Geodesy and Geodynamics 2012,3 ( 1) :34 -38 http://www. jgg09. com Doi:10.3724/SP.J.1246.2012.00034

Correlation of coordinate transformation parameters

1 2 1 1 Du Lan , Zhang Hanwei , Zhou Qingyong and Wang Ruopu 'Institute of Surveying and Mapping, lnforTTUHion Engirreering University, Zhengzhou 450052, China 2 School of Surveying & Innd Informatwn Engirreering, Henan Polytecknic University, ]i.oozuo 454000, China

Abstract: Coordinate transformation parameters between two spatial Cartesian coordinate systems can be solved from the positions of non-colinear corresponding points. Based on the characteristics of translation, rotation and zoom components of the transformation, the complete solution is divided into three steps. Firstly, positional vectors are regulated with respect to the centroid of sets of points in order to separate the translation compo­ nents. Secondly, the scale coefficient and rotation matrix are derived from the regulated positions independent­ ly and correlations among transformation model parameters are analyzed. It is indicated that this method is ap­ plicable to other sets of non-position data to separate the respective attributions for transformation parameters. Key words: coordinate transformation model ; Bursa model; orthnormal matrix ; singular value decomposition ( SVD) ; correlation

(R1 ,R2 ,R3 ) T, which is called seven-parameter trans­ 1 Introduction formation model. For the small values of hoth the scale coefficient and the angles , which are common in space Coordinate transformations include translation , rotation geodesy, we define and zoom. On the basis of non -colinear corresponding point from two sets of spatial Cartesian coordinate sys­ >.(1 +D) and R =(I +R') (2) tem, inverse solution of transformation parameters is widely used in geodesy, photogrannnetry, computer vi­ and equation is expressed as ( Bursa model) [ ll : sion, etc.

I x2 Let X and denote a pair of points' and they X,= T +X1 +DX1 +R' X1 (3) satisfy where

X, =T+J.RX1 (1) -R, Where T = ( T,, T , T,) T and R = R, (R )R (R, )R, R,l 1 1 1 R' = [ ;, 0 -R, ( R ) , they are the rotation matrix and the translation 3 -R, R, 0 vector, respectively; and A is a scale. The rotation parameter is expressed by Euler angles is an anti-symmetric matrix, and D is scale factor. When there are no less than three non -colinear cor­ Received :2012-01-12; Accepted :2012-01-19 responding points , the seven transformation parameters Corresponding author: Tel: + 86-0371-81636076; E-mail: Lan. du09@ gmail. com; can he computed by the least-squares adjustment[']. This work was supported by the National Natural Science Fowulation of However, in practice , the strong correlations among China( 41174ll25 ,41174026) transformation parameters often make some solved pa- No.1 Du Lan, et al. Correlation of coordinate transformation parameters 35 rameters not accurate[3 J. In reference[ 4] , the param­ eters are divided into several groups, and the general­ ized correlation coefficients between the groups are an­ 2 alyzed. With a sampling area of 100 X 100 km , it is concluded that translation parameters have strong cor­ relation with rotation and scale parameters, while the latter two are un-correlated. For generalized correlation coefficients are merely suitable to numerical validation, more rigorous certification are needed. (a) Spatial relation between points in tbe original set In this paper, the author proposes a new method to parameter inversion in coordinates transformation. U­ sing this method, the consistent conclusions with refer­ ence [ 4] are drawn. Furthermore, this method can be used in some other kinds of non-positional analysis, such as baseline vectors or attitude matrix, in terms of fusion of multi -source information. (b) Spatial relation between points in tbe original set witb respect to centroid 2 Position sensitivity to coordinate Figure 1 Spatial relation between point set transformation parameters (6) There are two sets of position vectors 1xli l and 1x2i l

