Solution of the weighted symmetric similarity transformations based on quaternions

Abstract

A new method through Gauss–Helmert model of adjustment is presented for the solution of the similarity transformations, either 3D or 2D, in the frame of errors-in-variables (EIV) model. EIV model assumes that all the variables in the mathematical model are contaminated by random errors. Total least squares estimation technique may be used to solve the EIV model. Accounting for the heteroscedastic uncertainty both in the target and the source coordinates, that is the more common and general case in practice, leads to a more realistic estimation of the transformation parameters. The presented algorithm can handle the heteroscedastic transformation problems, i.e., positions of the both target and the source points may have full covariance matrices. Therefore, there is no limitation such as the isotropic or the homogenous accuracy for the reference point coordinates. The developed algorithm takes the advantage of the quaternion definition which uniquely represents a 3D rotation matrix. The transformation parameters: scale, translations, and the quaternion (so that the rotation matrix) along with their covariances, are iteratively estimated with rapid convergence. Moreover, prior least squares (LS) estimation of the unknown transformation parameters is not required to start the iterations. We also show that the developed method can also be used to estimate the 2D similarity transformation parameters by simply treating the problem as a 3D transformation problem with zero (0) values assigned for the z-components of both target and source points. The efficiency of the new algorithm is presented with the numerical examples and comparisons with the results of the previous studies which use the same data set. Simulation experiments for the evaluation and comparison of the proposed and the conventional weighted LS (WLS) method is also presented.

Appendices

Appendix

1.1 A Partial derivatives of scaled rotation matrix

If we write the scaled rotation matrix $k\mathbf {R}$ in terms of the scaled quaternion elements $q_0$, $q_1$, $q_2$, $q_3$ in following explicit form

$$\begin{aligned} k\mathbf {R}= \begin{bmatrix} q_0^2 + q_1^2 -q_2^2 - q_3^2&\quad 2(q_1 q_2 - q_0 q_3)&\quad 2(q_1 q_3 + q_0 q_2)\\ 2(q_1 q_2 + q_0 q_3)&\quad q_0^2 - q_1^2 +q_2^2 - q_3^2&\quad 2(q_2 q_3 - q_0 q_1)\\ 2(q_1 q_3 - q_0 q_2)&\quad 2(q_2 q_3 + q_0 q_1)&\quad q_0^2 - q_1^2 -q_2^2+q_3^2\\ \end{bmatrix}\nonumber \\ \end{aligned}$$

(A.1)

then the partial derivatives matrices $\mathbf {Q}_i$ of matrix $k\mathbf {R}$ w.r.t. $q_j$ (for $j=0,\,1,\,2,\,3$) read:

$$\begin{aligned} \mathbf {Q}_0= & {} \frac{\partial k \mathbf {R}}{\partial q_0}=2 \begin{bmatrix} q_0&\quad -q_3&\quad q_2\\ q_3&\quad q_0&\quad -q_1\\ -q_2&\quad q_1&\quad q_0\\ \end{bmatrix} \end{aligned}$$

(A.2)

$$\begin{aligned} \mathbf {Q}_1= & {} \frac{\partial k \mathbf {R}}{\partial q_1}=2 \begin{bmatrix} q_1&\quad -q_2&\quad q_3\\ q_2&\quad -q_1&\quad -q_0\\ q_3&\quad q_0&\quad -q_1\\ \end{bmatrix}\end{aligned}$$

(A.3)

$$\begin{aligned} \mathbf {Q}_2= & {} \frac{\partial k \mathbf {R}}{\partial q_2}=2 \begin{bmatrix} -q_2&\quad q_1&\quad q_0\\ q_1&\quad q_2&\quad q_3\\ -q_0&\quad q_3&\quad -q_2\\ \end{bmatrix} \end{aligned}$$

