High-precision local three-dimensional coordinate conversion method and device

Abstract

The present invention relates to the field of positioning, and discloses a high-precision local three-dimensional coordinate conversion method and device thereof. The high-precision local three-dimensional coordinate conversion method comprises: selecting a number of common points, which have three-dimensional rectangular coordinates of two sets of spatial coordinate systems at the same time; using the Bursa-Wolf model and the two sets of coordinates of the common points to construct a conversion equation; taking into account the random errors of the two sets of coordinates and the pathological nature of the model, solving the conversion parameter estimation based on the minimum norm criterion, and using the conversion parameter estimation to perform coordinate conversion of different coordinate systems. Compared with the traditional least squares method that only considers the random errors in one set of coordinate systems, the present invention not only considers the random errors of the two sets of coordinates at the same time, but also takes into account the disturbance caused by the model pathological nature due to the small conversion area, so that the conversion parameter estimation accuracy is higher.

Description

Technical Field

[0001] The present invention relates to the field of positioning, and in particular to a high-precision local three-dimensional coordinate conversion technology. Background Art

[0002] Modern geodetic applications often involve the conversion of three-dimensional spatial coordinates between different coordinate systems. For example, in GPS baseline network adjustment, the baseline vector determined by GPS baseline solution is based on the WGS84 coordinate system, while the points in the network may also contain coordinates of the target coordinate system (such as the city coordinate system). In this case, the coordinates of the points need to be converted from the WGS84 coordinate system to the target coordinate system. The essence of three-dimensional spatial coordinate conversion is to use the coordinates of the common points of the two coordinate systems and the first set of coordinates of the non-common points to infer the second set of coordinates of the non-common points. The conversion process includes two steps: (1) estimating the conversion parameters based on the two sets of coordinates of the common points (inversion); (2) inferring the second set of coordinates of the non-common points based on the first set of coordinates of the non-common points and the conversion parameters (forward). At present, the commonly used conversion models include the Bursa-Wolf model, the Molodensky model, and the Veis model.

[0003] If only one set of errors at the common points is considered, the transformation model can be expressed as a simple linear Gauss-Markov model, in which case the least squares method can be used to estimate the transformation parameters. In reality, the common points contain errors in both coordinate systems. For this situation, relatively mature solutions are currently available in the prior art, the main idea of ​​which is to use the total least squares criterion instead of the least squares criterion for solution. Furthermore, the Bursa-Wolf model is applicable to global coordinate transformations, requiring that the points be evenly distributed around the globe. However, the actual GNSS engineering control network often covers only a few hundred to tens of square kilometers, or even smaller. Due to the poor network configuration, directly applying the Bursa-Wolf model to solve the transformation parameters will cause the translation parameters to be strongly correlated with the rotation parameters. In this case, a small coordinate error will lead to a huge deviation in the estimated transformation parameters. For this situation, the Tikhonov regularization method is currently used in the prior art to solve the three-dimensional coordinate transformation parameters of a local area, which can effectively reduce the model's pathological nature and improve the accuracy of parameter estimation.

[0004] Taking the Bursa-Wolf model as an example, the mathematical model of three-dimensional space coordinate transformation is

[0005]

[0006] In the formula, [X i Y i Z i ] Tis the three-dimensional rectangular coordinate of point i, subscripts I and II represent the first and second coordinate systems respectively; x0, y0 and z0 are translation parameters, κ is the scale factor, ω x ,ω y and ω z To select parameters, if there are g common points in the network, equation (1) can be rewritten as a matrix form:

[0007] l=Aξ (2)

[0008] Where,

[0009]

[0010] Since both sets of coordinates are obtained from actual measurements, they inevitably contain errors. Therefore, formula (2) should be rewritten as

[0011] le l =(AE A )ξ (4)

[0012] Where, e l and E A are the error vectors (matrices) of the observation vector l and the coefficient matrix A, and ξ is the conversion parameter. At present, the total least squares criterion is mainly used to solve equation (4), that is,

[0013]

[0014] Where, e A =vec(E A ), vec(·) is the matrix straightening operator. The total least squares estimate of the transformation parameters can be derived according to the Lagrange multiplier method. This algorithm takes into account the errors of the two sets of coordinates of the common point at the same time. It is rigorous in theory, but it is only applicable to large-scale three-dimensional coordinate transformations on a global scale. When performing local area coordinate transformation, the coefficient matrix A of the model is ill-conditioned due to the non-ideal configuration of the GPS network. At this time, it is difficult to obtain a stable transformation parameter estimate by either the least squares method or the total least squares method. In view of the ill-conditioning of the transformation model, the Tikhonov regularization method is currently used to estimate the parameters by imposing regularization constraints on the translation parameters (hereinafter referred to as the Tikhonov regularization algorithm). The criterion is

[0015]

[0016] In the formula, α and R are the regularization parameter and regularization matrix respectively, where I3 is the third-order identity matrix, and O is the zero matrix. Similarly, the formula for estimating its parameters can be derived using the Lagrange multiplier method. This method accounts for the ill-posed nature of the local region transformation model, and the accuracy of the transformation parameter estimates is somewhat improved compared to traditional least squares solutions. However, it does not account for errors in the coefficient matrix (i.e., errors in the second set of common point coordinates), making it theoretically less rigorous.

