Linear and cubic box splines for the body centered cubic lattice

Entezari, Alireza; Dyer, Ramsay; Möller, Torsten B.

doi:10.1109/visual.2004.65

Cited by 48 publications

(52 citation statements)

References 13 publications

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“…reversible conversion between the BCC and orthogonal lattices in 3-D is of practical interest in computer graphics [11]. The reference image is undistinguishable from the resampled image with the sinc filter.…”

Section: Resultsmentioning

confidence: 99%

H2O: Reversible Hexagonal-Orthogonal Grid Conversion by 1-D Filtering

Condat

Forster-Heinlein

Ville

2007

2007 IEEE International Conference on Image Processing

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In this work, we propose a new grid conversion algorithm between the hexagonal lattice and the orthogonal (a.k.a. Cartesian) lattice. The conversion process, named H 2 O, is easy to implement and is perfectly reversible using the same algorithm to return from one lattice to the other. The key observation of our approach is a decomposition of the lattice conversion as a sequence of shearing operations along three well-chosen directions. Hence, only 1-D fractional sample delay operators are required, which can be implemented by simple convolutions. The proposed algorithm combines reversibility and fast 1-D operations, together with high-quality resampled images.

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Section: Resultsmentioning

confidence: 99%

H2O: Reversible Hexagonal-Orthogonal Grid Conversion by 1-D Filtering

Condat

Forster-Heinlein

Ville

2007

2007 IEEE International Conference on Image Processing

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“…Its direction matrix Ξ is the operator T . Entezari et al [18] previously derived this kernel for d = 3. 3 .…”

Section: Algorithmmentioning

confidence: 98%

“…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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“…Earlier, a direct rendering paradigm of trivariate B-spline functions for large data sets with interactive rates was presented in the work in [38], where the rendering is conducted from a fixed viewpoint in two phases suitable for sculpting operations. Entezari et al [14] derive piecewise linear and piecewise cubic box spline reconstruction filters for data sampled on the body-centered cubic lattice. Given such a representation, they directly visualize isosurfaces.…”

Section: Classical Problem Statementmentioning

confidence: 99%

Direct Isosurface Visualization of Hex-Based High-Order Geometry and Attribute Representations

Martín

Cohen

Kirby

2012

IEEE Trans. Visual. Comput. Graphics

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Abstract-In this paper, we present a novel isosurface visualization technique that guarantees the accurate visualization of isosurfaces with complex attribute data defined on (un)structured (curvi)linear hexahedral grids. Isosurfaces of high-order hexahedralbased finite element solutions on both uniform grids (including MRI and CT scans) and more complex geometry representing a domain of interest that can be rendered using our algorithm. Additionally, our technique can be used to directly visualize solutions and attributes in isogeometric analysis, an area based on trivariate high-order NURBS (Non-Uniform Rational B-splines) geometry and attribute representations for the analysis. Furthermore, our technique can be used to visualize isosurfaces of algebraic functions. Our approach combines subdivision and numerical root finding to form a robust and efficient isosurface visualization algorithm that does not miss surface features, while finding all intersections between a view frustum and desired isosurfaces. This allows the use of viewindependent transparency in the rendering process. We demonstrate our technique through a straightforward CPU implementation on both complex-structured and complex-unstructured geometries with high-order simulation solutions, isosurfaces of medical data sets, and isosurfaces of algebraic functions.Index Terms-Isosurface visualization of hex-based high-order geometry and attribute representations, numerical analysis, roots of nonlinear equations, spline and piecewise polynomial interpolation.

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Linear and cubic box splines for the body centered cubic lattice

Abstract: Abstract

Cited by 48 publications

References 13 publications

H2O: Reversible Hexagonal-Orthogonal Grid Conversion by 1-D Filtering

H2O: Reversible Hexagonal-Orthogonal Grid Conversion by 1-D Filtering

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

Direct Isosurface Visualization of Hex-Based High-Order Geometry and Attribute Representations

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