Box splines defined

Boor, Carl de; Höllig, Klaus; Riemenschneider, S. D.

doi:10.1007/978-1-4757-2244-4_1

Cited by 60 publications

(106 citation statements)

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“…Box splines can be interpreted as generalizations of the well‐known tensor‐product B‐splines to various, not necessarily separable lattices. In ℝ d , a box spline is defined by m ≥ d vectors that are the columns of its matrix Ξ = [ ξ 1 , ξ 2 , …, ξ m ] [dBHR93]. The simplest box spline is constructed by m = d vectors and it is the normalized characteristic function of the parallelepiped spanned by ξ i :…”

Section: Box Splinesmentioning

confidence: 99%

Retailoring Box Splines to Lattices for Highly Isotropic Volume Representations

Csébfalvi

Rácz

2016

Computer Graphics Forum

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3D box splines are defined by convolving a 1D box function with itself along different directions. In volume visualization, box splines are mainly used as reconstruction kernels that are easy to adapt to various sampling lattices, such as the Cartesian Cubic (CC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC) lattices. The usual way of tailoring a box spline to a specific lattice is to span the box spline by exactly those principal directions that span the lattice itself. However, in this case, the preferred directions of the box spline and the lattice are the same, amplifying the anisotropic effects of each other. This leads to an anisotropic volume representation with strongly preferred directions. Therefore, in this paper, we retailor box splines to lattices such that the sets of vectors that span the box spline and the lattice are disjoint sets. As the preferred directions of the box spline and the lattice compensate each other, a more isotropic volume representation can be achieved. We demonstrate this by comparing different combinations of box splines and lattices concerning their anisotropic behavior in tomographic reconstruction and volume visualization.

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Section: Box Splinesmentioning

confidence: 99%

Retailoring Box Splines to Lattices for Highly Isotropic Volume Representations

Csébfalvi

Rácz

2016

Computer Graphics Forum

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“…In fact, it is a non-trivial problem for computing all complex vectors satisfying θ M ij = 1. In the following Lemma, we shows the generalized Fourier-Dedekind sums (15) can be converted into the 1-dimensional Fourier-Dedekind sums.…”

Section: Two-dimension Vector Partition Functionsmentioning

confidence: 99%

An explicit formulation for two dimensional vector partition functions

Xu¹

2008

Integer Points in Polyhedra—Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics

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“…For a probabilistic interpretation, the reader is referred to Karlin, Micchelli and Rinott (1986). The simplest examples of this type of spline are the so-called box splines, which are de ned with respect to very regular grids (see de Boor and H ollig 1982;de Boor, H ollig and Riemenschneider 1993). We have chosen our space of tent functions because it is the starting point for these two approaches to spline construction.…”

Section: Multivariate Splines and Triangulationsmentioning

confidence: 99%

Triogram Models

Hansen¹,

Kooperberg

Sardy

1998

Journal of the American Statistical Association

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In this paper we introduce the Triogram method for function estimation using piecewise linear, bivariate splines based on an adaptively constructed triangulation. We illustrate the technique for bivariate regression and log-density estimation and indicate how our approach can be applied directly to model bivariate functions in the broader context of an extended linear model. The entire estimation procedure is invariant under a ne transformations and is a natural approach for modeling data when the domain of the predictor variables is a polygonal region in the plane. Although our examples deal exclusively with estimating bivariate functions, the use of Triograms for modeling twofactor interactions in ANOVA decompositions of functions depending on more than two variables is straightforward. Software for tting these models will be available in Version 4 of S.

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Box splines defined

Cited by 60 publications

References 0 publications

Retailoring Box Splines to Lattices for Highly Isotropic Volume Representations

Retailoring Box Splines to Lattices for Highly Isotropic Volume Representations

An explicit formulation for two dimensional vector partition functions

Triogram Models

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