[0017] To sum up, in the existing technology, when performing local three-dimensional space coordinate transformation, the traditional transformation algorithm based on the least squares criterion (or the overall least squares criterion) is difficult to obtain stable transformation parameter estimates due to the pathological nature of the model. Moreover, since the coordinates of the two sets of frames inevitably contain errors, the transformation parameter estimates are further distorted.

[0018] Therefore, there is an urgent need for a high-precision three-dimensional coordinate transformation method that can take into account both the pathological properties of the transformation model and the random errors of the two sets of frame coordinates. Summary of the Invention

[0019] The purpose of the present invention is to provide a high-precision local three-dimensional coordinate transformation method and device, which takes into account the random errors of the two sets of coordinates and the disturbance caused by model pathology due to the small transformation area, so that the obtained transformation parameter estimation accuracy is higher.

[0020] To solve the above technical problems, the embodiments of the present invention disclose a high-precision local three-dimensional coordinate conversion method, comprising the following steps:

[0021] Selecting a plurality of common points from two sets of coordinate systems respectively, and obtaining the three-dimensional rectangular coordinates of the common points in the two sets of coordinate systems, the two sets of coordinate systems including a first set of coordinate systems and a second set of coordinate systems;

[0022] Constructing a transformation equation using the Bursa-Wolf model and the two sets of coordinates of the common point;

[0023] The conversion parameter estimation of the conversion equation is solved based on Tikhonov regularization and TSVD regularization respectively. and

[0024] Calculate the above and The mean square error matrix M Tikhonov and M TSVD ;

[0025] Compare the M Tikhonov and M TSVD The trace of M Tikhonov and M TSVD The conversion parameter estimate corresponding to the smaller trace in the traces is taken as the final conversion parameter estimate;

[0026] The coordinates of the first set of coordinate systems are converted to coordinates of the second set of coordinate systems according to the final conversion parameter estimates.

[0027] The embodiment of the present invention further discloses a high-precision local three-dimensional coordinate conversion device, which is characterized by comprising:

[0028] an acquisition module, configured to select a plurality of common points from two sets of coordinate systems, respectively, and acquire the three-dimensional rectangular coordinates of the common points in the two sets of coordinate systems, wherein the two sets of coordinate systems include a first set of coordinate systems and a second set of coordinate systems;

[0029] A construction module for constructing a transformation equation using the Bursa-Wolf model and the two sets of coordinates of the common point;

[0030] Solving module, used to solve the conversion parameter estimation of the conversion equation based on Tikhonov regularization and TSVD regularization respectively and

[0031] A calculation module is used to calculate the and The mean square error matrix M Tikhonov and M TSVD ;

[0032] Comparison module, used to compare the M Tikhonov and M TSVD The trace of M Tikhonov and M TSVD The conversion parameter estimate corresponding to the smaller trace in the traces is taken as the final conversion parameter estimate;

[0033] A conversion module is configured to convert the coordinates of the first set of coordinate systems into the coordinates of the second set of coordinate systems according to the final conversion parameter estimation.

[0034] Compared with the prior art, the main differences and effects of the embodiments of the present invention are:

[0035] The random errors of the two sets of coordinates are taken into account at the same time, and the disturbance caused by model ill-conditioning due to the small conversion area is taken into account, so the obtained conversion parameter estimates are more accurate.

[0036] Furthermore, the present application is a three-dimensional coordinate transformation algorithm that can simultaneously handle two sets of coordinate errors of common points and local transformation model pathologies. Compared with the overall least squares algorithm, it takes into account the pathologies of the local area transformation model, effectively weakens its pollution to parameter estimation, and improves the stability and accuracy of the estimation; compared with the Tikhonov regularization method, it can simultaneously handle two sets of coordinate errors of common points, the theory is more rigorous, and the accuracy of transformation parameter estimation is further improved.

