17,347 research outputs found
Least Squares Subdivision Surfaces
International audienceThe usual approach to design subdivision schemes for curves and surfaces basically consists in combining proper rules for regular configurations, with some specific heuristics to handle extraordinary vertices. In this paper, we introduce an alternative approach, called Least Squares Subdivision Surfaces (LS^3), where the key idea is to iteratively project each vertex onto a local approximation of the current polygonal mesh. While the resulting procedure have the same complexity as simpler subdivision schemes, our method offers much higher visual quality, especially in the vicinity of extraordinary vertices. Moreover, we show it can be easily generalized to support boundaries and creases. The fitting procedure allows for a local control of the surface from the normals, making LS^3 very well suited for interactive freeform modeling applications. We demonstrate our approach on diadic triangular and quadrangular refinement schemes, though it can be applied to any splitting strategies
Reversing subdivision rules: local linear conditions and observations on inner products
AbstractIn a previous work (Samavati and Bartels, Comput. Graphics Forum 18 (1998) 97–119) we investigated how to reverse subdivision rules using global least-squares fitting. This led to multiresolution structures that could be viewed as semiorthogonal wavelet systems whose inner product was that for finite-dimensional Cartesian vector space. We produced simple and sparse reconstruction filters, but had to appeal to matrix factorization to obtain an efficient, exact decomposition. We also made some observations on how the inner product that defines the semiorthogonality influences the sparsity of the reconstruction filters. In this work we carry the investigation further by studying biorthogonal systems based upon subdivision rules and local least-squares fitting problems that reverse the subdivision. We are able to produce multiresolution structures for some common univariate subdivision rules that have both sparse reconstruction and decomposition filters. Three will be presented here – for quadratic and cubic B-spline subdivision and for the four-point interpolatory subdivision of Dyn et al. We observe that each biorthogonal system we produce can be interpreted as a semiorthogonal system with an inner product induced on the multiresolution that is quite different from that normally used. Some examples of the use of this approach on images, curves, and surfaces are given
Compression for Smooth Shape Analysis
Most 3D shape analysis methods use triangular meshes to discretize both the
shape and functions on it as piecewise linear functions. With this
representation, shape analysis requires fine meshes to represent smooth shapes
and geometric operators like normals, curvatures, or Laplace-Beltrami
eigenfunctions at large computational and memory costs.
We avoid this bottleneck with a compression technique that represents a
smooth shape as subdivision surfaces and exploits the subdivision scheme to
parametrize smooth functions on that shape with a few control parameters. This
compression does not affect the accuracy of the Laplace-Beltrami operator and
its eigenfunctions and allow us to compute shape descriptors and shape
matchings at an accuracy comparable to triangular meshes but a fraction of the
computational cost.
Our framework can also compress surfaces represented by point clouds to do
shape analysis of 3D scanning data
Generalized Debye Sources Based EFIE Solver on Subdivision Surfaces
The electric field integral equation is a well known workhorse for obtaining
fields scattered by a perfect electric conducting (PEC) object. As a result,
the nuances and challenges of solving this equation have been examined for a
while. Two recent papers motivate the effort presented in this paper. Unlike
traditional work that uses equivalent currents defined on surfaces, recent
research proposes a technique that results in well conditioned systems by
employing generalized Debye sources (GDS) as unknowns. In a complementary
effort, some of us developed a method that exploits the same representation for
both the geometry (subdivision surface representations) and functions defined
on the geometry, also known as isogeometric analysis (IGA). The challenge in
generalizing GDS method to a discretized geometry is the complexity of the
intermediate operators. However, thanks to our earlier work on subdivision
surfaces, the additional smoothness of geometric representation permits
discretizing these intermediate operations. In this paper, we employ both ideas
to present a well conditioned GDS-EFIE. Here, the intermediate surface
Laplacian is well discretized by using subdivision basis. Likewise, using
subdivision basis to represent the sources, results in an efficient and
accurate IGA framework. Numerous results are presented to demonstrate the
efficacy of the approach
Subdivision surface fitting to a dense mesh using ridges and umbilics
Fitting a sparse surface to approximate vast dense data is of interest for many applications: reverse engineering, recognition and compression, etc. The present work provides an approach to fit a Loop subdivision surface to a dense triangular mesh of arbitrary topology, whilst preserving and aligning the original features. The natural ridge-joined connectivity of umbilics and ridge-crossings is used as the connectivity of the control mesh for subdivision, so that the edges follow salient features on the surface. Furthermore, the chosen features and connectivity characterise the overall shape of the original mesh, since ridges capture extreme principal curvatures and ridges start and end at umbilics. A metric of Hausdorff distance including curvature vectors is proposed and implemented in a distance transform algorithm to construct the connectivity. Ridge-colour matching is introduced as a criterion for edge flipping to improve feature alignment. Several examples are provided to demonstrate the feature-preserving capability of the proposed approach
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