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.

Progressive integration of visibility constraints for implicit functions

“…Our work builds on results from [16,17]. We therefore summarize their relevant findings in Section 2.1 while introducing necessary notation.…”

“…We propose a fitting method for implicit functions that efficiently integrates visibility information. Like previous work [17], our method requires only that an approximation of the Euclidean distance to the model surface can be computed at a given point. However, our method is robust against errors in this approximation.…”

“…This relaxes the need for correct bounds and makes our method more easily applicable to practical implicit modeling techniques. Additionally, the previous works [16,17] are efficient in the sense that they compute and utilize the smallest set of function evaluation points to test if the free space constraints are satisfied (i.e. necessary and sufficient conditions).…”

“…For general implicit models, these methods require dense sampling of f x in the volume or along viewing rays, which can be prohibitively expensive. More efficient methods have recently been proposed for the special case when f x is either exactly the signed Euclidean distance function to the surface, or when f x can be bounded from above and below by known functions of the Euclidean distance [16,17]. However, there is no closed-form expression for the Euclidean distance to several implicit modeling techniques of interest, including set operations, blending and non-isometric deformations [1,3].…”

“…However, there is no closed-form expression for the Euclidean distance to several implicit modeling techniques of interest, including set operations, blending and non-isometric deformations [1,3]. Deriving the necessary upper and lower bounds can also be challenging [17] and the use of incorrect bounds can lead to an inaccurate fit (Fig. 1…”