A Robust Least Squares Solution to the Calibrated Two-View Geometry with Two Known Orientation Angles

Abstract

This paper proposes a robust least squares solution to the calibrated two-view geometry with two known orientation angles. Using the knowledge reduces the degrees of freedom (DoF) from five to three: one from a remaining angle and two from a translation vector. This paper determines that the three parameters are obtained by solving a minimization problem of the smallest eigenvalue containing the unknown angle. The proposed solution minimizes a new simple cost function based on the matrix determinant in order to avoid the complicated eigenvalue computation. The estimated parameters are optimal since the cost function is minimized under three DoFs. Experimental results of synthetic data show that the robustness of the proposed solution is up to 1.5$^\circ$ angle noise, which is approximately three times that of a conventional solution. Moreover, 60 point correspondences, fewer than half those in conventional solutions, are sufficient to reach the performance boundary.

Appendix

The proof of that the proposed 4-point algorithm including the 3-point algorithm is as follows.

Substituting three point correspondences into (17), we have

$$\begin{aligned} \begin{aligned} \frac{d}{d\theta } \mathrm {det}(\varvec{B}^T \varvec{B})&= \frac{d}{d\theta } \mathrm {det}(\varvec{A}^{T}\varvec{A}) \\&= \frac{d}{d\theta } \mathrm {det}(\varvec{A})^2 \\&= 2 \mathrm {det}(\varvec{A}) \frac{d}{d\theta } \mathrm {det}(\varvec{A}). \end{aligned} \end{aligned}$$

(20)

We can construct a system of polynomial equations as follows:

$$\begin{aligned} \left\{ \begin{aligned} f_3 (c,s)&= \mathrm {det}(\varvec{A}) \frac{d}{d\theta } \mathrm {det}(\varvec{A}) \Bigr |_{ \mathop {\scriptscriptstyle \sin \theta =s}\limits ^{\scriptscriptstyle \cos \theta =c,} } = 0, \\ g(c,s)&= c^2+s^2-1 = 0. \end{aligned} \right. \end{aligned}$$

(21)

The solutions of $\mathrm {Res}(f_3,g,c)=0$ include that of $\mathrm {Res}(f_1,g,c)=0$.