Cameras Seeing Cameras Geometry

Abstract

We study several theoretical aspects of both 2D and 3D intra multi-view geometry of calibrated cameras when all that they can reliably recognize is each other. Starting with minimal reconstructable configurations, we propose a method for obtaining the position-orientation structure of such camera ensembles, up to a global similarity. In the 3D setting we base our analysis on Rodrigues’ vector techniques familiar from mechanics and robotics. We also examine the average number of visible cameras and discuss some kinematic aspects of the problem.

Appendix: How Many Cameras does Each Camera See on Average?

Here we treat the number of sighting of other cameras in a sphere as a function of the distance to the center, orientation and FOV. In the two-dimensional case we consider a point $z_0$ in a circle of radius r at distance from its center $d\in [0,r]$. It is convenient to introduce polar coordinates $\rho \in [0,r]$, $\theta \in [0,2\pi )$ choosing $z_0$ as the origin. The circle’s boundary is viewed from the perspective of $z_0$ as a curve with polar equation (see [14] for a derivation)

$$\begin{aligned} \rho (\theta ) = d\cos {\theta }+ \sqrt{r^2-d^2\!\sin ^2\!{\theta }} \end{aligned}$$

(37)

and it is straightforward to obtain the area of the disc segment sliced by the viewing angle $\theta \in [a,b]$ at $z_0$ as

$$\begin{aligned} A = \int \limits _{a}^{b}\int \limits _0^{\rho (\theta )}{\rho \,\mathrm{d}\rho \,\mathrm{d}\theta }. \end{aligned}$$

Choosing a polar orientation $\phi \in [0,2\pi )$ for the camera at $z_0$ and denoting the width of the field of view $FOV = 2\delta$, one obtains the above integral as a a function of the parameters $\phi$ and d, or more conveniently $\epsilon = d/r$ (keeping r and $\delta$ fixed), namely

$$\begin{aligned} A(\phi , \epsilon ) = \frac{r^2}{2}\!\int \limits _ {\phi \!-\!\delta }^{\phi \!+\!\delta } {\left( 1\!+\epsilon ^2\cos {2\theta }\!+2\epsilon \cos {\theta }\sqrt{1\!-\!\epsilon ^2\sin ^2\!{\theta }}\right) \!\mathrm{d}\theta }. \end{aligned}$$

The first two terms are trivial, while for the third one we use integration by parts, finally arriving at

$$\begin{aligned} \displaystyle A(\phi , \epsilon )= & {} \delta r^2 + \left( \frac{\epsilon r}{2}\right) ^2\sin (2\theta ) \Big |_{\phi \!-\!\delta }^{\phi \!+\!\delta } \nonumber \\&+ \frac{r^2}{2}\left( \epsilon \sin {\theta }\sqrt{1-\epsilon ^2\!\sin ^2\theta }+\arcsin {\epsilon \sin {\theta }}\right) \Big |_{\phi \!-\!\delta }^{\phi \!+\!\delta } \end{aligned}$$

(38)

e.g. on the boundary of the unit circle $\epsilon \!= \!r\! =\! 1$ one has

$$\begin{aligned} A = 2\delta + \cos { 2\delta } \cos {2\phi } \end{aligned}$$

while for an arbitrarily placed camera pointing towards the center (note that we always assume $\delta \in [0,\pi ]$)

$$\begin{aligned} A = \delta + {\tilde{\delta }} + \sin {{\tilde{\delta }}}(\cos {\delta }+\cos {{\tilde{\delta }}}), \qquad {\tilde{\delta }} = \arcsin {\epsilon \sin {\delta }}. \end{aligned}$$

Using formula (38) one obtains an estimate for the geometric probability

$$\begin{aligned} P = A/ A_0,\qquad A_0=\pi r^2 \end{aligned}$$

that in the case of the uniform distribution corresponds to the relative number of agents seen by the camera at $z_0$. The rotational symmetry allows us to work with the above-chosen range for the parameters $\epsilon \in [0, 1]$, $\phi \in [0,2\pi )$ and then integrate $P(\phi ,\epsilon )$ dividing by $2\pi$ in order to obtain the geometric probability and thus, the average number of cameras seen by an arbitrary camera

$$\begin{aligned} \langle {P} \rangle = \frac{1}{2\pi ^2 r^2} \int \limits _{0}^{1}\!\int \limits _{0}^{2\pi } A(\phi , \epsilon )\, \mathrm{d}\phi \,\mathrm{d}\epsilon = \frac{\delta }{\pi }=\frac{FOV}{2\pi } \end{aligned}$$

(39)

where we use the periodicity of the trigonometric terms in (38) with respect to $\phi$ and get the same result as in (21). Note that since the nonzero contribution to (39) does not depend explicitly on $\epsilon$, the above relation holds for any point on the circle. This result may also be derived using symmetries of circles and spheres, which is preferable especially in the 3D case where explicit calculations are nontrivial except for a camera positioned at the center: then the observed domain becomes a spherical sector with volume proportional, with a factor equal to one third of the radius, to the solid viewing angle

$$\begin{aligned} FOV = 2\pi (1-\cos {\delta }). \end{aligned}$$

For an arbitrary position and orientation however averaging factors out the complexity just as in the planar case (or the physical setting of a charged spherical shell) and formula (27) still holds as if the camera was at the center.