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39
The Flyspeck Project
"... Abstract. This article gives an introduction to a longterm project called Flyspeck, whose purpose is to give a formal verification of the ..."
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Cited by 47 (3 self)
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Abstract. This article gives an introduction to a longterm project called Flyspeck, whose purpose is to give a formal verification of the
Linear and Cubic Box Splines for the Body Centered Cubic Lattice
 In Proceedings of the IEEE Conference on Visualization
, 2004
"... In this paper we derive piecewise linear and piecewise cubic box spline reconstruction filters for data sampled on the body centered cubic (BCC) lattice. We analytically derive a time domain representation of these reconstruction filters and using the Fourier sliceprojection theorem we derive their ..."
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Cited by 41 (8 self)
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In this paper we derive piecewise linear and piecewise cubic box spline reconstruction filters for data sampled on the body centered cubic (BCC) lattice. We analytically derive a time domain representation of these reconstruction filters and using the Fourier sliceprojection theorem we derive their frequency responses. The quality of these filters, when used in reconstructing BCC sampled volumetric data, is discussed and is demonstrated with a raycaster. Moreover, to demonstrate the superiority of the BCC sampling, the resulting reconstructions are compared with those produced from similar filters applied to data sampled on the Cartesian lattice.
Practical box splines for reconstruction on the body centered cubic lattice
 IEEE Trans. Vis. Comput. Graphics
, 2008
"... Abstract—We introduce a family of box splines for efficient, accurate, and smooth reconstruction of volumetric data sampled on the bodycentered cubic (BCC) lattice, which is the favorable volumetric sampling pattern due to its optimal spectral sphere packing property. First, we construct a box spli ..."
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Cited by 29 (5 self)
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Abstract—We introduce a family of box splines for efficient, accurate, and smooth reconstruction of volumetric data sampled on the bodycentered cubic (BCC) lattice, which is the favorable volumetric sampling pattern due to its optimal spectral sphere packing property. First, we construct a box spline based on the four principal directions of the BCC lattice that allows for a linear C 0 reconstruction. Then, the design is extended for higher degrees of continuity. We derive the explicit piecewise polynomial representations of the C 0 and C 2 box splines that are useful for practical reconstruction applications. We further demonstrate that approximation in the shiftinvariant space—generated by BCClattice shifts of these box splines—is twice as efficient as using the tensorproduct Bspline solutions on the Cartesian lattice (with comparable smoothness and approximation order and with the same sampling density). Practical evidence is provided demonstrating that the BCC lattice not only is generally a more accurate sampling pattern, but also allows for extremely efficient reconstructions that outperform tensorproduct Cartesian reconstructions. Index Terms—BCC, box splines, discrete/continuous representations, optimal regular sampling. Ç 1
Kissing Numbers, Sphere Packings, and Some Unexpected Proofs
 NOTICES AMER. MATH. SOC
, 2004
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Flyspeck i: Tame graphs
 International Joint Conference on Automated Reasoning, volume 4130 of LNCS
, 2006
"... Abstract. We present a verified enumeration of tame graphs as defined in Hales ’ proof of the Kepler Conjecture and confirm the completeness of Hales ’ list of all tame graphs while reducing it from 5128 to 2771 graphs. 1 ..."
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Cited by 14 (1 self)
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Abstract. We present a verified enumeration of tame graphs as defined in Hales ’ proof of the Kepler Conjecture and confirm the completeness of Hales ’ list of all tame graphs while reducing it from 5128 to 2771 graphs. 1
Computer Assisted Proof of Optimal Approximability Results
, 2002
"... We obtain computer assisted proofs of several spherical volume inequalities that appear in the analysis of semidefinite programming based approximation algorithms for Boolean constraint satisfaction problems. These inequalities imply, in particular, that the performance ratio achieved by the MAX 3S ..."
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Cited by 13 (4 self)
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We obtain computer assisted proofs of several spherical volume inequalities that appear in the analysis of semidefinite programming based approximation algorithms for Boolean constraint satisfaction problems. These inequalities imply, in particular, that the performance ratio achieved by the MAX 3SAT approximation algorithm of Karloff and Zwick is indeed 7/8, as conjectured by them, and that the performance ratio of the MAX 3CSP algorithm of the author is indeed ½. Other results are also implied. The computer assisted proofs are obtained using a system called REALSEARCH written by the author. This system uses interval arithmetic to produce rigorous proofs that certain collections of constraints in real variables have no real solution.
