71 research outputs found
Consistent Computation of First- and Second-Order Differential Quantities for Surface Meshes
Differential quantities, including normals, curvatures, principal directions,
and associated matrices, play a fundamental role in geometric processing and
physics-based modeling. Computing these differential quantities consistently on
surface meshes is important and challenging, and some existing methods often
produce inconsistent results and require ad hoc fixes. In this paper, we show
that the computation of the gradient and Hessian of a height function provides
the foundation for consistently computing the differential quantities. We
derive simple, explicit formulas for the transformations between the first- and
second-order differential quantities (i.e., normal vector and principal
curvature tensor) of a smooth surface and the first- and second-order
derivatives (i.e., gradient and Hessian) of its corresponding height function.
We then investigate a general, flexible numerical framework to estimate the
derivatives of the height function based on local polynomial fittings
formulated as weighted least squares approximations. We also propose an
iterative fitting scheme to improve accuracy. This framework generalizes
polynomial fitting and addresses some of its accuracy and stability issues, as
demonstrated by our theoretical analysis as well as experimental results.Comment: 12 pages, 12 figures, ACM Solid and Physical Modeling Symposium, June
200
Robust Discontinuity Indicators for High-Order Reconstruction of Piecewise Smooth Functions
In many applications, piecewise continuous functions are commonly
interpolated over meshes. However, accurate high-order manipulations of such
functions can be challenging due to potential spurious oscillations known as
the Gibbs phenomena. To address this challenge, we propose a novel approach,
Robust Discontinuity Indicators (RDI), which can efficiently and reliably
detect both C^{0} and C^{1} discontinuities for node-based and cell-averaged
values. We present a detailed analysis focusing on its derivation and the
dual-thresholding strategy. A key advantage of RDI is its ability to handle
potential inaccuracies associated with detecting discontinuities on non-uniform
meshes, thanks to its innovative discontinuity indicators. We also extend the
applicability of RDI to handle general surfaces with boundaries, features, and
ridge points, thereby enhancing its versatility and usefulness in various
scenarios. To demonstrate the robustness of RDI, we conduct a series of
experiments on non-uniform meshes and general surfaces, and compare its
performance with some alternative methods. By addressing the challenges posed
by the Gibbs phenomena and providing reliable detection of discontinuities, RDI
opens up possibilities for improved approximation and analysis of piecewise
continuous functions, such as in data remap.Comment: 37 pages, 37 figures, submitted to Computational and Applied
Mathematics (COAM
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