[0037] The specification of this application records a large number of technical features, which are distributed in various technical solutions. If all possible combinations of technical features of this application (i.e., technical solutions) are to be listed, the specification will be too lengthy. In order to avoid this problem, the various technical features disclosed in the above-mentioned invention content of this application, the various technical features disclosed in the various embodiments and examples below, and the various technical features disclosed in the accompanying drawings can be freely combined with each other to form various new technical solutions (these technical solutions are all deemed to have been recorded in this specification), unless such a combination of technical features is technically infeasible. For example, in one example, feature A+B+C is disclosed, and in another example, feature A+B+D+E is disclosed. Features C and D are equivalent technical means that play the same role. Technically, only one of them can be used, and it is impossible to use them at the same time. Feature E can be technically combined with feature C. Then, the solution of A+B+C+D should not be considered as having been recorded because it is technically infeasible, while the solution of A+B+C+E should be considered as having been recorded. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 1 is a flow chart of a high-precision local three-dimensional coordinate conversion method in the first embodiment of the present invention;

[0039] Figure 2 It is a flow chart of a preferred embodiment of the first embodiment of the present invention;

[0040] Figure 3 It is a structural schematic diagram of a high-precision local three-dimensional coordinate conversion device in the third embodiment of the present invention. DETAILED DESCRIPTION

[0041] In the following description, many technical details are provided to help readers better understand this application. However, those skilled in the art will understand that even without these technical details and the various changes and modifications based on the following embodiments, the technical solutions claimed in the claims of this application can be implemented.

[0042] To make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0043] A first embodiment of the present invention relates to a high-precision local three-dimensional coordinate transformation method. Figure 1 This is a flow chart of the high-precision local three-dimensional coordinate transformation method.

[0044] Specifically, if Figure 1 As shown, the high-precision local three-dimensional coordinate conversion method includes the following steps:

[0045] In step 101 , a plurality of common points are selected from two sets of coordinate systems respectively, and the three-dimensional rectangular coordinates of the common points in the two sets of coordinate systems are obtained. The two sets of coordinate systems include a first set of coordinate systems and a second set of coordinate systems.

[0046] In this embodiment, preferably, the three-dimensional rectangular coordinates of g common points are selected from the two sets of coordinate systems respectively. and the cofactor matrix Q of the two sets of coordinates I ,Q II Among them, [X i Y i Z i ] T is the three-dimensional rectangular coordinate of point i, and the subscripts I and II represent the first and second coordinate systems respectively.

[0047] Then, the process proceeds to step 102 , where a conversion equation is constructed using the Bursa-Wolf model and the two sets of coordinates of the common point.

[0048] Furthermore, preferably, in step 102, the following sub-steps are included:

[0049] Constructing an observation vector l and a coefficient matrix A according to the two sets of coordinates of the common point and the cofactor matrix of the two sets of coordinates;

[0050] Calculate the cofactor matrix Q of the observation vector l and the coefficient matrix A l , Q A ; Wherein, the conversion equation is:

[0051] le l =(AE A )ξ

[0052] Among them, e l and E A are the error vector matrices of the observation vector l and the coefficient matrix A respectively, and ξ is the conversion parameter.

[0053] Furthermore, preferably, in the above step of “constructing the observation vector l and the coefficient matrix A according to the two sets of coordinates of the common point and the cofactor matrix of the two sets of coordinates”, the observation vector l and the coefficient matrix A are constructed according to the following formula:

[0054]

[0055]

[0056] Wherein, g is the number of common points selected from the two sets of coordinate systems; i=1, 2,…, g; subscripts I and II represent the first set of coordinate systems and the second set of coordinate systems in the two sets of coordinate systems.

[0057] Furthermore, preferably, in the above "calculating the co-factor matrix Q of the observation vector l and the coefficient matrix A l , Q A In the step of ", the co-factor matrix Q of the observation vector l and the coefficient matrix A is calculated according to the following formula l , Q A :

[0058] Q l =Q II

[0059] Q A =JQ I J T

[0060]

[0061]

[0062] Wherein, g is the number of common points selected from the two sets of coordinate systems; subscripts I and II represent the first set of coordinate systems and the second set of coordinate systems in the two sets of coordinate systems.

[0063] Furthermore, preferably, after step 102, the following steps are further included:

[0064] Q l Perform Cholesky decomposition to get Q l =G T G, where G is an upper triangular matrix;

[0065] Generate a new observation vector l' and coefficient matrix A', where l' = Gl, A' = GA;

[0066] Then, step 103 is entered to solve the conversion parameter estimation of the conversion equation based on Tikhonov regularization and TSVD regularization respectively. and

[0067] In step 103, the estimation formulas of the conversion parameters are derived based on Tikhonov regularization and TSVD regularization, specifically,

[0068] Furthermore, preferably, in step 103, the conversion parameter estimation of the conversion equation is solved based on Tikhonov regularization. The steps include the following sub-steps:

[0069] According to the new observation vector l' and the coefficient matrix A', the regularization parameter α is calculated based on the Tikhonov regularization method;

[0070] Let i = 0, take

[0071] calculate Wherein, m is the number of rows of the coefficient matrix A. In this embodiment, preferably, m=3g;

[0072] calculate

[0073] judge Is it true, where ε is the predetermined threshold for algorithm iteration termination, and ε>0;

[0074] If so, calculate calculate Will Rearrange into a matrix And output

[0075] If not, set i=i+1 and return to perform the calculation steps.