Reconstruction Schemes for High Quality Raycasting of the BodyCentered Cubic Grid
"... The bodycentered cubic (BCC) grid has received attention in the volume visualization community recently due to its ability to represent the same data with almost 30% fewer samples as compared to the Cartesian cubic (CC) grid. In this paper we present several resampling strategies for raycasting BCC ..."
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Cited by 12 (5 self)
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The bodycentered cubic (BCC) grid has received attention in the volume visualization community recently due to its ability to represent the same data with almost 30% fewer samples as compared to the Cartesian cubic (CC) grid. In this paper we present several resampling strategies for raycasting BCC grids. These strategies range from 2D interpolation in planes to piecewise linear (barycentric) interpolation in a tetrahedral decomposition of the grid to trilinear and sheared trilinear interpolation. We compare them to raycasting with comparable resampling techniques in the commonly used CC grid in terms of computational complexity and visual quality. 1
Scanning the structure of illknown spaces: Part 2. Principles of construction of physical space
 in Kybernetes: The International Journal of Systems and Cybernetics
"... Abstract. An abstract lattice of empty set cells is shown to be able to account for a primary substrate in a physical space. Spacetime is represented by ordered sequences of topologically closed Poincaré sections of this primary space. These mappings are constrained to provide homeomorphic structure ..."
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Cited by 12 (11 self)
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Abstract. An abstract lattice of empty set cells is shown to be able to account for a primary substrate in a physical space. Spacetime is represented by ordered sequences of topologically closed Poincaré sections of this primary space. These mappings are constrained to provide homeomorphic structures serving as frames of reference in order to account for the successive positions of any objects present in the system. Mappings from one to the next section involve morphisms of the general structures, standing for a continuous reference frame, and morphisms of objects present in the various parts of this structure. The combination of these morphisms provides spacetime with the features of a nonlinear generalized convolution. Discrete properties of the lattice allow the prediction of scales at which microscopic to cosmic structures should occur. Deformations of primary cells by exchange of empty set cells allow a cell to be mapped into an image cell in the next section as far as mapped cells remain homeomorphic. However, if a deformation involves a fractal transformation to objects, there occurs a change in the dimension of the cell and the homeomorphism is not conserved. Then the fractal kernel stands for a ”particle ” and the reduction
BCCsplines: Generalization of Bsplines for the bodycentered cubic lattice
 Journal of WSCG
, 2008
"... Recently, the Bspline family of reconstruction filters has been generalized for the hexagonal lattice, which is optimal for sampling 2D circularly bandlimited signals. In this paper, we extend this generalization to the bodycentered cubic (BCC) lattice, which is optimal for sampling spherically b ..."
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Cited by 11 (4 self)
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Recently, the Bspline family of reconstruction filters has been generalized for the hexagonal lattice, which is optimal for sampling 2D circularly bandlimited signals. In this paper, we extend this generalization to the bodycentered cubic (BCC) lattice, which is optimal for sampling spherically bandlimited 3D signals. We call the obtained new reconstruction filters BCCsplines. Although the explicit analytical formulas are not defined yet, we evaluate the discrete approximation of these filters in the frequency domain in order to analyze their performance in a volumerendering application. Our experimental results show that the BCCsplines can be superior over the box splines previously proposed for the BCC lattice.
On Visual Quality of Optimal 3D Sampling and Reconstruction
"... This paper presents a user study of the visual quality of an imaging pipeline employing the optimal bodycentered cubic (BCC) sampling lattice. We provide perceptual evidence supporting the theoretical expectation that sampling and reconstruction on the BCC lattice offer superior imaging quality ove ..."
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Cited by 8 (4 self)
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This paper presents a user study of the visual quality of an imaging pipeline employing the optimal bodycentered cubic (BCC) sampling lattice. We provide perceptual evidence supporting the theoretical expectation that sampling and reconstruction on the BCC lattice offer superior imaging quality over the traditionally popular Cartesian cubic (CC) sampling lattice. We asked 12 participants to choose the better of two images: one image rendered from data sampled on the CC lattice and one image that is rendered from data sampled on the BCC lattice. We used both synthetic and CT volumetric data, and confirm that the theoretical advantages of BCC sampling carry over to the perceived quality of rendered images. Using 25 % to 35 % fewer samples, BCC sampled data result in images that exhibit comparable visual quality to their CC counterparts.