( i = 1, 2, · · ·, N) , and the parameters of the transla­ as the new sets of position with normalized coordi­ tion, rotation and zoom. In order to decouple the po­ nates. Substitute equations ( 4 ) - ( 6 ) into equation sition sensitivity to coordinate transformation parame­ ( 1 ) , the new relation between 1x;, ~ and 1X~i ~ IS ters, two corresponding centroids are introduced to derive the translation information, while the rotation (7) and scale information are left in the normalized position sets. The translation transformation is eliminated , as 2. 1 Centroidal regulation shown in figure 1 ( b) . Note that the position errors of the new set of points Suppose the position errors are independent, and have have changed slightly due to the errors in the original the equal precision , then define centroid coordinates. To equation ( 6) , the error prop­ agations satisfy (4)

(8) where, X 1c and X2c are the centroid coordinates of

1xli l and 1x2i l ' respectively. Where m is the standard error. Obviously, the normal­ 1:xli l is supposed to completely match 1x2i l after ized coordinates will retain the same precisions with the the transformation if the coordinate errors are negli­ original ones if the corresponding points are enough. gible, hence, the two centroids overlap. (Fig. 1 (a) ) and satisfy equation ( 1 ) . It can be written as 2. 2 Dependence of translation on rotation and zoom (5) After the normalization with respect to centroid , the original sets of points have been divided into two parts, 36 Geodesy and Geodynamics Vol. 3 the centroids and the normalized coordinates , while the tation matrix itself to be taken into calculation as a tranalation information is completely absorbed by the whole is much more universal and acceptable. corresponding centroids ( compared with fignre I ( a) Herein, the optimal rotation matrix R is derived di­ and (b)). recdy through singnlar value decomposition ( SVD) as With the known estimates of both k and A, we have well as the properties of orthogonal and rotation matri­ the following relation for the translation parameters 1' ces. Some other derivation methods based on the clas­ from equation ( 5 ) sical least-squares algorithm can be seen in references [2] and [6]. (9) First, we construct the 3 x N position matrices from the normalized coordinates , namely, X and Y Obviously , 1' have a linear relation with the combina­ tion of R and A. Therefore , if all the seven parameters x = (x;, ... x;,. .. x;N) ( 11) are fitted by the least-squares, the strong correlations Y = (x;, ... x;,. .. x~) will exist in the estimated parameters of translation and the combination of rotation and zoom. and they satisfy equation ( 7 )

3 Decoupling of rotation and zoom (12)

Then construct the 3 X 3 matrix of S by 3. 1 Scale estimation

Due to the module-retained property nf rotation matrix, S=XYT ( 13) we have I RX;; I = I x;; I . Now compute the 2-nonn of equation ( 7 ) , we have Let

N ' 2 (14) LIX2;1 j:-1 (10) Be an SVD of the real square matrix S, where U and V are orthogonal matrices and r is diagonal. Which shows that the optimal scale is independent to From equation (14) , we have the optimal rotation parameters. uTsv= r 3. 2 Rotation matrix estimation with orthogonality ( 15) { vTsTu=rT constraint