[0076] Furthermore, preferably, the step of "calculating the regularization parameter α" includes the following sub-steps:

[0077] Select a positive integer γ, where 0<γ<5

[0078] Let i = 1;

[0079] Calculate α(i)=(i-1)·0.001;

[0080] Calculate conversion parameter estimates and residuals

[0081] Calculating distance

[0082] Determine whether i>1000γ+1 holds;

[0083] If so, calculate d is The index corresponding to the minimum value in the array is recorded as j, and α = α(j) is output;

[0084] If not, set i=i+1 and return to the step of calculating α(i)=(i-1)·0.001.

[0085] Furthermore, preferably, in step 103, the conversion parameter estimation of the conversion equation is solved based on TSVD regularization. The steps include the following sub-steps:

[0086] Select a p value where 0 < p < 1;

[0087] According to the new coefficient matrix A' and p, based on the TSVD regularization method, the truncation coefficient k and the truncation coefficient matrix A are calculated. k and its cofactor matrix

[0088] Let i = 0, take

[0089] calculate Where m is the number of rows of the coefficient matrix A, and n is the number of columns of the coefficient matrix A;

[0090] calculate

[0091] calculate

[0092] judge Is it true, where ε is the predetermined threshold for algorithm iteration termination, and ε>0;

[0093] If so, calculate calculate Will Rearrange into a matrix And output

[0094] If not, set i=i+1 and return to perform the calculation steps.

[0095] Furthermore, preferably, the step of "calculating the truncation coefficient k" includes the following sub-steps:

[0096] Perform singular value decomposition on A' and get

[0097]

[0098] in, and is an orthogonal matrix, is a diagonal matrix with diagonal element λ i is the singular value of A', 1≤i≤n, u i and v iare the i-th columns of U and V respectively;

[0099] Calculate the sum of singular values Let j = 1;

[0100] calculate

[0101] Determine whether H>p holds;

[0102] If yes, then output k=j;

[0103] If not, set j = j + 1 and return to perform the calculation steps.

[0104] Furthermore, preferably, in the above "calculating the coefficient matrix A after truncation k and its cofactor matrix In the step of ", the truncated coefficient matrix A is calculated according to the following formula k and its cofactor matrix

[0105]

[0106]

[0107] U k =(u1,u2,…,u k )

[0108] Among them, U k is the matrix consisting of the first k columns of U.

[0109] In addition, it should be noted that in other embodiments of the present invention, other ill-conditioned model estimation criteria other than Tikhonov regularization and TSVD regularization may be adopted, such as Liu estimation criterion; other nonlinear equation solving algorithms other than Gauss-Newton method may be adopted, such as Lagrange method; other methods other than the minimum mean square error criterion may be adopted to determine the regularization parameter α and the truncation parameter k, such as GCV method, L curve method, etc.

[0110] Then enter step 104, calculate the and The mean square error matrix M Tikhonov and M TSVD .

[0111] Then, step 105 is entered to compare the M Tikhonov and M TSVD The trace of M Tikhonov and M TSVD The conversion parameter estimate corresponding to the smaller trace in the traces is the final conversion parameter estimate.

[0112] In step 105, an optimal regularization method is selected based on a minimum mean square error trace criterion.

[0113] Then, the process proceeds to step 106 , in which the coordinates of the first set of coordinate systems are converted into the coordinates of the second set of coordinate systems according to the final conversion parameter estimates.

[0114] This process ends thereafter.

[0115] In summary, the embodiments of the present invention select several common points that have three-dimensional rectangular coordinates in two sets of spatial coordinate systems; construct a transformation equation using the Bursa-Wolf model and the two sets of coordinates of the common points; and simultaneously account for the random errors in the two sets of coordinates and the model's pathological properties. Based on the minimum norm criterion, the transformation parameters are solved and used to perform coordinate transformations between different coordinate systems. Compared to traditional least squares methods that only consider random errors in one set of coordinate systems, the present invention not only considers the random errors in both sets of coordinates, but also accounts for disturbances caused by model pathological properties due to the smaller transformation area, resulting in a more accurate estimate of the transformation parameters.

[0116] The present application is a three-dimensional coordinate transformation algorithm that can simultaneously handle two sets of coordinate errors of common points and the pathological nature of the local transformation model. Compared with the total least squares algorithm, it takes into account the pathological nature of the local area transformation model, effectively weakens its contamination of parameter estimation, and improves the stability and accuracy of the estimation. Compared with the Tikhonov regularization method, it can simultaneously handle two sets of coordinate errors of common points, has a more rigorous theory, and further improves the accuracy of transformation parameter estimation.

