An analytical solution to optimal focal distance in catadioptric imaging systems

Catadioptric imaging systems are important in many computer vision and robotics applications. This work addresses the issue of optimally setting the focal distance of a lens based camera in a catadioptric imaging system in order to acquire a best focused image. To this end, understanding the spatial distribution of virtual feature formed by mirror reflection is important. It is known that the virtual features of infinite range of scene depth are limited to a finite depth extent, which is named a caustic volume. In this work we further find that for a variety of quadric mirror based catadioptric systems, when the objects are located at a certain distance to the system, the corresponding virtual features can be considered to be located on the caustic volume boundary. We verify this property with real catadioptric images. Based on this property, an analytical solution is derived for the optimal focal distance setting, which can only be calculated by software simulation or numerical approaches in previous work. This solution is compared with a numerical solution of previous method and is also verified by a simulation of the optical process.