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Research Papers

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

[+] Author and Article Information
Duane Storti, Chris Finley, Mark Ganter

Department of Mechanical Engineering, University of Washington, Seattle, WA 98195-2600

J. Comput. Inf. Sci. Eng 10(2), 021012 (Jun 08, 2010) (8 pages) doi:10.1115/1.3428736 History: Received June 05, 2009; Revised March 29, 2010; Published June 08, 2010; Online June 08, 2010

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.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

(a) 1D signed distance sample values; (b) connect the dots violates a unit gradient; (c) extend the lines produces the simplest valid candidate for signed distance and models, 1D implicit object corresponding to 3.8<X<5.1; (d) valid signed distance function modeling the union of intervals 3.8<X<4.2 and 4.9<X<5.1, features with a size smaller than the sample spacing mandate careful consideration; and (e) signed distance function modeling a union of three intervals derived from the same sample data

Grahic Jump Location
Figure 2

(a) 45 deg wedges above sample data (from Fig. 1) provide upper bounds; i.e., signed distance is excluded from the shaded regions. (b) 45 deg wedges below sample data provide lower bounds excluding signed distance from shaded regions. (c) Minimum value from Fig. 2 provides a global upper bound. (d) Maximum value from Fig. 2 provides a global lower bound. (e) (Lowerbound, upperbound) provides an interval extension of the signed distance. The shaded region shows allowed variation in signed distance corresponding to the sample data.

Grahic Jump Location
Figure 3

Illustration of global bounds based on a uniform 11×11×11 sampling of a torus distance field. The outer bound surface (shown at the left) is cut away to reveal the torus (center), which is in turn cut away to reveal the inner bound surface (shown at the right).

Grahic Jump Location
Figure 4

Width of the global interval SDF on a quadrant of the midplane of the torus. White circles indicate the boundary, the black circle indicates the skeleton, and the black dots indicate SDF sample points. Darkest regions, where interval half-width exceeds one-third, are surrounded by lighter regions, where interval half-width exceeds one-fourth. Note that these significant interval widths only occur near skeletal points (specifically those that do not have nearby sample points), which are isolated from the boundary for objects with smooth surfaces.

Grahic Jump Location
Figure 5

Reliable octree classification based on a 1283 grid of SDF data for a torus. (a) Rendering of torus consisting of surface cubes shaded based on an estimate of surface normal based on connection coefficient estimate of SDF gradient. (b) Cut-away view illustrating a multiresolution decomposition/classification. Lighter shading indicates cells that were fully classified at a lower resolution.

Grahic Jump Location
Figure 6

Reliable octree classification based on a 1283 grid of SDF data for a talus. (a) Rendering of talus consisting of surface cubes shaded based on an estimate of surface normal based on connection coefficient estimate of SDF gradient. (b) Cut-away view illustrating a multiresolution decomposition/classification.

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