[0117] In order to better understand the technical solution of this specification, a preferred embodiment is described below. The details listed in this preferred embodiment are mainly for ease of understanding and are not intended to limit the scope of protection of this application.

[0118] The technical solution of this preferred embodiment is as follows:

[0119] Several common points are selected, each with three-dimensional rectangular coordinates in two spatial coordinate systems. A transformation equation is constructed using the Bursa-Wolf model and the two sets of coordinates of the common points. Based on the minimum norm criterion, the transformation parameters are solved, taking into account both the random errors in the two coordinate systems and the model's pathological properties. These parameters are then used to perform the coordinate transformation between the different coordinate systems. Compared to traditional least squares methods, which only consider the random errors in one coordinate system, this method not only considers the random errors in both coordinate systems, but also accounts for disturbances caused by model pathological properties due to the smaller transformation area. This results in a more accurate estimate of the transformation parameters. Figure 2 This is a schematic diagram of the overall process of this preferred embodiment. Specifically, Figure 2 As shown, the specific steps are as follows:

[0120] Step 1: Select the three-dimensional rectangular coordinates of g common points from the two coordinate systems respectively and the cofactor matrix Q of the two sets of coordinates I ,Q II , the subscripts I and II represent the first and second sets of coordinate systems; a small value ε>0 is selected as the threshold for terminating the algorithm iteration;

[0121] Step 2: Construct the coefficient matrix A and observation vector l according to formula (6-1);

[0122]

[0123] Step 3: Calculate the cofactor matrix Q of the observation vector l and the coefficient matrix A according to formula (6-2) l ,Q A ;

[0124]

[0125] Step 4: Q l Perform Cholesky decomposition to get Q l =G T G, where G is an upper triangular matrix. Generate a new coefficient matrix and observation vector according to formula (6-3)

[0126]

[0127] Step 5: Calculate the regularization parameter α according to the algorithm shown in Table 1 below;

[0128]

[0129] Table 1 Solving the regularization parameter α

[0130] Step 6: L, A, Q l ,Q A ,ε,α are used as input and the Tikhonov regularization solution is calculated according to the algorithm shown in Table 2 below.

[0131]

[0132]

[0133] Table 2 Solution steps

[0134] Step 7: L, A, Q l ,Q A ,ε,α are used as input and the algorithm shown in Table 3 is used to calculate The mean square error matrix M Tikhonov ;

[0135]

[0136]

[0137] Table 3 M Tikhonov Solution steps

[0138] Step 8: Select a p value (0<p<1), and use A' and p as input to calculate the truncation parameter k according to the algorithm shown in Table 4 below; perform singular value decomposition on A to obtain Where, and is an orthogonal matrix, is a diagonal matrix with diagonal element λ i (1≤i≤n) is the singular value of A, u i and v i are the i-th column of U and V respectively.

[0139]

[0140]

[0141] Table 4 Solution for truncation parameter k

[0142] Step 9: Calculate the truncated coefficient matrix A according to formula (6-4) k and its cofactor matrix

[0143]

[0144] Step 10: With l, A k ,Q l , ε is used as input and the TSVD regularization solution is calculated according to the algorithm shown in Table 5 below.

[0145]

[0146]

[0147] Table 5 Solution steps

[0148] Step 11: Calculate according to the algorithm shown in Table 6 below The mean square error matrix M TSVD ;

[0149]

[0150]

[0151] Table 6 M TSVD Solution steps

[0152] Step 12: Compare M Tikhonov and M TSVD The final conversion parameter estimate is the conversion parameter estimate corresponding to the smaller trace, that is, if tr(M Tikhonov )<tr(M TSVD ),but otherwise, Where tr(·) is the matrix trace operator.

[0153] (1) When the regularization method selected is Tikhonov regularization, the corresponding parameter estimation criterion is

[0154]

[0155] Comparing equations (5), (6) and the above equation, it can be found that the above equation takes into account the errors of the two sets of coordinates and the ill-conditioned nature of the model, effectively making up for the shortcomings of the total least squares method and the Tikhonov regularization method. In the prior art, a parameter estimation formula is given when R is a unit matrix. The present invention extends it to the case where R is a general symmetric (semi-) positive definite matrix. Assuming that the parameters are solved in an iterative form, the parameter estimation for the i+1th iteration is

[0156]

[0157] Where, and are the parameter estimates at the i-th iteration, is the parameter estimate correction number for the i+1th iteration, which can be estimated using the following formula:

[0158]

[0159] Where,

[0160]

[0161]

[0162] in, is the residual matrix of the prediction coefficient of the i-th iteration, and its straightened vector can be determined by the following formula:

[0163]

[0164] (2) When the regularization method selected is TSVD regularization method, perform singular value decomposition on the coefficient matrix

[0165]

[0166] Where, and is an orthogonal matrix, is a diagonal matrix with diagonal element λ i (1≤i≤n) is the singular value of A. When the coefficient matrix is ​​ill-conditioned, the singular values ​​of the tail part are very close to zero, and these singular values ​​will pollute the parameter estimation. The basic idea of ​​TSVD regularization is to cut off the small singular values ​​of the coefficient matrix. Assuming that only the first k larger singular values ​​are taken, the coefficient matrix can be reconstructed.

