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
a Volume data b Level set 700
c Level
set 1,100
d Level
set 1,400
Fig. 13.9. A 63 by 63 by 92 density volume and three level sets. The data are from the Visible Human Project National Library of Medicine, 2003 and are rendered with
Kitware’s VolView.
that a simplification algorithm, such as the one presented in Section 6.2.4, will be always successful. There are, therefore, many interesting challenges in this
area for future research.
13.3.3 Time-Varying Data
Often, we are interested in data varying with time. For example, the wind velocity on Earth, measured through time, describes a time-varying function
on a two-manifold, the sphere. We may view time as another dimension of space, converting d-dimensional time-varying data to
d + 1-dimensional data. We then denoise the data through time by constructing a hierarchy of
d + 1-dimensional MS complexes. For the example above, we will need three-dimensional MS complexes. Four-dimensional data also arise in prac-
tice. For instance, researchers are currently simulating solid propellant rockets Heath and Dick, 2000. The temperature, pressure, and velocity are computed
for a time-interval at every point inside the rocket. Viewing time as space, we obtain a four-dimensional data set for which we need a four-dimensional
MS complex. Once again, generalizing the MS complex to higher dimensions seems to be a rich avenue for future research.
13.3.4 Medial Axis Simplification
In two dimensions, the medial axis is the locus of all centers of circles inside a closed planar 1-manifold that touch the boundary of the manifold in two or
more points Blum, 1967. The medial axis has been used heavily as a de-
a b
Fig. 13.10. The dashed medial axis of the solid polygon a is ill-conditioned as a small perturbation changes the resulting axis dramatically b.
scriptor of shapes for pattern recognition, solid modeling, mesh generation, and pocket machining. This descriptor, however, is ill-conditioned, as a small
perturbation in the data changes the description radically. I illustrate the sen- sitivity of the medial axis with an example in Figure 13.10. By restating the
problem in terms of persistence, we may be able to denoise the data and, in turn, simplify the medial axis, obtaining a robust description of the data.
We can extend the definition of the medial axis to n-dimensional manifolds by using n-dimensional spheres, instead of circles. The definition remains
sensitive to noise in all dimensions and therefore still requires a method for simplification.
13.4 Surface Reconstruction
Another direction for future work is using persistence for surface reconstruc- tion. I introduced this problem as an example of a topological question in
Chapter 1. We may employ the control persistence gives us over the topology of a space to reconstruct surfaces from sampled points.
Figure 13.11 shows a single-click reconstruction of the bunny surface. Note that I selected a complex with a tunnel. The bunny was not sampled on its base
across the two black felts it rests on, as a laser range-finder scanner was used for acquiring the samples. A good reconstruction, therefore, has two holes or
a single tunnel. Such knowledge, however, is not always available.
I believe that a successful reconstruction algorithm must be interactive, it- erative, and adaptive. Abstractly, we wish to identify a coordinate
l, p such that the complex K
l ,p
contains a reconstruction of the point set. We may enrich the solution space by computing radii for the points. For example, we can es-
timate the local curvature at each point, assigning the inverse curvature as the radius of the point. We then recompute the filtration with the new radii. Sta-
