Minimal Free Space Constraints for Implicit Distance Bounds

Abstract

A general approach for fitting implicit models to sensor data is to optimize an objective function measuring the quality of the fit. The objective function often involves evaluating the model’s implicit function at several points in space. When the model is expensive to evaluate, the number of points can become a bottleneck, making the use of volumetric information, such as free space constraints, challenging. When the model is the Euclidean distance function to its surface, previous work has been able to integrate free space constraints in the optimization problem, such that the number of distance computations is linear in the scene’s surface area. Here, we extend this work to only require the model’s implicit function to be a bound of the Euclidean distance. We derive necessary and sufficient conditions for the model to be consistent with free space. We validate the correctness of the derived constraints on implicit model fitting problems that benefit from the use of free space constraints.