On Discrete Killing Vector Fields and Patterns on Surfaces

Ben‐Chen, Mirela; Butscher, Adrian; Solomon, Justin; Guibas, Leonidas J.

doi:10.1111/j.1467-8659.2010.01779.x

Cited by 64 publications

(92 citation statements)

References 27 publications

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“…To compute infinitesimal intrinsic symmetries, Ben Chen et al [3] propose a method to compute approximate Killing vector fields on triangle meshes, which are tangent vector fields on the surface that generate isometric transformations. There are two methods that allow for the computation of partial infinitesimal symmetries.…”

Section: Related Workmentioning

confidence: 99%

“…There are two methods that allow for the computation of partial infinitesimal symmetries. Solomon et al [26] partition the shape into intrinsically symmetric building blocks and can be thought of as an extension of the method by Ben Chen et al [3] to the partial case. While this method allows to "untangle" the global information into parts, it does not allow the detection of symmetries on a specific scale.…”

Section: Related Workmentioning

confidence: 99%

“…In contrast to the method of Ben-Chen et al [3], we constraint the measured stretching to the shape part U instead of the entire shape.…”

Section: Detecting R-symmetriesmentioning

confidence: 99%

“…To find the r-slippage directions, we aim to minimize the energy given in Equation (3). Consider the identity mapping f 0 (x) = x, x ∈ U.…”

Section: Detecting R-symmetriesmentioning

confidence: 99%

“…Based on the observation that a map of a single point pair and its tangent space is sufficient to recover an isometry, they propose a lowdimensional representation to efficiently compute a mapping f : V → M that minimizes the stretch given in Equation (3). Here, V is grown to the largest possible size that allows a mapping with metric stretching at most .…”

Section: Discrete R-symmetries By Isometric Region Growingmentioning

confidence: 99%

See 4 more Smart Citations

Characterization of Partial Intrinsic Symmetries

Shehu

Brunton

Wuhrer

et al. 2015

Computer Vision - ECCV 2014 Workshops

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Abstract. We present a mathematical framework and algorithm for characterizing and extracting partial intrinsic symmetries of surfaces, which is a fundamental building block for many modern geometry processing algorithms. Our goal is to compute all "significant" symmetry information of the shape, which we define as r-symmetries, i.e., we report all isometric self-maps within subsets of the shape that contain at least an intrinsic circle or radius r. By specifying r, the user has direct control over the scale at which symmetry should be detected. Unlike previous techniques, we do not rely on feature points, voting or probabilistic schemes. Rather than that, we bound computational efforts by splitting our algorithm into two phases. The first detects infinitesimal r-symmetries directly using a local differential analysis, and the second performs direct matching for the remaining discrete symmetries. We show that our algorithm can successfully characterize and extract intrinsic symmetries from a number of example shapes.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

“…In contrast to the method of Ben-Chen et al [3], we constraint the measured stretching to the shape part U instead of the entire shape.…”

Section: Detecting R-symmetriesmentioning

confidence: 99%

“…To find the r-slippage directions, we aim to minimize the energy given in Equation (3). Consider the identity mapping f 0 (x) = x, x ∈ U.…”

Section: Detecting R-symmetriesmentioning

confidence: 99%

Section: Discrete R-symmetries By Isometric Region Growingmentioning

confidence: 99%

See 3 more Smart Citations

Characterization of Partial Intrinsic Symmetries

Shehu

Brunton

Wuhrer

et al. 2015

Computer Vision - ECCV 2014 Workshops

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Generalized Metric Energies for Continuous Shape Deformation

Esturo

Rössl

Theisel

2014

Mathematical Methods for Curves and Surfaces

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Extending Discrete Exterior Calculus to a Fractional Derivative

Crum

Levine

Gillette

2019

Computer-Aided Design

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Fractional partial differential equations (FDEs) are used to describe phenomena that involve a "non-local" or "longrange" interaction of some kind. Accurate and practical numerical approximation of their solutions is challenging due to the dense matrices arising from standard discretization procedures. In this paper, we begin to extend the well-established computational toolkit of Discrete Exterior Calculus (DEC) to the fractional setting, focusing on proper discretization of the fractional derivative. We define a Caputo-like fractional discrete derivative, in terms of the standard discrete exterior derivative operator from DEC, weighted by a measure of distance between p-simplices in a simplicial complex. We discuss key theoretical properties of the fractional discrete derivative and compare it to the continuous fractional derivative via a series of numerical experiments.

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On Discrete Killing Vector Fields and Patterns on Surfaces

Abstract: Abstract

Cited by 64 publications

References 27 publications

Characterization of Partial Intrinsic Symmetries

Characterization of Partial Intrinsic Symmetries

Generalized Metric Energies for Continuous Shape Deformation

Extending Discrete Exterior Calculus to a Fractional Derivative

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