Data-dependent fairing of subdivision surfaces

In this paper we present a new algorithm for solving the data dependent fairing problem for subdivision surfaces, using Catmull-Clark surfaces as an example. Earlier approaches to subdivision surface fairing encountered problems with singularities in the parametrization of the surface. We address these issues through the use of the characteristic map parametrization, leading to well defined membrane and bending energies even at irregular vertices. Combining this approach with ideas from data-dependent energy operators we are able to express the associated nonlinear stiffness matrices for Catmull-Clark surfaces as linear combinations of precomputed energy matrices. This machinery also provides exact, inexpensive gradients and Hessians of the new energy operators. With these the nonlinear minimization problem can be solved in a stable and efficient way using Steihaug's Newton/CG trust-region method. We compare properties of linear and nonlinear methods through a number of examples and report on the performance of the algorithm

Approximation of Surfaces by Normal Meshes

This thesis introduces a novel geometry processing pipeline based on unconstrained spherical parameterization and normal remeshing. We claim three main contributions: First we show how to increase the stability of Normal Mesh construction, while speeding it up by decomposing the process into two stages: parameterization and remeshing. We show that the remeshing step can be seen as resampling under a small perturbation of the given parameterization. Based on this observation we describe a novel algorithm for efficient and stable (interpolatory) normal mesh construction via parameterization perturbation. Our second contribution is the introduction of Variational Normal Meshes. We describe a novel algorithm for encoding these meshes, and use our implementation to argue that variational normal meshes have a higher approximation quality than interpolating normal meshes, as expected. In particular we demonstrate that interpolating normal meshes have about 60 percent higher Hausdorff approximation error for the same number of vertices than our novel variational normal meshes. We also show that variational normal meshes have less aliasing artifacts than interpolatory normal meshes. The third contribution is on creating parameterizations for unstructured genus zero meshes. Previous approaches could only avoid collapses by introducing artificial constraints or continuous reprojections, which are avoided by our method. The key idea is to define upper bound energies that are still good approximations. We achieve this by dividing classical planar triangle energies by the minimum distance to the sphere center. We prove that these simple modifaction provides the desired upper bounds and are good approximations in the finite element sense. We have implemented all algorithms and provide example results and statistical data supporting our theoretical observations

Fast Generation of Randomized Low Discrepancy Point Sets

We introduce two novel techniques for speeding up the generation of digital $(t,s)$-sequences. Based on these results a new algorithm for the construction of Owen's randomly permuted $(t,s)$-sequences is developed and analyzed. An implementation of the new techniques is available at http://www.cs.caltech.edu/~ilja/libseq/index.htm

Fast Generation of Randomized Low-Discrepancy Point Sets

We introduce two novel techniques for speeding up the generation of digital (t,s)-sequences. Based on these results a new algorithm for the construction of Owen's randomly permuted (t,s)-sequences is developed and analyzed. An implementation is available at http://www.mcqmc.org/Software.html

Variational normal meshes

Hierarchical representations of surfaces have many advantages for digital geometry processing applications. Normal meshes are particularly attractive since their level-to-level displacements are in the local normal direction only. Consequently, they only require scalar coefficients to specify. In this article, we propose a novel method to approximate a given mesh with a normal mesh. Instead of building an associated parameterization on the fly, we assume a globally smooth parameterization at the beginning and cast the problem as one of perturbing this parameterization. Controlling the magnitude of this perturbation gives us explicit control over the range between fully constrained (only scalar coefficients) and unconstrained (3-vector coefficients) approximations. With the unconstrained problem giving the lowest approximation error, we can thus characterize the error cost of normal meshes as a function of the number of nonnormal offsets---we find a significant gain for little (error) cost. Because the normal mesh construction creates a geometry driven approximation, we can replace the difficult geometric distance minimization problem with a much simpler least squares problem. This variational approach reduces magnitude and structure (aliasing) of the error further. Our method separates the parameterization construction into an initial setup followed only by subsequent perturbations, giving us an algorithm which is far simpler to implement, more robust, and significantly faster

Variational Normal Meshes

this article, we propose a novel method to approximate a given mesh with a normal mesh. Instead of building an associated parameterization on the fly, we assume a globally smooth parameterization at the beginning and cast the problem as one of perturbing this parameterization. Controlling the magnitude of this perturbation gives us explicit control over the range between fully constrained (only scalar coefficients) and unconstrained (3-vector coefficients) approximations. With the unconstrained problem giving the lowest approximation error, we can thus characterize the error cost of normal meshes as a function of the number of nonnormal offsets---we find a significant gain for little (error) cost. Because the normal mesh construction creates a geometry driven approximation, we can replace the difficult geometric distance minimization problem with a much simpler least squares problem. This variational approach reduces magnitude and structure (aliasing) of the error further. Our method separates the parameterization construction into an initial setup followed only by subsequent perturbations, giving us an algorithm which is far simpler to implement, more robust, and significantly faste

Data-dependent fairing of subdivision surfaces

In this paper we present a new algorithm for solving the data dependent fairing problem for subdivision surfaces, using Catmull-Clark surfaces as an example. Earlier approaches to subdivision surface fairing encountered problems with singularities in the parametrization of the surface. We address these issues through the use of the characteristic map parametrization, leading to well defined membrane and bending energies even at irregular vertices. Combining this approach with ideas from data-dependent energy operators we are able to express the associated nonlinear stiffness matrices for Catmull-Clark surfaces as linear combinations of precomputed energy matrices. This machinery also provides exact, inexpensive gradients and Hessians of the new energy operators. With these the nonlinear minimization problem can be solved in a stable and efficient way using Steihaug's Newton/CG trust-region method. We compare properties of linear and nonlinear methods through a number of examples and report on the performance of the algorithm

Variational normal meshes

Hierarchical representations of surfaces have many advantages for digital geometry processing applications. Normal meshes are particularly attractive since their level-to-level displacements are in the local normal direction only. Consequently, they only require scalar coefficients to specify. In this article, we propose a novel method to approximate a given mesh with a normal mesh. Instead of building an associated parameterization on the fly, we assume a globally smooth parameterization at the beginning and cast the problem as one of perturbing this parameterization. Controlling the magnitude of this perturbation gives us explicit control over the range between fully constrained (only scalar coefficients) and unconstrained (3-vector coefficients) approximations. With the unconstrained problem giving the lowest approximation error, we can thus characterize the error cost of normal meshes as a function of the number of nonnormal offsets---we find a significant gain for little (error) cost. Because the normal mesh construction creates a geometry driven approximation, we can replace the difficult geometric distance minimization problem with a much simpler least squares problem. This variational approach reduces magnitude and structure (aliasing) of the error further. Our method separates the parameterization construction into an initial setup followed only by subsequent perturbations, giving us an algorithm which is far simpler to implement, more robust, and significantly faster

Caltech

We present a simple technique for easing the computation of spherical parameterizations by a simple modification of traditional planar parameterization methods: our spherical energies differ from the usual planar quadratic energies only by multiplication by a simpl