23,514 research outputs found
Representing and Understanding Non-Manifold Objects
Solid Modeling is a well-established field. The significance of the contributions of this field is visible in the availability of abundant commercial and free modeling tools for the applications of CAD, animation, visualization etc.
There are various approaches to modeling shapes. A common problem to all of them however, is the handling of non-manifold shapes. Manifold shapes are shapes with the property of topological ``smoothness'' at the local neighbourhood of every point. Objects that contain one or more points that lack this smoothness are all considered non-manifold. Non-manifold objects form a huge catagory of shapes. In the field of solid modeling, solutions typically limit the application domain to manifold shapes. Where the occurrence of non-manifold shapes is inevitable, they are often processed at a high cost. The lack of understanding on the nature of non-manifold shapes is the main cause of it. There is a tremendous gap between the well-established mathematical theories in topology and the materialization of such knowledge in the discrete combinatorial domain of computer science and engineering. The motivation of this research is to bridge this gap between the two.
We present a characterization of non-manifoldness in 3D simplicial shapes. Based on this characterization, we propose data structures to address the applicational needs for the representation of 3D simplicial complexes with mixed dimensions and non-manifold connectivities, which is an area that is greatly lacking in the literature. The availability of a suitable data structure makes the structural analysis of non-manifold shapes feasible. We address the problem of non-manifold shape understanding through a structural analysis that is based on decomposition
Faster ASV decomposition for orthogonal polyhedra using the Extreme Vertices Model (EVM)
The alternating sum of volumes (ASV) decomposition is a widely used
technique for converting a B-Rep into a CSG model. The obtained CSG
tree has convex primitives at its leaf nodes, while the contents of
its internal nodes alternate between the set union and difference
operators.
This work first shows that the obtained CSG tree T can also be
expressed as the regularized Exclusive-OR operation among all the
convex primitives at the leaf nodes of T, regardless the structure and
internal nodes of T. This is an important result in the case in which
EVM represented orthogonal polyhedra are used because in this model
the Exclusive-OR operation runs much faster than set union and
difference operations. Therefore this work applies this result to EVM
represented orthogonal polyhedra. It also presents experimental
results that corroborate the theoretical results and includes some
practical uses for the ASV decomposition of orthogonal polyhedra.Postprint (published version
On the optimality of shape and data representation in the spectral domain
A proof of the optimality of the eigenfunctions of the Laplace-Beltrami
operator (LBO) in representing smooth functions on surfaces is provided and
adapted to the field of applied shape and data analysis. It is based on the
Courant-Fischer min-max principle adapted to our case. % The theorem we present
supports the new trend in geometry processing of treating geometric structures
by using their projection onto the leading eigenfunctions of the decomposition
of the LBO. Utilisation of this result can be used for constructing numerically
efficient algorithms to process shapes in their spectrum. We review a couple of
applications as possible practical usage cases of the proposed optimality
criteria. % We refer to a scale invariant metric, which is also invariant to
bending of the manifold. This novel pseudo-metric allows constructing an LBO by
which a scale invariant eigenspace on the surface is defined. We demonstrate
the efficiency of an intermediate metric, defined as an interpolation between
the scale invariant and the regular one, in representing geometric structures
while capturing both coarse and fine details. Next, we review a numerical
acceleration technique for classical scaling, a member of a family of
flattening methods known as multidimensional scaling (MDS). There, the
optimality is exploited to efficiently approximate all geodesic distances
between pairs of points on a given surface, and thereby match and compare
between almost isometric surfaces. Finally, we revisit the classical principal
component analysis (PCA) definition by coupling its variational form with a
Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can
handle cases that go beyond the scope defined by the observation set that is
handled by regular PCA
Virtual Delamination Testing through Non-Linear Multi-Scale Computational Methods: Some Recent Progress
This paper deals with the parallel simulation of delamination problems at the
meso-scale by means of multi-scale methods, the aim being the Virtual
Delamination Testing of Composite parts. In the non-linear context, Domain
Decomposition Methods are mainly used as a solver for the tangent problem to be
solved at each iteration of a Newton-Raphson algorithm. In case of strongly
nonlinear and heterogeneous problems, this procedure may lead to severe
difficulties. The paper focuses on methods to circumvent these problems, which
can now be expressed using a relatively general framework, even though the
different ingredients of the strategy have emerged separately. We rely here on
the micro-macro framework proposed in (Ladev\`eze, Loiseau, and Dureisseix,
2001). The method proposed in this paper introduces three additional features:
(i) the adaptation of the macro-basis to situations where classical
homogenization does not provide a good preconditioner, (ii) the use of
non-linear relocalization to decrease the number of global problems to be
solved in the case of unevenly distributed non-linearities, (iii) the
adaptation of the approximation of the local Schur complement which governs the
convergence of the proposed iterative technique. Computations of delamination
and delamination-buckling interaction with contact on potentially large
delaminated areas are used to illustrate those aspects
A Statistical Toolbox For Mining And Modeling Spatial Data
Most data mining projects in spatial economics start with an evaluation of a set of attribute variables on a sample of spatial entities, looking for the existence and strength of spatial autocorrelation, based on the Moran’s and the Geary’s coefficients, the adequacy of which is rarely challenged, despite the fact that when reporting on their properties, many users seem likely to make mistakes and to foster confusion. My paper begins by a critical appraisal of the classical definition and rational of these indices. I argue that while intuitively founded, they are plagued by an inconsistency in their conception. Then, I propose a principled small change leading to corrected spatial autocorrelation coefficients, which strongly simplifies their relationship, and opens the way to an augmented toolbox of statistical methods of dimension reduction and data visualization, also useful for modeling purposes. A second section presents a formal framework, adapted from recent work in statistical learning, which gives theoretical support to our definition of corrected spatial autocorrelation coefficients. More specifically, the multivariate data mining methods presented here, are easily implementable on the existing (free) software, yield methods useful to exploit the proposed corrections in spatial data analysis practice, and, from a mathematical point of view, whose asymptotic behavior, already studied in a series of papers by Belkin & Niyogi, suggests that they own qualities of robustness and a limited sensitivity to the Modifiable Areal Unit Problem (MAUP), valuable in exploratory spatial data analysis
Principal Nested Spheres for Time Warped Functional Data Analysis
There are often two important types of variation in functional data: the
horizontal (or phase) variation and the vertical (or amplitude) variation.
These two types of variation have been appropriately separated and modeled
through a domain warping method (or curve registration) based on the Fisher Rao
metric. This paper focuses on the analysis of the horizontal variation,
captured by the domain warping functions. The square-root velocity function
representation transforms the manifold of the warping functions to a Hilbert
sphere. Motivated by recent results on manifold analogs of principal component
analysis, we propose to analyze the horizontal variation via a Principal Nested
Spheres approach. Compared with earlier approaches, such as approximating
tangent plane principal component analysis, this is seen to be the most
efficient and interpretable approach to decompose the horizontal variation in
some examples
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