[0167]

[0168] Where, D k is a diagonal matrix with diagonal elements of . Replace A in formula (4) with A k ,have

[0169]

[0170] Since small singular values ​​are truncated, A k Rank deficiency, its rank is rank(A k )=k<n. The following gives The derivation of A k The relationship between and A is shown as follows:

[0171]

[0172] Where U k is the matrix consisting of the first k columns of U. k Vectorize it

[0173]

[0174] According to the error propagation law, we can get

[0175]

[0176] If the parameters are estimated iteratively, Can be rewritten as

[0177]

[0178] Where,

[0179]

[0180] ξ=ξ (i) +δξ

[0181] The current guidelines are

[0182]

[0183]

[0184] According to the above criteria, the Lagrangian function can be constructed

[0185]

[0186] Apply the above formula to e y , λ,δξ calculate the partial derivative and obtain by algebraic derivation

[0187]

[0188] Where,

[0189]

[0190] Because A k Rank deficiency, Also rank deficiency, so the formula There are countless solutions. To obtain a unique solution, the minimum norm is introduced

[0191]

[0192] The minimum norm solution can be obtained

[0193]

[0194] Where, The superscript '-' indicates pseudo-inverse.

[0195] (III) Basis for choosing between Tikhonov regularization and TSVD regularization: When Tikhonov regularization is used, the mean square error of the transformation parameters can be calculated by the following steps.

[0196] Calculate the unit weight variance After several iterations, At this point, the prediction residual can be calculated

[0197]

[0198]

[0199] Compute biased unit weighted variance

[0200]

[0201] Construct the coefficient matrix and observation values ​​after residual correction

[0202]

[0203] In summary, the embodiments of this application have the following technical features:

[0204] (1) The coordinate errors of the model's ill-conditioned and common points during local area transformation are considered simultaneously;

[0205] (2) The estimation formula of the transformation parameters and their accuracy evaluation formula were derived based on Tikhonov regularization and TSVD regularization respectively;

[0206] (3) Select the optimal regularization method based on the minimum mean square error trace criterion.

[0207] Compared with the total least squares algorithm, the implementation method of the present application takes into account the pathological nature of the local area conversion model, effectively weakens its pollution to the parameter estimation, and improves the stability and accuracy of the estimation; compared with the Tikhonov regularization method, the present invention can simultaneously process two sets of coordinate errors of the common point, the theory is more rigorous, and the accuracy of the conversion parameter estimation is further improved.

[0208] Each method embodiment of the present invention can be implemented in software, hardware, firmware, etc. Regardless of whether the present invention is implemented in software, hardware, or firmware, the instruction code can be stored in any type of computer-accessible memory (e.g., permanent or modifiable, volatile or non-volatile, solid or non-solid, fixed or removable media, etc.). Similarly, the memory can be, for example, Programmable Array Logic (PAL), Random Access Memory (RAM), Programmable Read Only Memory (PROM), Read-Only Memory (ROM), Electrically Erasable Programmable Read Only Memory (EEPROM), magnetic disk, optical disk, Digital Versatile Disc (DVD), etc.

[0209] A second embodiment of the present invention relates to a high-precision local three-dimensional coordinate conversion device. Figure 3 It is a structural diagram of the high-precision local three-dimensional coordinate conversion device.

[0210] Specifically, if Figure 3 As shown, the high-precision local three-dimensional coordinate conversion device includes:

[0211] an acquisition module, configured to select a plurality of common points from two sets of coordinate systems, respectively, and acquire the three-dimensional rectangular coordinates of the common points in the two sets of coordinate systems, wherein the two sets of coordinate systems include a first set of coordinate systems and a second set of coordinate systems;

[0212] A construction module for constructing a transformation equation using the Bursa-Wolf model and the two sets of coordinates of the common point;

[0213] Solving module, used to solve the conversion parameter estimation of the conversion equation based on Tikhonov regularization and TSVD regularization respectively and

[0214] A calculation module is used to calculate the and The mean square error matrix M Tikhonov and M TSVD ;

[0215] Comparison module, used to compare the M Tikhonov and M TSVD The trace of M Tikhonov and M TSVD The conversion parameter estimate corresponding to the smaller trace in the traces is taken as the final conversion parameter estimate;

[0216] A conversion module is configured to convert the coordinates of the first set of coordinate systems into the coordinates of the second set of coordinate systems according to the final conversion parameter estimation.

