In scientific research and industrial applications, the questions of computing, characterizing and tracking important features in the input data arise with noticeable frequency. Feature detection is a critical step in automated data processing such as mesh generation. In high-dimensional data analysis, there is a strong demand for feature classification and tracking methods. Finally, in scientific visualization and simulation of real-world phenomena, modeling small-scale features is often essential for visual realism. All these questions motivated a substantial amount of research in geometric modeling, computational geometry, computational topology and computer animation.
The first topic we investigate in regard to these questions is feature processing for automated mesh generation. Specifically, for a shape provided as a collection of possibly improperly meeting and/or self-intersecting patches, we construct a watertight mesh that has many of these representation errors fixed. To achieve that, we augment the existing Delaunay meshing algorithm for piecewise smooth domains with a preprocessing step that recomputes the 1-skeleton of the complex, producing a valid input for the mesher.
For shapes provided in the form of a point cloud data, researchers are often interested in analyzing high-level features, which are traditionally computed using topological methods from simplicial complexes built on top of the point clouds. This motivates the second topic of our research: geometrical and topological queries on arbitrary simplicial complexes.
The first problem we tackle is maintaining geometry-aware homological features under changes in the underlying simplicial complex. We capitalize on the theory of persistent homology, which provides a principled way to describe features. Building upon the existing matrix framework, we propose a method for tracking a chosen generator so that it undergoes only local changes. Another question we address is speeding up miscellaneous queries about the homology of a simplicial complex. We annotate the simplices of the complex with binary signatures that allow efficient characterizations
of homology cycles. Using this method, we improve the running time of existing algorithms for computing a shortest homology basis and the shortest cycle in a homology class in 1-dimensional homology. We also provide a more implementation-friendly annotation algorithm,
which, though having a worse running time, blends nicely into the persistence homology theory and can be considered as a generalization of the persistence algorithm.
The final topic of our research in regards to geometric features is relatively independent of the previous two. We switch our attention to the field of physically-based simulation of natural phenomena and present methods to realistically model small-scale features that improve the simulation realism. Our contribution is the physical model for bubbles in a liquid foam. We propose to approximate the foam structure using a space-filling diagram, which allows us
to easily infer connectivity between bubbles and efficiently simulate their interaction, handling foam behaviour such as coalescing and instability of multi-bubble clusters. We demonstrate efficiency of our approach by providing simulation examples that
plausibly imitate real-world bubble effects.