Niklas Peinecke

Laplace-spectra as fingerprints for shape matching

This paper introduces a method to extract fingerprints of any surface or solid object by taking the eigenvalues of its respective Laplace-Beltrami operator. Using an objects spectrum (i.e. the family of its eigenvalues) as a fingerprint for its shape is motivated by the fact that the related eigenvalues are isometry invariants of the object. Employing the Laplace-Beltrami spectra (not the spectra of the mesh Laplacian) as fingerprints of surfaces and solids is a novel approach in the field of geometric modeling and computer graphics. Those spectra can be calculated for any representation of the geometric object (e.g. NURBS or any parametrized or implicitly represented surface or even for polyhedra). Since the spectrum is an isometry invariant of the respective object this fingerprint is also independent of the spatial position. Additionally the eigenvalues can be normalized so that scaling factors for the geometric object can be obtained easily. Therefore checking if two objects are isometric needs no prior alignment (registration/localization) of the objects, but only a comparison of their spectra. With the help of such fingerprints it is possible to support copyright protection, database retrieval and quality assessment of digital data representing surfaces and solids.

Global medical shape analysis using the Laplace-Beltrami spectrum

This paper proposes to use the Laplace-Beltrami spectrum (LBS) as a global shape descriptor for medical shape analysis, allowing for shape comparisons using minimal shape preprocessing: no registration, mapping, or remeshing is necessary. The discriminatory power of the method is tested on a population of female caudate shapes of normal control subjects and of subjects with schizotypal personality disorder.

Laplace spectra as fingerprints for image recognition

In the area of image retrieval from data bases and for copyright protection of large image collections there is a growing demand for unique but easily computable fingerprints for images. These fingerprints can be used to quickly identify every image within a larger set of possibly similar images. This paper introduces a novel method to automatically obtain such fingerprints from an image. It is based on a reinterpretation of an image as a Riemannian manifold. This representation is feasible for gray value images and color images. We discuss the use of the spectrum of eigenvalues of different variants of the Laplace operator as a fingerprint and show the usability of this approach in several use cases. Contrary to existing works in this area we do not only use the discrete Laplacian, but also with a particular emphasis the underlying continuous operator. This allows better results in comparing the resulting spectra and deeper insights in the problems arising. We show how the well known discrete Laplacian is related to the continuous Laplace-Beltrami operator. Furthermore, we introduce the new concept of solid height functions to overcome some potential limitations of the method.

Generating Tactile Textures using Periodicity Analysis

In this work we present an approach to generate a tactile representation from an image that may be a photograph or a scan of cloth. A large class of fabrics can be generated by repetition of a parallelogram primitive. That means there are two non-collinear directions where the pattern is periodically repeated. The method we present is based on analysing the repetitive structure of the sample, that is to find the two principle directions of repetition, and building a model from that analysis. We compare our method to a technique developed by Gang Huang.

Mass Density Laplace-Spectra for Image Recognition

Modern multimedia applications generate vast amounts of image data. With the availability of cheap photo hardware and affordable rendering software even more such data is being collected. In order to manage huge collections of image data one needs short representations of the data sets, or to be more precise invariant features being appropriate to identify a specific voxel data set using just a few numbers. This paper describes a variation of a method introduced by Reuter, Wolter and Peinecke based on the computation of the spectrum of the Laplace operator for the image for generating an invariant feature vector - a fingerprint. Oppose to previous techniques interpreting the image as a height function we make use of the representation of the image as a density function. We discuss the use of the spectrum of eigenvalues of the Laplace mass density operator as a fingerprint and show the usability of this approach in several cases. Instead of using the discrete Laplace-Kirchhoff operator the approach presented in this paper is based on the continuous Laplace operator allowing better results in comparing the resulting spectra and deeper insights into the problems arising when comparing two spectra generated using discrete Laplacians.