[0217] This embodiment is a device embodiment corresponding to the first embodiment, and this embodiment can be implemented in conjunction with the first embodiment. The relevant technical details mentioned in the first embodiment are still valid in this embodiment, and to reduce repetition, they are not repeated here. Accordingly, the relevant technical details mentioned in this embodiment can also be applied to the first embodiment.

[0218] It should be noted that the modules mentioned in the various device embodiments of the present invention are all logical modules. Physically, a logical module can be a physical module, a part of a physical module, or a combination of multiple physical modules. The physical implementation of these logical modules themselves is not the most important. The combination of functions implemented by these logical modules is the key to solving the technical problems proposed by the present invention. In addition, in order to highlight the innovative aspects of the present invention, the above-mentioned device embodiments of the present invention do not introduce modules that are not closely related to solving the technical problems proposed by the present invention. This does not mean that other modules do not exist in the above-mentioned device embodiments.

[0219] It should be noted that those skilled in the art should understand that the implementation functions of the modules shown in the embodiments of the above-mentioned devices can be understood with reference to the relevant descriptions of the corresponding methods. The functions of the modules shown in the embodiments of the above-mentioned devices can be implemented by a program (executable instruction) running on a processor, or by a specific logic circuit. If the above-mentioned devices of the embodiments of this specification are implemented in the form of software function modules and sold or used as independent products, they can also be stored in a computer-readable storage medium. Based on this understanding, the technical solutions of the embodiments of this specification, or the part that contributes to the prior art, can be embodied in the form of a software product, which is stored in a storage medium and includes several instructions for enabling a computer device (which can be a personal computer, a server, or a network device, etc.) to execute all or part of the methods described in each embodiment of this specification. The aforementioned storage medium includes various media that can store program codes, such as a USB flash drive, a mobile hard disk, a read-only memory (ROM), a magnetic disk or an optical disk. In this way, the embodiments of this specification are not limited to any specific combination of hardware and software.

[0220] It should be noted that in this patent application, relational terms such as first and second, etc., are used solely to distinguish one entity or operation from another, and do not necessarily require or imply any actual relationship or order between these entities or operations. Furthermore, the terms "comprise," "include," or any other variations thereof are intended to encompass non-exclusive inclusion, such that a process, method, article, or apparatus comprising a list of elements includes not only those elements but also other elements not explicitly listed, or elements inherent to such process, method, article, or apparatus. Without further limitation, an element specified by the phrase "comprising a" does not preclude the presence of additional identical elements in the process, method, article, or apparatus comprising the element. In this patent application, reference to performing an action in accordance with an element means performing the action in accordance with at least that element, including two situations: performing the action in accordance with that element alone, and performing the action in accordance with that element and other elements. Expressions such as "plurality," "multiple times," and "many" include "two," "twice," "two kinds," and "more than two," "more than two times," and "more than two kinds."

[0221] All documents mentioned in this application are considered to be included in their entirety in the disclosure of this application so that they can be used as a basis for modification when necessary. In addition, it should be understood that after reading the above disclosure of this application, those skilled in the art may make various changes or modifications to this application, and these equivalent forms also fall within the scope of protection claimed in this application.

Claims

1. A high-precision local three-dimensional coordinate transformation method, characterized in that: The following steps are involved: Selecting a plurality of common points from two sets of coordinate systems respectively, and obtaining the three-dimensional rectangular coordinates of the common points in the two sets of coordinate systems, the two sets of coordinate systems including a first set of coordinate systems and a second set of coordinate systems; Constructing a transformation equation using the Bursa-Wolf model and the two sets of coordinates of the common point; The conversion parameter estimation of the conversion equation is solved based on Tikhonov regularization and TSVD regularization respectively. and Calculate the above and The mean square error matrix M Tikhonov and M TSVD ; Compare the M Tikhonov and M TSVD The trace of M Tikhonov and M TSVD The conversion parameter estimate corresponding to the smaller trace in the traces is taken as the final conversion parameter estimate; transforming the coordinates of the first set of coordinate systems into coordinates of the second set of coordinate systems according to the final transformation parameter estimates; The conversion parameter estimation of the conversion equation is solved based on Tikhonov regularization The steps include the following sub-steps: According to the new observation vector l' and coefficient matrix A', the regularization parameter α is calculated based on the Tikhonov regularization method; Let i = 0, take calculate Where m is the number of rows in the coefficient matrix A; calculate judge Is it true, where ε is the predetermined threshold for algorithm iteration termination, and ε>0; If so, calculate calculate Will Rearrange into a matrix And output If not, set i=i+1 and return to perform the calculation Steps; The conversion parameter estimation of the conversion equation is solved based on TSVD regularization The steps include the following sub-steps: Select a p value where 0 < p < 1; According to the new coefficient matrix A' and p, based on the TSVD regularization method, calculate the truncation coefficient k and the truncated coefficient matrix A k and its cofactor matrix Let i = 0, take calculate Where m is the number of rows of the coefficient matrix A, and n is the number of columns of the coefficient matrix A; calculate calculate judge Is it true, where ε is the predetermined threshold for algorithm iteration termination, and ε>0; If so, calculate calculate Will Rearrange into a matrix And output If not, set i=i+1 and return to perform the calculation steps.

