A Bayesian Approach to Manifold Topology Reconstruction

In this paper, we investigate the problem of statistical reconstruction of piecewise linear manifold topology. Given a noisy, probably undersampled point cloud from a one- or two-manifold, the algorithm reconstructs an approximated most likely mesh in a Bayesian sense from which the sample might have been taken. We incorporate statistical priors on the object geometry to improve the reconstruction quality if additional knowledge about the class of original shapes is available. The priors can be formulated analytically or learned from example geometry with known manifold tessellation. The statistical objective function is approximated by a linear programming / integer programming problem, for which a globally optimal solution is found. We apply the algorithm to a set of 2D and 3D reconstruction examples, demon-strating that a statistics-based manifold reconstruction is feasible, and still yields plausible results in situations where sampling conditions are violated

A Bayesian Approach to Manifold Topology Reconstruction

In this paper, we investigate the problem of statistical reconstruction of piecewise linear manifold topology. Given a noisy, probably undersampled point cloud from a one- or two-manifold, the algorithm reconstructs an approximated most likely mesh in a Bayesian sense from which the sample might have been taken. We incorporate statistical priors on the object geometry to improve the reconstruction quality if additional knowledge about the class of original shapes is available. The priors can be formulated analytically or learned from example geometry with known manifold tessellation. The statistical objective function is approximated by a linear programming / integer programming problem, for which a globally optimal solution is found. We apply the algorithm to a set of 2D and 3D reconstruction examples, demon-strating that a statistics-based manifold reconstruction is feasible, and still yields plausible results in situations where sampling conditions are violated

Curve Reconstruction, the Traveling Salesman Problem, and Menger's Theorem on Length

We give necessary and sufficient regularity conditions under which the curve reconstruction problem is solved by a traveling salesman tour or path, respectively. For the proof we have to generalize a theorem of Menger [12], [13] on arc lengt

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

TSP-Based Curve Reconstruction in Polynomial Time

An instance of the curve reconstruction problem is a finite sample set $V$ of an unknown curve $\gamma$. The task is to connect the points in $V$ in the order in which they lie on $\gamma$. Giesen~\cite{SCG99*207} showed recently that the Traveling Salesman tour of $V$ solves the reconstruction problem under fairly week assumptions on $\gamma$ and $V$. We extend his result along three dimensions. We weaken the assumptions, give an alternate proof, and show that in the context of curve reconstruction, the Traveling Salesman tour can be constructed in polynomial time

TSP-Based Curve Reconstruction in Polynomial Time

An instance of the curve reconstruction problem is a nite sample V of an unknown curve and the task is to connect the points in V in the order in which they lie on . Giesen [Gie99] showed recently that the Traveling Salesman tour of V solves the reconstruction problem under fairly week assumptions on and V . We extend his result along three dimensions. We weaken the assumptions, we give an alternate proof, and we show that in the context of curve reconstruction the Traveling Salesman tour can be constructed in polynomial time