Discrete bisector function and Euclidean skeleton in 2D and 3D

International audienceWe propose a new definition and an exact algorithm for the discrete bisector function, which is an important tool for analyzing and filtering Euclidean skeletons. We also introduce a new thinning algorithm which produces homotopic discrete Euclidean skeletons. These algorithms, which are valid both in 2D and 3D, are integrated in a skeletonization method which is based on exact transformations, allows the filtering of skeletons, and is computationally efficient

Digital objects in rhombic dodecahedron grid

Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system

Thoughts on 3D Digital Subplane Recognition and Minimum-Maximum of a Bilinear Congruence Sequence

International audienceIn this paper we take first steps in addressing the 3D Digital Subplane Recognition Problem. Let us consider a digital plane P : 0 ≤ ax + by − cz + d < c (w.l.o.g. 0 ≤ a ≤ b ≤ c) and a finite subplane S of P dened as the points (x, y, z) of P such that (x, y) ∈ [x0, x1] × [y0, y1]. The Digital Subplane Recognition Problem consists in determining the characteristics of the subplane S in less than linear (in the number of voxels) complexity. We discuss approaches based on remainder values ax+by+d c , (x, y) ∈ [x0, x1] × [y0, y1] of the subplane. This corresponds to a bilinear congruence sequence. We show that one can determine if the sequence contains a value in logarithmic time. An algorithm to determine the minimum and maximum of such a bilinear congruence sequence is also proposed. This is linked to leaning points of the subplane with remainder order conservation properties. The proposed algorithm has a complexity in, if m = x1 −x0 < n = y1 −y0, O(m log (min(a, c − a)) or O(n log (min(b, c − b)) otherwise

Optimal Consensus Set for Digital Line Fitting

International audienceThis paper presents a new method for fitting a digital line to a given set of points in a 2D image in the presence of noise by maximizing the number of inliers, namely the consensus set. By using a digital line model instead of a continuous one, we show that we can generate all possible consensus sets for digital line fitting. We present a deterministic algorithm that efficiently searches the optimal solution with the time complexity O(N2 logN) and the space complexity O(N) where N is the number of points

Discrete bisector function and Euclidean skeleton

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Shear based Bijective Digital Rotation in Triangular Grids

In this paper we are proposing a way to perform bijective digital rotations on a triangular cell grid. The method is based on a decomposition of a rotation into shear transforms and on a way to transform the original triangle centroids in order to have a regular grid. The rotation method works for any angle and achieves an average distance between the digital rotated point and the continuous rotated point of about 0.5 (with 1.0 the side of a triangle and a distance between two neighboring triangle centroids of 0.57)

Author manuscript, published in &quot;16th Discrete Geometry for Computer Imagery, Nancy: France (2011)&quot; DOI: 10.1007/978-3-642-19867-0_30 Optimal Consensus set for Annulus Fitting

Abstract. An annulus is defined as a set of points contained between two circles. This paper presents a method for fitting an annulus to a given set of points in a 2D images in the presence of noise by maximizing the number of inliers, namely the consensus set, while fixing the thickness. We present a deterministic algorithm that searches the optimal solution(s) within a time complexity of O(N 4), N being the number of points