Many tasks in geometry processing and physical simulation benefit from multiresolution hierarchies. One important characteristic across a variety of applications is that coarser layers strictly encage finer layers, nesting one another. Existing techniques such as surface mesh decimation, voxelization, or contouring distance level sets do not provide sufficient control over the quality of the output surfaces while maintaining strict nesting. We propose a solution that enables use of application-specific decimation and quality metrics. The method constructs each next-coarsest level of the hierarchy, using a sequence of decimation, flow, and contact-aware optimization steps. From coarse to fine, each layer then fully encages the next while retaining a snug fit. The method is applicable to a wide variety of shapes of complex geometry and topology. We demonstrate the effectiveness of our nested cages not only for multigrid solvers, but also for conservative collision detection, domain discretization for elastic simulation, and cage-based geometric modeling.
Input triangle mesh cMCF flow Intrinsic reverse flow Output tetrahedral mesh matching input Figure 1: The triangle mesh of the Hand forms a closed surface, but contains nearly 2000 intersecting triangle pairs. Our method flows the surface according to conformalized mean-curvature flow (cMCF) until all self-intersections are removed. Then we reverse the flow so that shape intrinsics are restored but self-intersections are avoided. Finally we can tet-mesh inside this surface and map the mesh so that it matches the original surface. We may then solve PDEs, such as this biharmonic function. AbstractDecades of research have culminated in a robust geometry processing pipeline for surfaces. Most steps in this pipeline, like deformation, smoothing, subdivision and decimation, may create self-intersections. Volumetric processing of solid shapes then becomes difficult, because obtaining a correct volumetric discretization is impossible: existing tet-meshing methods require watertight input. We propose an algorithm that produces a tetrahedral mesh that overlaps itself consistently with the self-intersections in the input surface. This enables volumetric processing on self-intersecting models. We leverage conformalized mean-curvature flow, which removes self-intersections, and define an intrinsically similar reverse flow, which prevents them. We tetrahedralize the resulting surface and map the mesh inside the original surface. We demonstrate the effectiveness of our method with applications to automatic skinning weight computation, physically based simulation and geodesic distance computation.
a b s t r a c tThe main goal of the paper is to introduce methods that compute Bézier curves faster than Casteljau's method does. These methods are based on the spectral factorization of an n  n Bernstein matrix, B e n ðsÞ ¼ P n G n ðsÞP À1 n , where P n is the n  n lower triangular Pascal matrix. To that end, we first calculate the exact optimum positive value t in order to transform P n into a scaled Toeplitz matrix (how to do so is a problem that was partially solved by Wang and Zhou (2006) [6]). Then, fast Pascal matrix-vector multiplications are combined with polynomial evaluations to compute Bézier curves. Nevertheless, when n increases, we need more precise Pascal matrix-vector multiplications to achieve stability in the numerical results. We see here that a Pascal matrix-vector product, combined with a polynomial evaluation and some affine transforms of the vectors of coordinates of the control points, will yield a method that can be used to efficiently compute a Bézier curve of degree n, n 6 60.
We propose new quasi-interpolators for the continuous reconstruction of sampled images, combining a narrowly supported piecewise-polynomial kernel and an efficient digital filter. In other words, our quasi-interpolators fit within the generalized sampling framework and are straightforward to use. We go against standard practice and optimize for approximation quality over the entire Nyquist range, rather than focusing exclusively on the asymptotic behavior as the sample spacing goes to zero. In contrast to previous work, we jointly optimize with respect to all degrees of freedom available in both the kernel and the digital filter. We consider linear, quadratic, and cubic schemes, offering different tradeoffs between quality and computational cost. Experiments with compounded rotations and translations over a range of input images confirm that, due to the additional degrees of freedom and the more realistic objective function, our new quasi-interpolators perform better than the state of the art, at a similar computational cost.
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