Camera Placement Optimization in Object Localization Systems

Abstract

This paper focuses on the placement of cameras in order to achieve the highest possible localization accuracy with a multi-camera system. The cameras have redundant fields of view. They have to be placed according to some natural constraints but user defined constraints are allowed as well. A camera model is described and the components causing the localization errors are identified. Some localization accuracy measures are defined for any number of cameras. The multi-camera placement is analytically formulated using the expanded measures for multiple cameras. An example of placing two cameras is shown and the generalizations into higher dimensional parameter spaces are examined. There are publications where camera placement algorithms are formulated or compared. We make an attempt to examine the analytical solution of this problem in case of different objective functions.

Multi-camera sensing for robust perception in robotics

“…First, we observe that the upper bound µ * on Problem 1's optimal value f * provided by the convex relaxation and the lower bound f * ≥ f (s g ) provided by the greedy solution are remarkably close across the majority of our experiments. In light of inequality (15), this shows that (i) the heuristic greedy method is remarkably effective in finding highquality sensor arrangements in these experiments, and (ii) the convex relaxation's optimal value µ * is a remarkably sharp approximation of Problem 1's optimal value, enabling us to certify the optimality of the greedy method's solutions. Indeed, examining Fig.…”

“…In SLAM, information-theoretic objectives are commonly applied in active SLAM [12], feature selection [13], and dynamic sensor selection [14]. Previous works that have explored information-theoretic sensor placement deal with bespoke sensors like radar [11], localization with fixed sensors [15], or object detection tasks [16]. The recent literature on co-design of complex systems presents a framework for optimization with highly-coupled multidisciplinary constraints and objectives [17].…”

“…for any feasible s in (12). The significance of inequality (15) is that it enables us to use the optimal value µ * of Problem 2 (which we can compute efficiently) in order to bound the suboptimality f * − f (s) of any feasible solution s of Problem 1. In particular, as we will see in Section V, this will provide a practical means of verifying the (global) optimality of candidate solutions of Problem 1.…”

“…The entire OASIS procedure is summarized in Algorithm 1. In brief, our method applies sequential greedy set maximization to obtain a feasible solution s g of Problem 1, and then solves the convex relaxation Problem 2 (using the Frank-Wolfe method) to obtain an upper bound µ * on Problem 2's optimal value that we can use to bound s g 's suboptimality via (15). As we show in Section V, this simple approach enables us to recover certifiably optimal solutions of the sensor arrangement problem (12) in practice.…”