L _∞ minimization in geometric reconstruction problems

Abstract

We investigate the use of the L∞ cost function in geometric vision problems. This cost function measures the maximum of a set of model-fitting errors, rather than the sum-of-squares, or L 2 cost function that is commonly used (in least-squares fitting). We investigate its use in two problems; multiview triangulation and motion recovery from omnidirectional cameras, though the results may also apply to other related problems. It is shown that for these problems the L∞ cost function is significantly simpler than the £2 cost. In particular L∞ minimization involves finding the minimum of a cost function with a single local (and hence global) minimum on a convex parameter domain. The problem may be recast as a constrained minimization problem and solved using commonly available software. The optimal solution was reliably achieved on problems of small dimension.