Interval Extensions of Signed Distance Functions: iSDF-reps and Reliable Membership Classification

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 such as magnetic resonance, computed tomography, and positron emission tomography. Techniques recently developed for automated segmentation convert intensity to signed distance, and the voxel structure imposes the uniform sampling grid. The specification of the signed distance function (SDF) throughout the ambient space would provide an implicit and function-based representation (f-rep) model that uniquely specifies the object, and we refer to this particular f-rep as the signed distance function representation (SDF-rep). However, 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 the upper and lower bounds on the allowed variation in signed distance, which combine to produce interval-valued extensions of the signed distance function. We employ an interval extension of the signed distance function as an interval SDF-rep that defines the range of object geometries that are consistent with the sampled SDF data. The particular interval extensions considered include a tight global extension and more computationally efficient local extensions that provide useful criteria for root exclusion/isolation. To illustrate a useful application of the interval bounds, we present a reliable approach to top-down octree membership classification for uniform samplings of signed distance functions.

Equations and Interval Computations for Some Fractals

“…Algebraic objects such as spheres, quadrics, and tori, are represented with polynomial functions (e.g. f (x, y, z) = (x − c x ) 2 + (y − c y ) 2 + (z − c z ) 2 − r 2 ≤ 0 for a ball of known radius r centered at a given point (c x , c y , c z )) or with signed distance functions 5 (e.g. for the polygon).…”

“…In other words, d Z 1 (x) is the signed distance functions to stripes 2Z +[1,2]. The function for the complement is the even function(5). Then, define vertical stripes A and A , horizontal stripes B and B , diagonal stripes C and C (e.g.…”