Computer aided geometric design, computer graphics, hardware for surface rendering, image morphing, computer algebra, handwriting recognition

Morphing

Peisheng Gao and I developed an algorithm that can perform morphs automatically with no user interaction.

For example, my picture (on the left) and Peisheng's picture (on the right) are here morphed into the middle image. Our algorithm performed this automatically, with no anchor points.

This is another example of a completely automatic morph.

This morph required the user to specify one anchor point.

This morph required the user to specify five anchor points.

Image reconstruction

Xiaohua Yu, Bryan Morse and myself just completed an algorithm for image reconstruction based on data-dependent triangulation. The technique is able to magnify images with far less aliasing than is often found doing bilinear or bicubic reconstruction. For example, the following image of a butterfly was magnified four times using bicubic reconstruction (on the left) and our algorithm (on the right).

Inspired by Tor Dokken's dissertation, Jianmin Zheng, Kris Klimaszewski, Tor Dokken and I found a way to approximately implicitize a rational surface using a monoid surface. Highly accurate approximations can be obtained using low-degree algebraic surfaces that avoid the problems of phantom branches and singularities that can plague implicit surfaces.

David Sewell did a Masters Thesis on non-uniform Catmull-Clark surfaces. Whereas Catmull-Clark surfaces generalize uniform B-splines to control nets of arbitrary topology, we can now generalize non-uniform B-splines. This allows features such as creases to be defined by adjusting the knot spacings. David also developed a method for approximating a metaball using Catmull-Clark surfaces.

I am currently studying algorithms for **implicitizing** curves and surfaces, continuing work on the ideas in ``Implicitization using moving curves and surfaces,'' *Computer Graphics Annual Conference Series*, 1995, 301--308. David Cox at Amherst has been working with me on proving that two moving lines of degree n_{1} and n_{2} in t (which sum to the degree of the curve) can serve as a basis for the ideal generated by the parametric equations of the curve. Cox discovered that a form of this "new" result was actually published by Hilbert in 1890!

Kris Klimaszewski completed an excellent dissertation on **ray tracing**. Using his adaptive grid spatial indexing, his algorithm ray traced this 80,000 triangle car in 64 seconds. (512X512 resolution, 60 Specmark workstation).

Mel Spencer wrote a dissertation on root finding of polynomials in Bernstein form.

Check out the figures in the paper by Eugene Greenwood and myself on shape blending of B-splines.

Alan Zundel completed a nice dissertation on surface intersections. He devised an algorithm for "loop destruction" - intelligently splitting surfaces so as to remove closed intersection loops from the interior of patches.

Mu Hong completed a master's thesis on surface reconstruction from contours. This presents an automatic algorithm for fitting a piecewise triangular surface over a set of branching contours. The underlying idea is to search for the surface with overall minimal surface area. This heuristic has previously proven good in the non-branching case, and it has now proven good in the branching case, also.