Principal component and Voronoi skeleton alternatives for curve reconstruction from noisy point sets

Surface reconstruction from noisy point samples must take into consideration the stochastic nature of the sample -- In other words, geometric algorithms reconstructing the surface or curve should not insist in following in a literal way each sampled point -- Instead, they must interpret the sample as a “point cloud” and try to build the surface as passing through the best possible (in the statistical sense) geometric locus that represents the sample -- This work presents two new methods to find a Piecewise Linear approximation from a Nyquist-compliant stochastic sampling of a quasi-planar C1 curve C(u) : R → R3, whose velocity vector never vanishes -- One of the methods articulates in an entirely new way Principal Component Analysis (statistical) and Voronoi-Delaunay (deterministic) approaches -- It uses these two methods to calculate the best possible tape-shaped polygon covering the planarised point set, and then approximates the manifold by the medial axis of such a polygon -- The other method applies Principal Component Analysis to find a direct Piecewise Linear approximation of C(u) -- A complexity comparison of these two methods is presented along with a qualitative comparison with previously developed ones -- It turns out that the method solely based on Principal Component Analysis is simpler and more robust for non self-intersecting curves -- For self-intersecting curves the Voronoi-Delaunay based Medial Axis approach is more robust, at the price of higher computational complexity -- An application is presented in Integration of meshes originated in range images of an art piece -- Such an application reaches the point of complete reconstruction of a unified mes

Hierarchic Voronoi Skeletons

Robust and time-efficient skeletonization of a (planar) shape, which is connectivity preserving and based on Euclidean metrics, can be achieved by first regularizing the Voronoi diagram (VD) of a shape's boundary points, i.e., by removal of noise-sensitive parts of the tessellation and then by establishing a hierarchic organization of skeleton constituents. Each component of the VD is attributed with a measure of prominence which exhibits the expected invariance under geometric transformations and noise. The second processing step, a hierarchic clustering of skeleton branches, leads to a multiresolution representation of the skeleton, termed skeleton pyramid

Voronoi Skeletons: Theory and Applications

The paper presents a novel method of robust skeletonization based on the Voronoi diagram (VD) of boundary points, which is characterized by correct Euclidean metrics and inherent preservation of connectivity. The regularization of the Voronoi medial axis (VMA) in the sense of Blum's prairie fire analogy is done by attributing each component of the VMA with a measure of prominence and stability. The resulting Voronoi skeletons (VSK) appear largely invariant with respect to typical noise conditions in the image and geometric transformations. Hierarchical clustering of the skeleton branches, the so-called skeleton pyramid, leads to further simplification of the skeleton. Several applications demonstrate the suitability of the Voronoi skeleton to higher order tasks such as object recognition. 1 Introduction During the last decades, skeletonization or thinning has been a constant research topic. The concept of skeletonization denotes a process, which transforms a 2D object into a 1D lin..

Voronoi Tessellation of Points with Integer Coordinates: Time-Efficient Implementation and Online Edge-List Generation

The Voronoi tessellation in the plane can be computed in a particularly time-efficient manner for generators with integer coordinates, such as typically acquired from a raster image. The Voronoi tessellation is constructed line by line during a single scan of the input image, simultaneously generating an edge-list data structure (DCEL) suitable for postprocessing by graph traversal algorithms. In contrast to the generic case, it can be shown that the topology of the grid permits the algorithm to run faster on complex scenes. Consequently, in Computer Vision applications, the computation of the Voronoi tessellation represents an attractive alternative to rasterbased techniques in terms of both computational complexity and quality of data structures. Index terms ---Tessellation, Computational Geometry, Delaunay triangulation, Voronoi diagram. 1. Introduction The concept of the Voronoi diagram (also termed Voronoi or Dirichlet tessellation) refers to one of the basic closest point probl..

Medial manifolds and hierarchical description of 2D and 3D objects with applications to MRI data of the human brain

A mathematical description and representation of 2D and 3D shape, capable of hierarchically decomposing complex objects, is at the focus of this paper. The development is based on a hierarchic extension to the Medial Axis Transformation (MAT). Our implementation of the hierarchic MAT (HMAT) combines full Voronoi tessellation generated by the set of border points with regularization procedures to obtain a hierarchy of geometrically and topologically correct medial manifolds. This hierarchy defines a skeleton pyramid allowing shape-driven decomposition in 2D and 3D. The proposed methodology is illustrated in 2D on a planar section through a 3D MRI data set of the human brain. The hierarchical decomposition indicates the process history of brain development. For further analysis, it is converted into a boundary representation. The 3D extension of the HMAT concept is tested and illustrated on synthetic objects. It is applied to the full 3D MRI data set to obtain a description of the sulci ..