We propose to use a single temporally coded light source such as an LED as the signal to be captured by the cameras for synchronization. The light signal is essentially a time-continuous binary-valued function denoted asDisplay Formula
4$f:R\u21a6{0,1}.$
It is sampled at $T\alpha $ by the $\alpha $’th camera, producing a time-discrete binary-valued sequenceDisplay Formula5$[fcam\alpha (n)]n=1N\alpha =[f(tn\alpha )]n=1N\alpha ={f[(n\u22121)\Delta T+t1\alpha ]}n=1N\alpha .$
Since $f$ is binary valued, it can be characterized by time instants where its function value rises from 0 to 1 or drops from 1 to 0. We term each of these instants a transition event. Let $\Phi $ denote a subsequence of all transition events in $f$,Display Formula6$\Phi =(\varphi 1,\varphi 2,\u2026,\varphi N).$
For each $\varphi k$, we haveDisplay Formula7$\u2200\u03f5>0,f(\varphi k\u2212\u03f5)\u2295f(\varphi k+\u03f5)=1,$
where ⊕ denotes the exclusive or operator, and $\u03f5$ is an arbitrarily small real positive number. For the $\alpha $’th camera, let $\Phi \alpha $ denote the transition events corresponding to $\Phi $. Obviously, as part of $T\alpha $, $\Phi \alpha $ can be expressed as follows:Display Formula8$\Phi \alpha =[tI\alpha (1)\alpha ,tI\alpha (2)\alpha ,\u2026,tI\alpha (N)\alpha ],$
where $I\alpha $ is a subsequence of $(1,2,\u2026,N\alpha )$. For each $I\alpha (k)$, it satisfiesDisplay Formula9$fcam\alpha [I\alpha (k)]\u2295fcam\alpha [I\alpha (k)\u22121]=1.$
This notation and their relationship are illustrated in Fig. 1.