Total absolute curvature as a tool for modelling curves and surfaces

Total absolute curvature (TAC) is proposed as a tool for modelling curves and surfaces from discrete two– or three–dimensional data

Accurate correction of surface noises of polygonal meshes

In this paper we propose a new algorithm for accurate correction of surface noises of polygonal meshes. It consists of three basic components: (a) feature-preserving pre-smoothing; (b) partitioning of feature and non-feature regions; (c) second-order predictor for non-feature regions and median filter for feature regions. The unique contributions of our approach include (a) an idea of partitioning an input surface into feature and non-feature regions so that different smoothing algorithms, which are best suited for either feature or non-feature regions can be, respectively, applied; (b) a second-order predictor that provides higher smoothing accuracy and better convergence on smoothly curved surfaces. In comparison with several existing algorithms, our algorithm is evaluated quantitatively in terms of surface normal and vertex distance error metrics. Numerical experiments indicate the effectiveness of our approach in the aspects of convergence and accuracy. Copyright © 2005 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48758/1/1441_ftp.pd

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Optimising triangulated polyhedral surfaces with self-intersections

We discuss an optimisation procedure for triangulated polyhedral surfaces (referred to as (2 - 3) D triangulations) which allows us to process self-intersecting surfaces. As an optimality criterion we use minimisation of total absolute extrinsic curvature (MTAEC) and as a local transformation - a diagonal flip, defined in a proper way for (2 - 3) D triangulations. This diagonal flip is a natural generalisation of the diagonal flip operation in 2D, known as Lawson's procedure. The difference is that the diagonal flip operation in (2 - 3)D triangulations may produce self-intersections. We analyze the optimisation procedure for (2 - 3) D closed triangulations, taking into account possible self-intersections. This analysis provides a general insight on the structure of triangulations, allows to characterise the types of self-intersections, as well as the conditions for global convergence of the algorithm. It provides also a new view on the concept of optimisation on the whole and is useful in the analysis of global and local convergence for other optimisation algorithms. At the end we present an efficient implementation of the optimality procedure for (2 - 3)D triangulations of the data, situated in the convex position, and conjecture possible results of this procedure for non-convex data

Optimising triangulated polyhedral surfaces with self-intersections

We discuss an optimisation procedure for triangulated polyhedral surfaces (referred to as (2 - 3) D triangulations) which allows us to process self-intersecting surfaces. As an optimality criterion we use minimisation of total absolute extrinsic curvature (MTAEC) and as a local transformation - a diagonal flip, defined in a proper way for (2 - 3) D triangulations. This diagonal flip is a natural generalisation of the diagonal flip operation in 2D, known as Lawson's procedure. The difference is that the diagonal flip operation in (2 - 3)D triangulations may produce self-intersections. We analyze the optimisation procedure for (2 - 3) D closed triangulations, taking into account possible self-intersections. This analysis provides a general insight on the structure of triangulations, allows to characterise the types of self-intersections, as well as the conditions for global convergence of the algorithm. It provides also a new view on the concept of optimisation on the whole and is useful in the analysis of global and local convergence for other optimisation algorithms. At the end we present an efficient implementation of the optimality procedure for (2 - 3)D triangulations of the data, situated in the convex position, and conjecture possible results of this procedure for non-convex data

Constructing Regular Triangulation via Local Transformations: Theoretical and Practical Advances

Ph.DDOCTOR OF PHILOSOPH

The polyhedral Gauss map and discrete curvature measures in geometric modelling.

The Work in this thesis is concentrated on the study of discrete curvature as an important geometric property of objects, useful in describing their shape. The main focus is on the study of the methods to measure the discrete curvature on polyhedral surfaces. The curvatures associated with a polyhedral surface are concentrated around its vertices and along its edges. An existing method to evaluate the curvature at a vertex is the Angle Deficit, which also characterises vertices into flat, convex or saddle. In discrete surfaces other kinds of vertices are possible which this method cannot identify. The concept of Total Absolute Curvature (TAC) has been established to overcome this limitation, as a measure of curvature independent of the orientation of local geometry. However no correct implementation of the TAC exists for polyhedral surfaces, besides very simple cases.For two-dimensional discrete surfaces in space, represented as polygonal meshes, the TAC is measured by means of the Polyhedral Gauss Map (PGM) of vertices. This is a representation of the curvature of a vertex as an area on the surface of a sphere. Positive and negative components of the curvature of a vertex are distinguished as spherical polygons on the PGM. Core contributions of this thesis are the methods to identify these polygons and give a sign to them. The PGM provides a correct characterisation of vertices of any type, from basic convex and saddle types to complex mixed vertices, which have both positive and negative curvature in them.Another contribution is a visualisation program developed to show the PGM using 3D computer graphics. This program helps in the understanding and analysis of the results provided by the numerical computations of curvature. It also provides interactive tools to show the detailed information about the curvature of vertices.Finally a polygon simplification application is used to compare the curvature measures provided by the Angle Deficit and PGM methods. Various sample meshes are decimated using both methods and the simplified results compared with the original meshes. These experiments show how the TAC can be used to more effectively preserve the shape of an object. Several other applications that benefit in a similar way with the use of the TAC as a curvature measure are also proposed