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Publications
Publications (44)
We propose an analytical definition of discrete circles in the hexagonal grid. Our approach is based on a non-constant thickness function. We determine the thickness using the (edge and vertex) flake model. Both types of circles are connected. We prove that edge flake circles are without simple points for integer radii. Incremental generation algor...
Each 3D rotation can be decomposed into three 2D rotations parametrized by three Euler angles. Each of the three 2D rotations can be expressed as a sequence of three 2D shears along coordinate axes, leading to a decomposition of a 3D rotation into nine (beam) shears in total. We define a 3D digitized rotation using nine digitized beam shears, i.e.,...
Three-dimensional surface reconstruction is a well-known task in medical imaging. In procedures for intervention or radiation treatment planning, the generated models should be accurate and reflect the natural appearance. Traditional methods for this task, such as Marching Cubes, use smoothing post processing to reduce staircase artifacts from mesh...
A grid unfolding without refinement method for a new sub-class of polycubes, called Manhattan Towers with a H-convex base, is proposed. Such polycubes can be seen both as a Manhattan Tower and as an orthostack. A direct extension of this algorithm to Up-and-Down Orthoterrains is also presented.
The bisector function is an important tool for analyzing and filtering Euclidean skeletons. In this paper, we are proposing a new way to compute 2D and 3D discrete bisector function based on annuli. From a continuous point of view, a point that belongs to the medial axis is the center of a maximal ball that hits the background in more than one poin...
We define a new compact coordinate system in which each integer triplet addresses a voxel in the BCC grid, and we investigate some of its properties. We propose a characterization of 3D discrete analytical planes with their topological features (in the Cartesian and in the new coordinate system) such as the interrelation between the thickness of th...
In this paper, a new bijective digital rotation algorithm for the hexagonal grid is proposed. The method is based on an decomposition of rotations into shear transforms. It works for any angle with an hexagonal centroid as rotation center and is easily invertible. The algorithm achieves an average distance between the digital rotated point and the...
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 i...
In this article, we suggest a grid-unfolding of level 1 Menger polycubes of arbitrary size with L holes along the x-axis, M the y-axis and N the z-axis. These polycubes can have a high genus, and most vertices are of degree 6. The unfolding is based mainly on the inner faces (that do not lie on the outer most envelope) except for some outer faces t...
In this paper, a new bijective reflection algorithm in two dimensions is proposed along with an associated rotation. The reflection line is defined by an arbitrary Euclidean point and a straight line passing through this point. The reflection line is digitized and the 2D space is paved by digital perpendicular (to the reflection line) straight line...
In this paper we are proposing a new way to compute a discrete bisector function, which is an important tool for analyzing and filtering Euclidean skeletons. From a continuous point of view, a point that belongs to the medial axis is the center of a maximal ball that hits the background in more than one point. The maximal angle between those points...
In this paper, we propose an algorithm to build discrete spherical shell having integer center and real-valued inner and outer radii on the face-centered cubic (FCC) grid. We address the problem by mapping it to a 2D scenario and building the shell layer by layer on hexagonal grids with additive manufacturing in mind. The layered hexagonal grids ge...
This paper presents a method for fitting digital hyperspheres to a given set of nD points in an image in the presence of noise by maximizing the number of inliers, namely the consensus set. The digital Hyperspheres are defined using the k-Flake Digitization models We present an algorithm, that provides optimal fitting solutions for digital k-Flake...
In this paper we take rst 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 nite 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 ch...
This paper presents a method for fitting a nD fixed width spherical shell to a given set of nD points in an image in the presence of noise by maximizing the number of inliers, namely the consensus set. We present an algorithm, that provides the optimal solution(s) within a time complexity O(N n+1 log N) for dimension n, N being the number of points...
In this paper we introduce a notion of digital implicit surface in arbitrary dimensions. The digital implicit surface is the result of a morphology inspired digitization of an implicit surface {x ∈ ℝn : f(x) = 0} which is the boundary of a given closed subset of ℝn
, {x ∈ ℝn : f(x) ≤ 0}. Under some constraints, the digital implicit surface has some...
Given a digital straight line D of known characteristics (a, b, c), and given two arbitrary discrete points A(x a , y a) and B(x b , y b) of it, we are interested in com-puting the characteristics of the digital straight segment (DSS) of D delimited by the endpoints A and B. Our method is based entirely on the remainder subsequence S = {ax − c mod...
A digital annulus is defined as a set of grid points lying between two circles sharing an identical center and separated by a given width. This paper deals with the problem of fitting a digital annulus to a given set of points in a 2D bounded grid. More precisely, we tackle the problem of finding a digital annulus that contains the largest number o...
This paper presents a method for fitting 4-connected digital circles to a given set of points in 2D images in the presence of noise by maximizing the number of inliers, namely the optimal consensus set, while fixing the thickness. Our approach has a O(n
3log n) time complexity and O(n) space complexity, n being the number of points, which is lower...
This paper exploits the problem of fitting special forms of annuli that correspond to 4-connected digital circles to a given set of points in 2D images in the presence of noise by maximizing the number of inliers, namely the consensus set. We prove that the optimal solutions can be described by solutions with three points on the annulus boundary. T...
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 solutio...
This article presents a new method for fitting a digital line or plane to a given set of points in a 2D or 3D image in the presence of noise by maximizing the number of inliers, namely the consensus set. By using a digital model instead of a continuous one, we show that we can generate all possible consensus sets for model fitting. We present a det...
This paper presents a method for fitting a digital plane to a given set of points in a 3D image in the presence of outliers. We present a new method that uses a digital plane model rather than the conventional continuous model. We show that such a digital model allows us to efficiently examine all possible consensus sets and to guarantee the soluti...
We propose a new definition and an 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 313, are integrated in a skeletonization metho...
We present in this article a parallelization of a discrete radiosity method, based on scene division and achieved on a cluster
of workstations. This method is based on discretization of surfaces into voxels and not into patches like most of the radiosity
methods do. Voxels are stocked into visibility lists representing the space partition into disc...
In this paper we present three different parallelizations of a discrete radiosity method achieved on a cluster of workstations.
This radiosity method has lower complexity when compared with most of the radiosity algorithms and is based on the discretization
of surfaces into voxels and not into patches. The first two parallelizations distribute the...
We 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 method which produces homotopic discrete Euclidean skeletons.
Unlike previouly proposed approaches, this method is still valid in 3D.