There are several parameterizations of the rotation, but where L =L T , for they are the diagonal matrix, SO different parameters will affect the algorithm and accu­ racy[SJ . Euler angles are usually adopted due to the UTSV=VTSTU (16) clear geometrical meanings , while they need complex trigonometric arithmetic and may have potential mathe­ Now we cao substitute the relation of X aod Y, the def­ matical singularity. It is convenient to construct linear inition of matrix S, and the equations ( 12) and ( 13) , equations by the elements of rotation matrix, but inevi­ into equation ( 16) tably some nonlinear constraints are needed. Quatemi­ on is the most applicable to the rigid dynamics if in­ A.UTXXTRTV =A.VTRXXTU ( 17) volved. Otherwise, the advantages will discount be­ cause the non-typical arithmetic of quaternion is con­ Note that the scale can be canceled in equation siderably hard to understand. On the contrary, the ro- ( 17 ) , aud then left multiple by U aud right multiple No.I Du Lan, et al. Correlation of coordinate transformation parameters 37 by yrR for the both two sides of equation ( 17 ) , we nates of the original set of points. However, from the 2- have norm of differential nf equation ( 14) , we have II {}S II = II {)I, II , which indicates SVD would not magnify xxr = ( UVTR)XXT ( UVTR) (18) the impact of noises and can ensure the precision of the estimated rotation matrix[SJ. Where xxr .. 0 ' because the normalized coordinates 4) It would he convenient to assume that the points cannot he all at the centroids. Therefore, for the exist­ and their positional components are independent when ence of equation ( 18 ) , the prerequisite is , the position accuracy nf the two sets of points are taken into consideration simultaneously. For example, the UVTR=( ±1)1 (19) weighted coordinates of the two centroids can he re­ fined accounting for the individnal pnsitional compo­ Where I is 3 X 3 identity matrix. nent precision of the respective set of points in equation From the definition, the determinants of rotation ma­ ( 4) , whereas 3 -dimension position precision of each trix R and orthogonal matrices U and V are + 1 and point, instead of positional components, is weighted to ± 1 , respectively. So the optimal rotation matrix compute the scale in equation ( 10) . Similarly, matrix should he S can he modified as S = XMY" , where the pnsition er­ rors matrix M = diag ( llmi lim; ···,lim~) may •. {vur,det(VUr) =1 he determined by the set nf point which is relatively R = (20) - VUr,det(VUT) = -1 poor in precision, where mi denotes the position mean square error. Where det denotes the determinant nf a matrix. For compactness, equation ( 20) can he written as a 4 Contribution of multi -source data uniform expression Currendy , besides the pairs of 3 -dimension points, the 1 0 varieties of corresponding information increase gradual­ R=V 0 1 0 Jur (21) ly with the rapid development nf multi-sensors and ( 0 0 det( VUT) measurement techniques. For example, relative posi­ tioning of GNSS can provide high precision baseline Obviously , the optimal rotation matrix is independ­ vectors, and a 6-freedom sensor can obtain direcdy at­ ent to the optimal scale. titude matrix. Therefore , multi -source data fusion is a In the estimation of i and R , the rotation and scale privilege tool to take full advantage of different types of parameters are irrelative to each other. data regarding their respective contributions. Furthermore, we can discuss in details that: For example, baseline vectors obtained from relative 1 ) If there is only zoom between the two sets of positioning, linear-character extraction, etc, are sensi­ point, matrix S is symmetric , S = XYT = AXXT while tive to both rotation and zoom; attitude matrices ac­ the SVD is S = U I, yr = VI, yr. Therefore , the rota­ qnired from 6-freedom sensor-platforms and vision ro­ tion matrix is identity matrix with R = VVT =I. bots, are sensitive to rotation. Whether the above-men­ 2) All the rotation information is completely includ­ tioned analytical approach can be applied to all types of ed in matrix S, no matter whether the rotation matrix R corresponding information analysis , is determined by or quaternion is adopted as the parameterization of rota­ whether it is sensitive to translation, zoom and rota­ tions. In fact, quaternion-hased algorithm still needs to tion, furthermore , the contribution to transformation construct matrix S first , then a 4 x 4 matrix from linear parameters and calculation methods can he deter­ combination the elements of matrix S , and last , do mined. eigen-value decomposition to the 4 x4 matrix[']. To utilize these evident contributions to the correla­ 3) Matrix S is subject to the random errors in coordi- tions and solutions of transformation parameters , we 38 Geodesy and Geodynamics Vol. 3 should expand the corresponding calculations. considerably large part of the translation quantity with a Substituting sub-matrices dX and ax· constituted prior value before fitting seven-parameter transforma­ from pairs of baseline vectors , ll1 and ll,. attitude ma­ tion model. We can see that it suggests the similar de­ trices and into equation ( 12) , we have extended ma­ coupling effect as the centroid-regulation so that it also trices performs better than the direct seven-parameter fit, the numerical validation of which can be seen in reference X=(X dX IJ,.) [ 3] for more details. (22) Y=(Y ax· ll) 3 ) Of all the corresponding pairs of two coordinate systems , 3 -dimension points are sensitive to all the Then we reconstruct matrix S by X and Y. With the transformation motions; baseline vectors are independ­ SVD of S , the rotation matrix fl can be solved. ent from the translation ; attitude matrices are merely Similarly , we substitute the length constraints of sensitive to the rotation. We should optimize the solu­ baseline vectors into equation ( 10) , then tions of the transformation parameters from multi -source data according to their measurement accuracy and con­ tribution characteristics.

(23) References

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