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.

The Iterative Closest Point (ICP) method is widely used for fitting geometric models to sensor data. By formulating the problem as a minimization of distances evaluated at observed surface points, the method is computationally efficient and applicable to a rich variety of model representations. However, when the scene surface is only partially visible, the model can be ill-constrained by surface observations alone. Existing methods that penalize free space violations may resolve this issue, but require that the explicit model surface is available or can be computed quickly, to remain efficient. We introduce an extension of ICP that integrates free space constraints, while the number of distance computations remains linear in the scene's surface area. We support arbitrary shape spaces, requiring only that the distance to the model surface can be computed at a given point. We describe an implementation for range images and validate our method on implicit model fitting problems that benefit from the use of free space constraints.

This paper explores the use of continuous signed distance functions as an object representation for 3D vision. Popularized in procedural computer graphics, this representation defines 3D objects as geometric primitives combined with constructive solid geometry and transformed by nonlinear deformations, scaling, rotation or translation. Unlike their discretized counterpart, that have become important in dense 3D reconstruction, the continuous distance function is not stored as a sampled volume, but as a closed mathematical expression. Through surveys and qualitative studies we argue that this representation can have several benefits for 3D vision, such as being able to describe many classes of indoor and outdoor objects at the order of hundreds of bytes per class, getting parametrized shape variations for free. As a distance function, the representation also has useful computational aspects by defining, at each point in space, the direction and distance to the nearest surface, and whether a point is inside or outside the surface.

Procedurally-defined implicit functions, such as CSG trees and recent neural shape representations, offer compelling benefits for modeling scenes and objects, including infinite resolution, differentiability and trivial deformation, at a low memory footprint. The common approach to fit such models to measurements is to solve an optimization problem involving the function evaluated at points in space. However, the computational cost of evaluating the function makes it challenging to use visibility information from range sensors and 3D reconstruction systems. We propose a method that uses visibility information, where the number of function evaluations required at each iteration is proportional to the scene area. Our method builds on recent results for bounded Euclidean distance functions by introducing a coarse-to-fine mechanism to avoid the requirement for correct bounds. This makes our method applicable to a greater variety of implicit modeling techniques, for which deriving the Euclidean distance function or appropriate bounds is difficult.