We have studied the fundamental building blocks as well as concrete algorithms and data structures of geometry processing from the perspective of extending them to higher dimension. Several directions that appeared immediately fruitful have been followed. Among several achievements, the following three may be considered the highlichts:
- A standard problem in engineering and science is fitting a line or plane (or a higher dimensional hyper-plane) to data. It is well established how to do that if the data are points. But what if the data themselves are lines, planes, or hyper-planes? While this problem is understood from the perspective of mathematical theory, the current solutions are complicated, slow in practice, and lack important and natural properties. We have worked on a new solution to the problem of fitting lines or planes to lines or planes. It is simple, fast, and, importantly, the solutions it provides are invariant to rigid transformations, a property so far missing.
- The most successful technique for reconstructing surfaces in three-dimensional space from point samples is Poisson Surface Reconstruction. We have extended this method to work for other, more general cases, such as also reconstructing curves in space or surfaces in higher-dimensional surfaces. This required introducing a new concept in the approach, the so-called exterior calculus, and lead to a more complicated optimization problem. We have worked on a multi-level approach for this problem that is both efficient and finds plausible solutions.
- An important tool in engineering are parameterizations, allowing to map data from one domain to another, or establish correspondence between different data sets. This problem is well-understood for two-dimensional data. A basic building block for almost all such methods are so-called Tutte embeddings, which can be computed by solving linear system and guarantee one-to-one mappings. This seemingly natural approach fails to extend to higher dimension in general. We have worked on an analysis and provide the first characterization of the situation in 3D.
