Interval Extensions of Signed Distance Functions: Uniform Samplings and the Range of Associated Implicit Objects

Abstract

This paper considers the problem of inferring the geometry of an object from values of the signed distance sampled on a uniform grid. The problem is motivated by the desire to effectively and efficiently model objects obtained by 3D imaging technology that is now ubiquitous in medical diagnostics. Recently developed techniques for automated segmentation convert intensity to signed distance, and the voxel structure imposes the uniform sampling grid. While specification of the signed distance function throughout the ambient space would provide an implicit model that uniquely specifies the object, a set of uniformly sampled signed distance values may uniquely determine neither the distance function nor the shape of the object. Here we employ essential properties of the signed distance to construct upper and lower bounds on the allowed variation in signed distance in 1, 2, and 3 dimensions. The bounds are combined to produce interval-valued extensions of the signed distance function including a tight global extension and more computationally efficient local bounds that provide useful criteria for root exclusion/isolation.