Surface Modeling

Extraction of quantitative information from models requires accurate reconstruction techniques. To this end, we have developed a reconstruction algorithm which generates a C¹ continuous surface representation from a uniformly distributed set of points in space. The first step is to construct a 3D Delaunay triangulation of the points, from which we extract an -solid, that is a subset of the tetrahedra in the 3D triangulation that form a solid shape, whose boundary interpolates the data points.

Given a sufficiently dense and uniform sampling of the surface, the -solid can be recovered automatically. The boundary of the -solid is a triangle-mesh interpolating the data points, and can be used for visualization and computation purposes. Notice that the -solid also provides a tetrahedralization of the object, which can be used (eventually after further subdivision) for FEM analysis.

To obtain a more compact representation, and when a derivative-continuous model is needed, we can build an A-patch representation of the model [4]. An A-patch surface is a collection of polynomial patches of low degree, whose continuity ( C¹ for degree three patches, C² for degree five patches) can be easily achieved. Each patch is defined as the zero-contour of Bernstein-Bézier polynomial, defined within a tetrahedron. We therefore build a tetrahedral mesh containing the surface, and then set the coefficients of the polynomials to approximate the data points to a given tolerance, with the desired continuity constraints.

We have developed two methods to reconstruct a smooth object's model: In the first method [3], we define a volumetric signed distance function based on the -solid. We then incrementally build a tetrahedral mesh that supports the A-patches. The mesh is adaptively refined until the desired approximation tolerance is satisfied. With the second approach [2], we first decimate the boundary of the -solid, then build a collection of tetrahedra (called simplicial hull, on both sides of the triangle mesh. A-patches are then built inside the simplicial hull, approximating data points and estimated surface normals. Simultaneous reconstruction of one or more scalar functions defined at the input points allows us to model and query material properties which are derived from the input CT or MRI data.

Figure 9(a) shows the tetrahedral mesh and the associated spline model of the femur. Figure  9(b) shows a reconstructed model for the knee from the VHP data set.

  

(a)	(b)

Figure 9: A-patch reconstruction of a femur and a knee