My research interests include shape representation, shape approximation, discrete differential geometry, proximity queries, proximity graphs, shape matching, and cartographic generalization.

In my PhD work, I have been concerned with the representation of a solid as a union of overlapping spheres. The spheres I considered are maximal and inscribed spheres in the solid (medial spheres) such that the sphere centres have a certain density distribution.

Given a polyhedron, we can compute a set of points within a user-chosen distance from the medial axis of the polyhedron. These points are centres of spheres that can be used to approximate the solid. We only consider significant sphere centres. In the image below, our approximate medial points are grouped into medial sheets (centre). On the right, we see the union of spheres computed with our method. We propose the use of this union of spheres as an alternative representation of the solids on the left.

This sphere-based shape representation allows the computation of curvature of the solid represented by the union of spheres without reconstruction of the solid's boundary. We can compute boundary curvature directly on the medial sheets (left) and then project this information onto the surface of the solid (right). This property illustrates the richness of our sphere-based shape representation.

When a solid can be represented tightly using a small number of spheres, the spheres can be used to perform approximate distance computations between solids. Alternative methods for the approximation of solids with spheres have considered the set of Voronoi spheres of a solid (Voronoi sphere centres are shown in the centre). Our sphere centres (shown on the right) have a more uniform density than Voronoi spheres. Methods that use Voronoi spheres for shape approximation require an expesive optimization step to redistribute sphere centres to achieve a tight and small sphape approximation.

In our work, we have shown that our spheres can be used directly, without an expensive optimization step, to offer a tight volumetric fit to solids. Furthermore, our spheres offer numerous advantages over Voronoi spheres as they allow for efficient updates when solids undergo deformation and because we can quickly measure their exact volumetric error. We have looked at the application of our sphere sets to the computation of approximate separation distance.

In research work towards my Master's degree, I designed an algorithm to compute a set of voxels intersected by the medial axis of a polyhedron. The AOF' measure evaluates the significance of medial points in a voxel and can be thresholded to simplify the medial axis, as illustrated in the image below.

I have worked for the Ministry of Natural Resources of Canada developing tools for the generalization of cartographic data using the medial axis. In the image below, we see a triangulation of a waterbody coloured according to its local width measured with respect to the medial axis of the waterbody. This measure can be used to determine which parts of the waterbody should be enlarged and which parts should be replaced with a curve when reducing the mapping scale.

I have also done computational geometry research related to the alpha-complex.