L-8Minimization in Geometric Reconstruction Problems

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 sumof-squares, or L2 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 L2 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. 1 The triangulation problem Let Pi; i = 1, . . . , n be a sequence of n known cameras, and xi be the image of some unknown point X in 3-space, both expressed in homogeneous coordinates. Thus, we write xi = PiX. The problem of computing the point X given the camera matrices Pi and the image points xi is known as the triangulation problem. In the absence of noise, the triangulation problem is trivial, involving only finding the intersection point of rays in space. When noise is present, however, the rays corresponding to back-projections of the image points do not intersect in a common point, and obtaining the best estimate of the point X is not always easy. The correct procedure is to find the point X that projects most nearly to the image points xi. In this context, the words “most nearly” are usually interpreted in a least-squares sense. Thus, we are required to find the point X that minimizes the cost function