatial coordinate conversion parameters.
Drawings
Fig. 1 is a flow chart of the technical scheme of the invention.
FIG. 2 is a diagram of an affine transformation simulation experiment according to the present invention 1 、b 1 、c 1 、a 2 、b 2 、c 2 WTLS and IWLS parameters for six parameters true error contrast graphs.
FIG. 3 is a histogram of true mean square error of parameters of the affine transformation simulation experiment using LS, WTLS and IWL according to the present invention, wherein (a) is a parameter a 1 Is true mean square error of (a); (b) For parameter b 1 Is true mean square error of (a); (c) For parameter c 1 Is true mean square error of (a); (d) For parameter a 2 Is true mean square error of (a); (e) For parameter b 2 Is true mean square error of (a); (f) For parameter c 2 Is a true mean square error of (c).
Fig. 4 is the number of iterations of two methods WTLS and IWLS in an affine transformation simulation experiment according to the present invention.
Fig. 5 shows the calculation time of two methods WTLS and IWLS in the affine transformation simulation experiment according to the present invention.
Detailed Description
In order to make the technical problems, technical schemes and beneficial effects to be solved more clear, the invention is further described in detail by combining the embodiments and the drawings. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. The technical scheme of the present invention is described in detail below with reference to examples and drawings, but the scope of protection is not limited thereto.
The invention mainly solves the problem of solving the parameters of the geospatial coordinate transformation model. Classical least squares cannot take into account the effect of model coefficient matrix errors, whose resulting parameters are approximate; accurate parameters can be obtained by using general WLS, but the whole iterative process is complex, time-consuming and low in calculation efficiency, and is not suitable for combining a large data set and image coordinate conversion.
The method adopts the classical least square method to realize the iterative computation of the WTLS, has simple and clear process and easy realization, can obtain the parameter value close to the accurate solution of the WTLS, greatly improves the computing efficiency, ensures the good balance between the computing efficiency and the estimation precision, and is proved to be reliable and effective by experiments. The following specific implementation process is as follows:
1. commonly used geospatial coordinate conversion model
The transformation model commonly used for geospatial coordinate transformation generally comprises a plane similar four-parameter coordinate transformation, affine transformation, seven-parameter coordinate transformation and other models, and the specific form of the transformation model is as follows:
(1) Plane similarity four-parameter coordinate conversion model:
in which x is 0 And y 0 For the translation parameters, the auxiliary parameters u=kcosa, w=ksina. Where k is the scale parameter and a is the rotation parameter. T, S the target coordinate system and the original coordinate system, respectively.
(2) Affine transformation (two-dimensional six parameters) transformation model:
parameter c in 1 And c 2 The translation parameters along the x-axis and the y-axis, respectively. Other parameters a 1 ，a 2 ，b 1 And b 2 Associated with the four physical parameters of the 2D linear transformation, which include two dimensions along the x-axis and the y-axis, one rotation and one non-perpendicularity parameter. T, S the target coordinate system and the original coordinate system, respectively.
(3) Seven-parameter coordinate conversion model:
in which x is 0 、y 0 And z 0 For translation parameters, k is a scale parameter, a x 、a y And a z Is a rotation parameter. T, S represent the target coordinate system and the original coordinate system, respectively.
The coordinate observation value inevitably contains random errors under the influence of external environment, instruments and other factors, so that the coordinate observation value is contained in the coefficient matrix and the observation vector of the three models.
2. Principle of the method
All three conversion models can be represented by a function model (1 a) and a random model (1 b):
L＝(A-E)θ+e(1a)
wherein A (n x t) is a coefficient matrix, L (n x 1) is an observation value vector, E (n x t) and delta (n x 1) are random errors in the coefficient matrix and the observation vector respectively, and theta (t x 1) is a parameter value to be solved; q (Q) A (nt×nt) and Q L (n x n) coefficient matrix and observed value vector L-coefficient matrix, respectively，σ 2 As unit weight variance, vec (·) represents the transform to the matrix.
The invention creatively carries out coefficient matrix error-free change on the equation (1 a) in the geospatial coordinate transformation model, and represents the transformation as follows:
L＝(A-E)θ+e＝Aθ+Δ(2a)
Δ＝e-Eθ(2b)
it can be seen that the error term E in the coefficient matrix is scaled by a factor θ and then transferred into E to form Δ. Applying the co-factor propagation law to equation (2 b) results in a co-factor matrix M of Δ.
M＝Q L +X T Q A X (3)
In the method, in the process of the invention, representing the direct product of the two matrices.
