At issue is the question of smoothness. Previously, tangent plane continuity was used as the smoothness criteria. Piecewise polynomial (or rational polynomial) surface patches were fit to the data so that adjacent patches were tangent plane continous along their common boundary. Unfortunately, surfaces constructed in this fashion have poor visual shape despite meeting the mathematical smoothness constraints. Curvature plots show that curvature in unequally distributed across these patches, resulting in surfaces that look like smooth polyhedra (at best).
One class of techniques that have been used to construct better looking surfaces are global optimization techniques. These techniques are global in the sense that all the data is considered when constructing each patch (this is in constrast to the more common definition of global optimization meaning finding global minimums instead of local minimums).
While global optimization techniques provide far better surfaces, this improvement comes at a high computational price. Even for small numbers of surface patches, these techniques require hours of CPU time to compute the surface. I am currently looking at local techniques that create surfaces with better shape. There are several thrusts to my research, but all seem to indicate that lower degree surface patches (having fewer degrees of freedom) more naturally give high quality surfaces.
To construct cubic interpolant surfaces we need second order data
at the vertices. When approximating known functions, this data
is reasonable to obtain. However, when fitting surfaces to
scattered data, we are normally lacking both the first and second
derivatives at the data points. Thus, one of the topics I am
researching are methods for estimating this derivative information
from data points containing only positional information.
Approximate Continuity
A second research topic to come from my cubic interpolant research
is that of approximate continuity. More formally, two
patches are said to join with approximate continous with
tolerance e if the angle between the surface normals of the
two patches at any point along the common boundary is less than e.
In my work the the cubic interpolant, I worked backwards to produce approximately continuous surfaces: I would
Part of my research involves developing a better test for approximate continuity. For cubic patches, we can determine the maximum discontinuity by finding the roots of a degree 18 polynomial. What I would like is to find a less expensive technique that is accurate to a prescribed tolerance.
With a slightly different thrust, I am trying to develop a construction
of approximately continuous surace patches. That is, given that we
want our patches to interpolate certain data and meet with approximate
continuity, how do we build patches that meet these constraints?
Global optimization techniques try to minimize energy functions over
an entire surface patch (or patches). These techniques are slow
because it is expensive to evaluate these functionals (and since you have
to vary parameters to minimize this value, you have to evaluate your
functional many times).
In addition to the ideas mentioned in the earlier sections, my interest
in shape also deals with how to test the quality of surface patches. Commonly
used techniques include shaded images, curvature plots, reflections lines,
and isophotes. But it is unclear what metrics should be used, and indeed,
different applications probably require different metrics.
Surface Quality
All of the above ties into an idea of surface quality.
Essentially, as humans we can recognize surfaces having good shape
when we see them. Unfortunately, this idea of "good shape" is
hard to define mathematically. What is clear is that it is a metric
of the entire surface patch (and probably the entire surface). This
is why tangent plane continuous techniques fail to produce surfaces
of good shape: they are only a metric applied to a differential area
along the boundaries between adjacent patches.