Volumetrics and Rendering of Geologic Bodies by Three-Dimensional Geometric Reconstruction from Cross Sections or Contour Lines

Abstract

In applied earth sciences a three-dimensional object may be given by a set of parallel cross sections. More specifically, the body is given by a set of contour lines, which may represent the intersection of its boundary surface(s) with the plane of the cross section thus distinguishing the body’s interior and exterior, or a given value of some property of the body. Graphically displaying the body’s geometry and topology, that is its internal structure in terms of contour lines discriminating the body’s interior and exterior, or quantifying some of its features, for example arbitrary intersections or volumes corresponding to given contours, essentially requires to reconstruct the corresponding boundary surfaces of the three-dimensional body.

A computer-aided method of geometric reconstruction of closed three-dimensional bodies from their cross sections or contour lines, respectively, is presented, the essentials of which are implicit representation of surfaces and bodies, and set-valued interpolation. This approach suggests to approximate the interior of a three-dimensional body as an entity and thus allows the recovery of bodies of arbitrary shape and topology. In particular, a cross section may comprise more than one closed contour line.

Set-valued interpolation is performed numerically by consecutively solving a sequence of univariate interpolation problems. These problems are solved by applying quadratic splines of class C ¹ and employing their favorable properties of local support, smoothness, accuracy, and preservation of monotonicity.

The method is automatic and does not require any interaction by the user. Its capability to reconstruct complex geologic bodies from their contour lines for rendering and volumetrics is exemplified using synthetic data.