In equation (2 b), Δ already includes the effect of E, at which time the LS constraint criterion Δ may be utilized T M -1 Delta = min the conversion parameters of equation (2 a), namely:
equation (4) is an iterative calculation formula of the present invention, called an iterative weighted least squares method (iteration weighted least squares, IWLS) with an estimated formula of unit weight variance:
3. the technical implementation flow of the invention
According to the technical principle of the invention, the IWLS implementation steps of the invention are described as follows:
(1) Forming a system according to a specific geospatial coordinate transformation model and coincident point data for parameter calculationNumber matrix A and observation vector L and corresponding co-factor matrix Q A And Q L 。
(2) Setting a tiny quantity epsilon, and estimating an initial value theta of a parameter by using a least square method 0 ；
(3) Will be a parameter theta 0 To make direct product changeFurther calculate the matrix m=q L +X T Q A X；
(4) Updating parameters using equation (4): θ= [ A ] T M- 1 A]- 1 A T M- 1 L；
(5) Calculate delta= |θ - θ 0 I, if delta>Epsilon, then θ is substituted for θ 0 Repeating steps (3) to (5) until delta<Epsilon ends the iterative calculation. The parameter θ obtained by the last calculation is the final conversion parameter.
The technical scheme of the invention is shown in the flow chart of figure 1.
4. Experiment one: four parameter coordinate conversion
4.1 Experimental data
The experimental data used 5 coincident points and table 1 is the coincident point coordinates of the two coordinate systems.
TABLE 1 coincident point coordinates
The corresponding weights are as follows:
synergistic factor array Q A 、Q L The composition is as follows:
in qx i And q yi Representing P S Is the inverse of the corresponding element of (a).
4.2 experimental results
We apply classical LS, existing WTLS and the IWLS method of the present invention to estimate the conversion parameters and unit weight variance. The reported WTLS rigorous algorithm solutions are basically consistent, but the WTLS optimization algorithm proposed by Jazaeri has advantages in convergence rate, and is an excellent WTLS method, so the WTLS algorithm proposed by Jazaeri is selected as a representative to be compared with the IWLS in the experimental scheme, and the iteration threshold epsilon=10 -10 The method comprises the steps of carrying out a first treatment on the surface of the The experimental results are presented in table 2, with the results of the four parameters kept at least 8 digits after the decimal point for comparison purposes.
Table 2 parameter estimation results
In Table 2, the maximum difference between WLS and IWLS is that the translation parameters differ by 0.002mm, which indicates that the four parameters and unit weight variances estimated by the algorithm of the invention are almost identical to the results of the WLS optimization algorithm of Jazaeri, the difference by 0.002mm is completely negligible, the variance is significantly reduced compared with classical LS, and the accuracy is higher than LS.
5. Experiment two, affine transformation simulation experiment
In order to further verify that the accuracy of the algorithm is equivalent to that of the existing algorithm, an affine transformation model is simulated and calculated 1000 times by using three methods of LS, WTLS and IWLS.
5.1 Experimental data
The simulation experiment takes the number d=20 of coincident points and the error-free coordinates of the original coordinate system are given in table 3. The true value of the target coordinate is calculated by the conversion model through the original coordinate and the parameter true value.
TABLE 3 original coordinate System coordinate values
In this experiment, the truth values for the six parameters were as follows:
θ＝[a 1 b 1 c 1 a 2 b 2 c 2 ] T
＝[2 -1 15 -2 1 20] T
5.2 Experimental procedure
(1) Simulating the real errors of the original coordinate system and the target coordinate system:
(2) Determining weights according to the true errors of the original coordinate system and the target coordinate system:
(3) Constructing a co-factor array Q according to a co-factor array form similar to experiment one A 、Q L 。
5.3 experimental results
The precision evaluation of the experimental result adopts three indexes, namely, the average value of the 1000 simulation parameter estimation results is compared with the true value; second, the true error of the parameters (true residual error, TRE), i.e.(i=1, 2, …, m); third, the parameter true mean square error (root mean square error of the RTE, RMS-TRE) is calculated as follows:
for ease of comparison, the true error values calculated by WTLS and IWLS are of opposite sign. As shown in fig. 2, the true error values calculated by the two iterative methods are morphologically very symmetrical, which means that the estimated parameter values are similar.
TABLE 4 truth values for parameters and average values for three method parameters
Although fig. 2 shows that the parameter values are very similar, the estimated parameters for each simulation of IWLS and WTLS are slightly different, which is reflected in the average of the estimated parameters listed in table 4. It is not easy to evaluate which iterative algorithm is close to the true value by means of the parameter average. For example, WTLS estimated c 1 The value is better than IWLS, and IWLS estimated b 2 The value is better than WTLS.
In fact, true mean square error is the most effective indicator for evaluating parameter accuracy. For visual comparison, the true mean square error of six conversion parameters estimated by three methods is represented by the histogram of fig. 3. As can be seen from fig. 3, the classical LS has the greatest true mean square error and the lowest accuracy, and the IWLS has the same accuracy as WTLS.
In order to compare the calculation efficiency of WTLS and IWLS, the total amount of the coordinate transformation overlap point is changed from 50 to 500, the step length is 50, all the calculations are calculated under the same environment, and the statistics of the iteration times and the calculation time are shown in fig. 4 and fig. 5, respectively.
