Both Goodman-and-Said and Foley-and-Opitz created hybrid patches to solve this interpolation problem. Basically, they fit cubic patches to the data, except that the center control point is a rational blend of three points. These three interior points are set so as to obtain cubic precision.
This technique produces significantly better surfaces. For example, if we fit these patches to a sampling of the Franke function, and compare it to the standard Clough-Tocher technique, we see a signficant improvement in both isophote plots and in shaded image.
Likewise, if we integrate this improved crossboundary method into the Clough-Tocher method, we see a similar improvement in surface quality. In the below image, the surface on the left is the Franke function; the center surface was constructed by the standard Clough-Tocher method; the one on the right was a modified Clough-Tocher method where we use the improved crossboundary function of Foley-Opitz.
To use this crossboundary technique in the parametric setting is a bit tricky. A simple Clough-Tocher split turns out to not be so simple, as you have to use a fourth degree patch instead of a third degree, and because this crossboundary technique is inherently a functional technique. Instead, we chose to develop a hybrid technique that involves a blend of the center point and blends of the boundary points.
As an example, we compare Triangular Gregory patches (another hybrid scheme) to our scheme. Both surfaces were fit to a cat data set. In the image below, the surface on the left is the Triangular Gregory surface, while the one on the right is our surface.
For details, see either Matthew Davidchuk's thesis, or the paper we have submitted to the Lillehammer conference.
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