2. The method according to claim 1, characterized in that The step of constructing the conversion equation using the Bursa-Wolf model and the two sets of coordinates of the common point includes the following sub-steps: Constructing an observation vector l and a coefficient matrix A according to the two sets of coordinates of the common point and the cofactor matrix of the two sets of coordinates; Calculate the cofactor matrix Q of the observation vector l and the coefficient matrix A l , Q A ; Wherein, the conversion equation is: he l =(AE A )ξ Among them, e l and E A are the error vector matrices of the observation vector l and the coefficient matrix A respectively, and ξ is the conversion parameter; After the step of constructing the conversion equation using the Bursa-Wolf model and the two sets of coordinates of the common point, the following steps are also included: Q l Perform Cholesky decomposition to get Q l =G T G, where G is an upper triangular matrix; Generate a new observation vector l' and coefficient matrix A', where l'=Gl, A'=GA.

3. The method according to claim 1, characterized in that The step of calculating the regularization parameter α includes the following sub-steps: Select a positive integer γ, where 0<γ<5 Let i = 1; Calculate α(i)=(i-1)·0.001; Calculate conversion parameter estimates and residuals Calculating distance Determine whether i>1000γ+1 holds; If so, calculate the index corresponding to the minimum value in the dis array, record it as j, and output α = α(j); If not, set i=i+1 and return to the step of calculating α(i)=(i-1)·0.

001.

4. The method according to claim 1, wherein The step of calculating the truncation coefficient k includes the following sub-steps: Perform singular value decomposition on A' and get in, and is an orthogonal matrix, is a diagonal matrix with diagonal element λ i is the singular value of A', 1≤i≤n, u i and v i are the i-th columns of U and V respectively; Calculate the sum of singular values Let j = 1; calculate Determine whether H>p holds; If yes, then output k=j; If not, set j = j + 1 and return to perform the calculation steps.

5. The method according to claim 4, characterized in that The coefficient matrix A after calculation of truncation k and its cofactor matrix In the step, the truncated coefficient matrix A is calculated according to the following formula k and its cofactor matrix U k =(u1,u2,…,u k ) Among them, U k is the matrix consisting of the first k columns of U.

6. The method according to claim 2, characterized in that In the step of constructing the observation vector l and the coefficient matrix A based on the two sets of coordinates of the common point and the cofactor matrix of the two sets of coordinates, the observation vector l and the coefficient matrix A are constructed according to the following formula: Wherein, g is the number of common points selected from the two sets of coordinate systems; i=1, 2,…, g; subscripts I and II represent the first set of coordinate systems and the second set of coordinate systems in the two sets of coordinate systems.

7. The method according to claim 2, characterized in that In the calculation of the co-factor matrix Q of the observation vector l and the coefficient matrix A l , Q A In the step of , the cofactor matrix Q of the observation vector l and the coefficient matrix A is calculated according to the following formula l , Q A : Q l =Q II Q A =JQ I J T Wherein, g is the number of common points selected from the two sets of coordinate systems; subscripts I and II represent the first set of coordinate systems and the second set of coordinate systems in the two sets of coordinate systems.

8. A high-precision local three-dimensional coordinate conversion device for executing the method according to claim 1, characterized in that: include: an acquisition module, configured to select a plurality of common points from two sets of coordinate systems, respectively, and acquire the three-dimensional rectangular coordinates of the common points in the two sets of coordinate systems, wherein the two sets of coordinate systems include a first set of coordinate systems and a second set of coordinate systems; A construction module for constructing a transformation equation using the Bursa-Wolf model and the two sets of coordinates of the common point; Solving module, used to solve the conversion parameter estimation of the conversion equation based on Tikhonov regularization and TSVD regularization respectively and A calculation module is used to calculate the and The mean square error matrix M Tikhonov and M TSVD ; Comparison module, used to compare the M Tikhonov and M TSVD The trace of M Tikhonov and M TSVD The conversion parameter estimate corresponding to the smaller trace in the traces is taken as the final conversion parameter estimate; A conversion module is configured to convert the coordinates of the first set of coordinate systems into the coordinates of the second set of coordinate systems according to the final conversion parameter estimation.

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