# Calculating Sparse and Dense Correspondences for Near-Isometric Shapes

Dissertation, University of Bonn, May 2018

## Bibtex

```@PHDTHESIS{burghard-2018-dissertation,
author = {Burghard, Oliver},
title = {Calculating Sparse and Dense Correspondences for Near-Isometric Shapes},
type = {Dissertation},
year = {2018},
month = may,
school = {University of Bonn},
abstract = {Comparing and analysing digital models are basic techniques of geometric shape processing. These
techniques have a variety of applications, such as extracting the domain knowledge contained in the
growing number of digital models to simplify shape modelling. Another example application is the
analysis of real-world objects, which itself has a variety of applications, such as medical
examinations, medical and agricultural research, and infrastructure maintenance. As methods to
digitalize physical objects mature, any advances in the analysis of digital shapes lead to progress
in the analysis of real-world objects.
Global shape properties, like volume and surface area, are simple to compare but contain only very
limited information. Much more information is contained in local shape differences, such as where
and how a plant grew. Sadly the computation of local shape differences is hard as it requires
knowledge of corresponding point pairs, i.e. points on both shapes that correspond to each other.
The following article thesis (cumulative dissertation) discusses several recent publications for the
computation of corresponding points:
- Geodesic distances between points, i.e. distances along the surface, are fundamental for several
shape processing tasks as well as several shape matching techniques. Chapter 3 introduces and
analyses fast and accurate bounds on geodesic distances.
- When building a shape space on a set of shapes, misaligned correspondences lead to points moving
along the surfaces and finally to a larger shape space. Chapter 4 shows that this also works the
other way around, that is good correspondences are obtain by optimizing them to generate a compact
shape space.
- Representing correspondences with a “functional map” has a variety of advantages. Chapter 5
shows that representing the correspondence map as an alignment of Green’s functions of the Laplace
operator has similar advantages, but is much less dependent on the number of eigenvectors used for
the computations.
- Quadratic assignment problems were recently shown to reliably yield sparse correspondences.
Chapter 6 compares state-of-the-art convex relaxations of graphics and vision with methods from
discrete optimization on typical quadratic assignment problems emerging in shape matching.},
url = {https://nbn-resolving.org/urn:nbn:de:hbz:5n-50900}
}
```