The iteration number results in fig. 4 show that both IWLS and WTLS have fast convergence speeds, and the iteration numbers of IWLS and WTLS do not differ much, so there is little difference in convergence rate between them. But the IWLS algorithm consumes less time than the WTLS algorithm, which can be clearly visualized in the computation time diagram (fig. 5).
From the trend in fig. 5, it is inferred that as the number of coincident points increases, the difference in computation time between IWLS and WTLS increases, because as the number of points increases, the dimension of the coefficient matrix increases more rapidly, and in this experiment, the computation efficiency of IWLS increases by about 2.85 times compared to WTLS when the number of coordinate points is 500.
In summary, two experiments show that when the method is applied to geospatial coordinate transformation, the method (IWLS) is basically equivalent to the existing strict algorithms such as WTLS algorithm in terms of resolving precision, but the method has higher computing efficiency, and particularly has more obvious advantage when processing a large amount of data.
The error term E in the coefficient matrix under the common coordinate conversion model is scaled to be a multiple of the conversion parameter and then combined with the error of the observed value vector, the errors can be mutually offset, the whole is regarded as a random error of the observed value vector, and the complexity of the random model is simplified.
The parameters of the simplified conversion model are directly calculated by applying the least square criterion, random errors in the original coordinate system and the target coordinate system are considered, calculation is concise, accuracy is high, fewer matrix operations are provided, and the calculation efficiency of the geospatial coordinate conversion parameters is effectively improved.
While the invention has been described in detail in connection with specific preferred embodiments thereof, it is not to be construed as limited thereto, but rather as a result of a simple deduction or substitution by a person having ordinary skill in the art to which the invention pertains without departing from the scope of the invention defined by the appended claims.
Claims (6)
1. The method for improving the solving efficiency of the geospatial coordinate transformation model parameters is characterized by comprising the following steps of:
1) According to the geospatial coordinate transformation model and coincident point data for parameter acquisition, a coefficient matrix A, an observation vector L and a corresponding co-factor matrix Q are formed A And Q L ；
2) Setting a tiny quantity epsilon, and estimating an initial value theta of a parameter by using a least square method 0 ；
3) Will be a parameter theta 0 To make direct product change Representing the direct product of the two matrices; further calculate the co-factor matrix m=q L +X T Q A X；
4) Constraint criterion delta using LS T M -1 Delta = min estimation rotationThe parameters are as follows:
5) Updating parameters using formula (IV): θ= [ A ] T M -1 A] -1 A T M -1 L；
6) Calculate delta= |θ - θ 0 I, if delta>Epsilon, then θ is substituted for θ 0 Repeating steps 3 to 5 until delta<Ending the iterative calculation; the parameter theta obtained after iteration convergence is the final conversion parameter;
the geospatial coordinate transformation model is represented by a functional model and a stochastic model, wherein the functional model is represented by formula (I a):
L＝(A-E)θ+e (Ιa)
the random model is shown as formula (I b):
wherein A (n x t) is a coefficient matrix, L (n x 1) is an observation value vector, E (n x t) and delta (n x 1) are random errors in the coefficient matrix and the observation vector respectively, and theta (t x 1) is a parameter value to be solved; q (Q) A (nt×nt) and Q L (n x n) is coefficient matrix and observed value vector L-form coefficient matrix, sigma respectively 2 As unit weight variance, vec (·) represents the transform to the matrix.
2. The method for improving the solving efficiency of the geospatial coordinate transformation model parameters according to claim 1, wherein the coefficient matrix is subjected to error-free change in the formula (i a), and the transformation is expressed as:
L＝(A-E)θ+e＝Aθ+Δ (Ⅱa)
Δ＝e-Eθ (Ⅱb)
the error term E in the coefficient matrix is scaled by a factor θ and then transferred into E to form Δ.
3. The method for improving the solving efficiency of the parameters of the geospatial coordinate transformation model according to claim 2, wherein the co-factor propagation law is applied to equation (IIb) to obtain a co-factor matrix M of delta;
M＝Q L +X T Q A X (Ⅲ)
in the method, in the process of the invention, representing the direct product of the two matrices.
4. The method for improving the solving efficiency of parameters of a geospatial coordinate conversion model according to claim 2, wherein the constraint criterion Δ of LS is used in step 4 T M -1 Delta=min the conversion parameters of the estimation equation (iia) yields formula (iv).
5. The method for improving the solving efficiency of the geospatial coordinate transformation model parameters according to claim 1, wherein the unit weight variance of the formula (iv) is estimated as:
6. the method for improving the parameter solving efficiency of a geospatial coordinate transformation model according to claim 1, wherein the geospatial coordinate transformation model is any one of a planar similar four-parameter coordinate transformation model, an affine transformation model and a seven-parameter coordinate transformation model.
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| EP1899678A1 (en) * | 2005-06-28 | 2008-03-19 | Scanalyse Pty Ltd | A system and method for measuring and mapping a surface relative to a reference |
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