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Parameterizations of Digital Surfaces Homeomorphic to a Sphere using Discrete Harmonic Functions
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2006 (English)In: Pattern Recognition Letters, ISSN 0167-8655, Vol. 27, no 16, 1934-1941 p.Article in journal (Refereed) Published
Abstract [en]

We discuss how to map simply connected digital surfaces to the unit sphere using discrete harmonic functions. This technique is well known for constructing parametrizations of surfaces onto simpler domains. A common problem is the creation of dense clusters of vertices, which leads to numerical instability of methods operating on this mapping. By an explicit calculation, we quantify the cluster density for a simple model example. This example shows that distances between mapped vertices can decrease exponentially in regions on the sphere. By numerical examples, we show that severe clustering often occurs for natural objects. A computationally inexpensive algorithm, based on a bijective transformation of the unit sphere, is suggested for post-processing of clusters. Experiments indicate that this algorithm often improves the mapping such that convergence of a non-linear optimization program is achieved. The program aims at optimizing the parameterization, making it useful for global shape analysis. Examples of approximations in terms of spherical harmonics functions are presented.

Place, publisher, year, edition, pages
2006. Vol. 27, no 16, 1934-1941 p.
Keyword [en]
shape, shape approximation, digital surface, surface parameterization
National Category
Physical Sciences
URN: urn:nbn:se:uu:diva-93658DOI: 10.1016/j.patrec.2006.05.003ISI: 000241718600007OAI: oai:DiVA.org:uu-93658DiVA: diva2:167201
Available from: 2005-11-16 Created: 2005-11-16 Last updated: 2011-05-13Bibliographically approved
In thesis
1. Global Shape Description of Digital Objects
Open this publication in new window or tab >>Global Shape Description of Digital Objects
2005 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Global formbeskrivning av digitala objekt
Abstract [en]

New methods for global shape description of three-dimensional digital objects are presented. The shape of an object is first represented by a digital surface where the faces are either triangles or quadrilaterals. Techniques for computing a high-quality parameterization of the surface are developed and this parameterization is used to approximate the shape of the object. Spherical harmonics are used as basis functions for approximations of the coordinate functions. Information about the global shape is then captured by the coefficients in the spherical harmonics expansions.

For a starshaped object it is shown how a parameterization can be computed by a projection from its surface onto the unit sphere. An algorithm for computing the position at which the centre of the sphere should be placed, is presented. This algorithm is suited for digital voxel objects. Most of the work is concerned with digital objects whose surfaces are homeomorphic to the sphere. The standard method for computing parameterizations of such surfaces is shown to fail on many objects. This is due to the large distortions of the geometric properties of the surface that often occur with this method. Algorithms to handle this problem are suggested. Non-linear optimization methods are used to find a mapping between a surface and the sphere that minimizes geometric distortion and is useful as a parameterization of the surface.

The methods can be applied, for example, in medical imaging for shape recognition, detection of shape deformations and shape comparisons of three-dimensional objects.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2005. viii+37 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 43
Applied mathematics, shape description, shape approximation, surface parameterization, digital object, spherical harmonics, digital surface, digital image, Tillämpad matematik
urn:nbn:se:uu:diva-6030 (URN)91-506-1833-4 (ISBN)
Public defence
2005-12-09, 2446, MIC, Uppsala, 13:15
Available from: 2005-11-16 Created: 2005-11-16Bibliographically approved

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