Calculating Sparse and Dense Correspondences for Near-Isometric Shapes

Dissertation, University of Bonn, May 2018
 

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.

Download: https://nbn-resolving.org/urn:nbn:de:hbz:5n-50900

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}
}