Provably good moving least squares

Kolluri, Ravikrishna

doi:10.1145/1198555.1198652

Cited by 67 publications

(72 citation statements)

References 32 publications

(19 reference statements)

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“…Moreover, the non-uniform distribution of the neighbours of m i within L σ (N i K ) ∈ R 2 often leads to an ill-conditioned matrix of the system (57). This fact has been recognised earlier in algorithms developed in the context of surface reconstruction from 3D laser scans where the surface points are acquired with experimental errors (see, for example, [34][35][36][37]). In the present case, the measurement errors are replaced by numerical errors due to integration of the vertex trajectories and, more importantly, possible errors introduced during a mesh refinement process (cf.…”

Section: Numerical Estimation Of Local Manifold Properties From Discrmentioning

confidence: 88%

An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

Branicki

Wiggins

2009

Physica D: Nonlinear Phenomena

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a b s t r a c tWe present a computational method for determining the geometry of a class of three-dimensional invariant manifolds in non-autonomous (aperiodically time-dependent) dynamical systems. The presented approach can be also applied to analyse the geometry of 3D invariant manifolds in threedimensional, time-dependent fluid flows. The invariance property of such manifolds requires that, at any fixed time, they are given by surfaces in R 3 . We focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in R 3 . The presented technique can be employed, in particular, to compute codimension one (invariant) stable and unstable manifolds of hyperbolic trajectories in 3D non-autonomous dynamical systems which are crucial in the Lagrangian transport analysis. The same approach can also be used to determine evolution of an orientable 'material surface' in a fluid flow. These developments represent the first step towards a non-trivial 3D extension of the so-called lobe dynamics -a geometric, invariant-manifold-based framework which has been very successful in the analysis of Lagrangian transport in unsteady, two-dimensional fluid flows. In the developed algorithm, the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise C 2 interpolating functions. The method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution. A variant of the advancing front technique is used for remeshing, whenever necessary. Such an approach allows for computationally efficient determination of highly convoluted, evolving geometry of codimension one invariant manifolds in unsteady threedimensional flows. We show that the developed method is capable of providing detailed information on the evolving Lagrangian flow structure in three dimensions over long periods of time, which is crucial for a meaningful 3D transport analysis.

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Section: Numerical Estimation Of Local Manifold Properties From Discrmentioning

confidence: 88%

An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

Branicki

Wiggins

2009

Physica D: Nonlinear Phenomena

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“…Most surface reconstruction methods roughly fall into two major categories: implicit surface methods [26,27] and Delaunay-based methods [28,29]. The most common approach to surface reconstruction is based on the Delaunay triangulation: the underlying idea is that, when the sampling is noise-free and dense enough, points close on the surface should also be close in space.…”

Section: The Related Workmentioning

confidence: 99%

Visualization of cracks by using the local Voronoi decompositions and distributed software

Pacevič

Kačeniauskas

Markauskas

2015

Advances in Engineering Software

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“…Moreover, it needs to adjust enveloping surfaces in the approximation mode iteratively because some parts of the surfaces are behind the original model surface. The techniques in [2], [3] were applied to point sets based on exponential weighting functions with no singularities in [12].…”

Section: Related Workmentioning

confidence: 99%

Analytic Solutions of Integral Moving Least Squares for Polygon Soups

Park¹,

Lee²,

Kim³

2012

IEEE Trans. Visual. Comput. Graphics

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This paper presents analytic solutions to the integral moving least squares (MLS) equations originally proposed by Shen et al. by choosing another specific weighting function that renders the numerator in the MLS equation unitless. In addition, we analyze the original method to show that their approximation surfaces (i.e., enveloping surfaces with nonzero values in the weighting function) often form zero isosurfaces near concavities behind the triangle-soup models. This paper also presents error terms for the integral MLS formulations against signed distance fields. Based on our analytic solutions, we show that our method provides both interpolation and approximation surfaces faster and more efficiently. Because our method computes solutions for integral MLS equations directly, it does not rely on numerical steps that might have numerical-accuracy issues. In particular, unlike the original method that deals with incorrect approximation surfaces by iteratively adjusting parameters, this paper proposes faster and more efficient approximations to surfaces without needing iterative routines. We also present computational efficiency comparisons, in which our method is 15-fold faster in computing integrations, even with conservative assumptions. Finally, we show that the surface normal vectors on the implicit surfaces formed by our analytic solutions are identical to the angle-weighted pseudonormal vectors.

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Provably good moving least squares

Cited by 67 publications

References 32 publications

An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

Visualization of cracks by using the local Voronoi decompositions and distributed software

Analytic Solutions of Integral Moving Least Squares for Polygon Soups

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