Using Many Cameras as One

Author : Pless

Year : 2003

Notes:

use a network of cameras as a single imaging device
generalized camera model: a pixel is equivalent to a raxel that defines a cone from which the pixel captures the light
here the raxels are simplified as a line that can be represented with Plucker vectors
Plücker representation is made of two vectors $q, q'$:
- $q$ is the direction of the line
- $q'$ is defined as $q' = q \times P$ for every $P$ that lies on the line
In the camera frame, the Plücker representation will simply be $(\pi^{-1}(u), [0, 0, 0]^T)$
A pair of Plücker vector intersects if $$ q_1 . q'_2 + q'_1 . q_2 = 0 $$
Fall back on the classic formula for perspective cameras (i.e. with $q'_1 = q'2 = 0$) $$ q_2^T R [t]{\times} q_1 = 0 $$
Come up with the optical flow equation, that depends on the angular velocity, the translationnal velocity and the depth: $$ \dot{q}=-(\vec{\omega} \times q)-\frac{(\vec{t} \times q) \times q}{\alpha} $$
Use the Fisher information matrix to determine which camera configuration is the best
the measurement is assumed to be the optical flow $\vec{u}$, it derives an expression of $h_i(\vec{t}, \vec{\omega}, \alpha, q, q') = \vec{u}$ that leads to an anlytical formula of the Fisher info matrix
Then a simulation is conducted on a uniform distribution over the parameters $\vec{t}, \vec{\omega}, \alpha_1, ..., \alpha_n$ with $n$ the number of pixels
The best configuration seems to be two aligned cameras facing opposite directions