Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice

Entezari, Alireza; Ville, Dimitri Van De; Möller, Torsten B.

doi:10.1109/tvcg.2007.70429

Cited by 69 publications

(91 citation statements)

References 25 publications

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“…The extremal properties in terms of surface-to-area ratio of the hexagonal (in 2D) and truncated octahedral (in 3D) tessellations imply that they may serve as optimal tool for achieving data compression [57]. Another similarity of these two tessellations is related to their topologically stability with respect to infinitesimal perturbations to the position of the lattice points [58].…”

Section: Introductionmentioning

confidence: 99%

Symmetry-Break in Voronoi Tessellations

Lucarini

2009

Symmetry

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Abstract:We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is α and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values α>0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with α<0.12. Basically, the same happens in the 3D case, where only the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. In both 2D and 3D cases, already for a moderate amount of Gaussian noise (α>0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the OPEN ACCESSSymmetry 2009, 1 22 perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomal...

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Section: Introductionmentioning

confidence: 99%

Symmetry-Break in Voronoi Tessellations

Lucarini

2009

Symmetry

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“…The advantage of using Z d as the lattice is that the choice of the d axes is straightforward, and that each of the d directional blurs is simple: for all x ∈ R d , x can be blurred with points of the form x ± u k . Meanwhile, it has been observed that box splines on the body-centered cubic lattice A * 3 can be expressed using four projected axes of the 4D hypercube [19]. We generalize this observation below.…”

Section: Blurringmentioning

confidence: 68%

“…A * d is affinely equivalent to the Kuhn triangulation [25], which partitions a unit cube into tetrahedra. Sampling and reconstructions on the BCC lattice have been studied previously by Entezari et al [18,19]. In the computer graphics community, A * d has been used by Perlin [33] for generating high-dimensional procedural noise, and by Kim [23] for interpolating high-dimensional splines.…”

Section: Lattices and The Permutohedral Latticementioning

confidence: 99%

Lattice-Based High-Dimensional Gaussian Filtering and the Permutohedral Lattice

2012

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High-dimensional Gaussian filtering is a popular technique in image processing, geometry processing and computer graphics for smoothing data while preserving important features. For instance, the bilateral filter, cross bilateral filter and non-local means filter fall under the broad umbrella of high-dimensional Gaussian filters. Recent algorithmic advances therein have demonstrated that by relying on a sampled representation of the underlying space, one can obtain speed-ups of orders of magnitude over the naïve approach. The simplest such sampled representation is a lattice, and it has been used successfully in the bilateral grid and the permutohedral lattice algorithms. In this paper, we analyze these lattice-based algorithms, developing a general theory of lattice-based high-dimensional Gaussian filtering. We consider the set of criteria for an optimal lattice for filtering, as it offers a good tradeoff of quality for computational efficiency, and evaluate the existing lattices under the criteria. In particular, we give a rigorous exposition of the properties of the permutohedral lattice and argue that it is the optimal lattice for Gaussian filtering. Lastly, we explore further uses of the permutohedral-lattice-based Gaussian filtering framework, showing that it can be easily adapted to perform mean shift filtering and yield improvement over the traditional approach based on a Cartesian grid.

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“…Nürnberger et al 2005;Rössl et al 2004;Sorokina and Zeilfelder 2007) or in trivariate box splines (see e.g. Entezari and Möller 2006;Entezari et al 2008Entezari et al , 2009Kim et al 2008;Remogna 2010b,c).…”

Section: Introductionmentioning

confidence: 99%

On trivariate blending sums of univariate and bivariate quadratic spline quasi-interpolants on bounded domains

Remogna¹,

Sablonnière²

2011

Computer Aided Geometric Design

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The aim of this paper is to investigate, in a bounded domain of R 3 , two blending sums of univariate and bivariate C 1 quadratic spline quasi-interpolants.The main problem consists in constructing the coefficient functionals associated with boundary generators, i.e. generators with supports not entirely inside the domain. In their definition, these functionals involve data points lying inside or on the boundary of the domain. Moreover, the weights of these functionals must be chosen so that the quasi-interpolants have the best approximation order and a reasonable infinite norm.We give their explicit constructions, infinite norms and error estimates. In order to illustrate the approximation properties of the proposed quasiinterpolants, some numerical examples are presented and compared with those obtained by some other trivariate quasi-interpolants given recently in the literature.

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Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice

Cited by 69 publications

References 25 publications

Symmetry-Break in Voronoi Tessellations

Symmetry-Break in Voronoi Tessellations

Lattice-Based High-Dimensional Gaussian Filtering and the Permutohedral Lattice

On trivariate blending sums of univariate and bivariate quadratic spline quasi-interpolants on bounded domains

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