(A.4)

$$\begin{aligned} \mathbf {Q}_3= & {} \frac{\partial k \mathbf {R}}{\partial q_3}=2 \begin{bmatrix} -q_3&\quad -q_0&\quad q_1\\ q_0&\quad -q_3&\quad q_2\\ q_1&\quad q_2&\quad q_3\\ \end{bmatrix} \end{aligned}$$

(A.5)

B Designation of covariance matrices for using 3D transformation algorithm to 2D problem

Let us denote the original ($2n\times 1$) vector of planar (2D) coordinates with $\mathbf {x}=[x_1\, y_1\,\ldots \,x_n\, y_n ]^T$. Then we can explicitly express the ($2n\times 2n$) covariance matrix $\varvec{\Sigma }_{xy}$ of planar coordinates as follows:

$$\begin{aligned} \varvec{\Sigma }_{xy}= \begin{bmatrix} \sigma _{x_1}^2&\quad \sigma _{x_1 y_1}&\quad \ldots&\quad \sigma _{x_1 x_n}&\sigma _{x_1 y_n}\\ \sigma _{y_1 x_1}&\quad \sigma _{y_1}^2&\quad \ldots&\quad \sigma _{y_1 x_n}&\sigma _{y_1 y_n}\\ \vdots&\vdots&\ddots&\vdots&\vdots \\ \sigma _{x_n x_1}&\quad \sigma _{x_n y_1}&\quad \ldots&\quad \sigma _{x_n}^2&\quad \sigma _{x_n y_n}\\ \sigma _{y_n x_1}&\quad \sigma _{y_n y_1}&\quad \ldots&\quad \sigma _{y_n x_n}&\quad \sigma _{y_n}^2\\ \end{bmatrix} \end{aligned}$$

(B.1)

The pseudo-3D coordinates of the above n-points is defined as the ($3n\times 1$) vector $\mathbf {x}^p =[x_1\, y_1\,0\,\ldots \, x_n\, y_n\, 0]^T$ where 0 values are assigned to the z-component of each point.

Then, we can easily design a ($3n\times 3n$) covariance matrix $\varvec{\Sigma }_{xyz}$ of the pseudo-3D coordinates in $\mathbf {x}^p$ simply extending the original covariance matrix as follows:

$$\begin{aligned} \varvec{\Sigma }_{xyz}= \begin{bmatrix} \sigma _{x_1}^2&\quad \sigma _{x_1 y_1}&\quad 0&\quad \ldots&\quad \sigma _{x_1 x_n}&\quad \sigma _{x_1 y_n}&\quad 0\\ \sigma _{y_1 x_1}&\quad \sigma _{y_1}^2&\quad 0&\quad \ldots&\quad \sigma _{y_1 x_n}&\quad \sigma _{y_1 y_n}&\quad 0\\ 0&\quad 0&\quad 1&\quad \mathbf {0}^T&\quad 0&\quad 0\quad&\quad 0\\ \vdots&\quad \vdots&\quad \mathbf {0}&\quad \ddots&\quad \vdots&\quad \vdots&\quad \mathbf {0}\\ \sigma _{x_n x_1}&\quad \sigma _{x_n y_1}&\quad 0&\quad \ldots&\quad \sigma _{x_n}^2&\sigma _{x_n y_n}&\quad 0\\ \sigma _{y_n x_1}&\quad \sigma _{y_n y_1}&\quad 0&\quad \ldots&\quad \sigma _{y_n x_n}&\quad \sigma _{y_n}^2&\quad 0\\ 0&\quad 0&\quad 0&\quad \mathbf {0}^T&\quad 0&\quad 0&\quad 1\\ \end{bmatrix}\nonumber \\ \end{aligned}$$

(B.2)

where the boldface $\mathbf {0}$ is a zero vector of size ($3\times 1$) in definition. Also note that instead of ones (1) on diagonal, one can use any other positive real small values (e.g., $10^{-12}$, depending on the application data) suitable for numerical computations, i.e., avoiding rank deficiency of the extended covariance matrix $\varvec{\Sigma }_{xyz}$.