Implementation The Linking Number Algorithm

onto the l, β plane, with the p axis coming out of the page. There are clear groups of persistent components, indicated by the horizontal lines across the graph. We hope to compare and contrast proteins using the graphs generated by this procedure. This idea is due to Thomas LaBean, from the Department of Computer Science at Duke University.

13.3 Denoising Density Functions

The second large class of applications of this work is denoising density func- tions. We use hierarchical MS complexes to eliminate noise in sampled data intelligently, changing the topology of the level-sets of the space by smooth- ing the geometry. In this section, I briefly describe future directions for such applications.

13.3.1 Terrain Simplification

In Chapter 9, I described algorithms for constructing the MS complex in two dimensions. I also provided evidence of the feasibility of this approach by implementing the algorithm for computing QMS complexes. My immediate plans are to complete this implementation. A hierarchy of two-dimensional MS complexes of a terrain gives us control over the level of detail in the rep- resentation. We may partition an increasingly smoother terrain into increas- ingly larger regions of uniform flow using the arcs of the MS complexes. Re- searchers may use this hierarchy to model natural phenomenon using multi- level adaptive refinement algorithms O’Callaghan and Mark, 1984. Inter- estingly, eliminating minima using persistent MS complexes corresponds to filling watersheds lakes incrementally Jenson and Domingue, 1988. Water- sheds need to be filled for computing water flow on terrains.

13.3.2 Iso-Surface Denoising

In three dimensions, volume data give rise to two-dimensional level sets or iso-surfaces . As before, inherent limitations of the data acquisition devices add noise to the data. The noise is often manifested as tiny bubbles near the main component of the iso-surface, as shown in Figure 13.9. It is trivial to compute a filtration of a volume grid by tetrahedralizing the volume and us- ing a three-dimensional manifold sweep. We need a three-dimensional MS complex, however, to modify the density values in a sensible fashion. A three- dimensional MS complex is more complicated than its two-dimensional coun- terpart, however, and is much harder to compute. Furthermore, it is not clear