Integration over curves and surfaces defined by the closest point mapping

We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wish to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation - initially derived to integrate over manifolds of codimension one - to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.Comment: Revised the pape

CMBfit: Rapid WMAP likelihood calculations with normal parameters

We present a method for ultra-fast confrontation of the WMAP cosmic microwave background observations with theoretical models, implemented as a publicly available software package called CMBfit, useful for anyone wishing to measure cosmological parameters by combining WMAP with other observations. The method takes advantage of the underlying physics by transforming into a set of parameters where the WMAP likelihood surface is accurately fit by the exponential of a quartic or sextic polynomial. Building on previous physics based approximations by Hu et.al., Kosowsky et.al. and Chu et.al., it combines their speed with precision cosmology grade accuracy. A Fortran code for computing the WMAP likelihood for a given set of parameters is provided, pre-calibrated against CMBfast, accurate to Delta lnL ~ 0.05 over the entire 2sigma region of the parameter space for 6 parameter ``vanilla'' Lambda CDM models. We also provide 7-parameter fits including spatial curvature, gravitational waves and a running spectral index.Comment: 14 pages, 8 figures, References added, accepted for publication in Phys.Rev.D., a Fortran code can be downloaded from http://space.mit.edu/home/tegmark/cmbfit

A Bayesian Approach to Manifold Topology Reconstruction

In this paper, we investigate the problem of statistical reconstruction of piecewise linear manifold topology. Given a noisy, probably undersampled point cloud from a one- or two-manifold, the algorithm reconstructs an approximated most likely mesh in a Bayesian sense from which the sample might have been taken. We incorporate statistical priors on the object geometry to improve the reconstruction quality if additional knowledge about the class of original shapes is available. The priors can be formulated analytically or learned from example geometry with known manifold tessellation. The statistical objective function is approximated by a linear programming / integer programming problem, for which a globally optimal solution is found. We apply the algorithm to a set of 2D and 3D reconstruction examples, demon-strating that a statistics-based manifold reconstruction is feasible, and still yields plausible results in situations where sampling conditions are violated

Discrete schemes for Gaussian curvature and their convergence

In this paper, several discrete schemes for Gaussian curvature are surveyed. The convergence property of a modified discrete scheme for the Gaussian curvature is considered. Furthermore, a new discrete scheme for Gaussian curvature is resented. We prove that the new scheme converges at the regular vertex with valence not less than 5. By constructing a counterexample, we also show that it is impossible for building a discrete scheme for Gaussian curvature which converges over the regular vertex with valence 4. Finally, asymptotic errors of several discrete scheme for Gaussian curvature are compared

Curvature estimation for meshes via algebraic quadric fitting

We introduce the novel method for estimation of mean and Gaussian curvature and several related quantities for polygonal meshes. The algebraic quadric fitting curvature (AQFC) is based on local approximation of the mesh vertices and associated normals by a quadratic surface. The quadric is computed as an implicit surface, so it minimizes algebraic distances and normal deviations from the approximated point-normal neighbourhood of the processed vertex. Its mean and Gaussian curvature estimate is then obtained as the respective curvature of its orthogonal projection onto the fitted quadratic surface. Experimental results for both sampled parametric surfaces and arbitrary meshes are provided. The proposed method AQFC approaches the true curvatures of the reference smooth surfaces with increasing density of sampling, regardless of its regularity. It is resilient to irregular sampling of the mesh, compared to the contemporary curvature estimators. In the case of arbitrary meshes, obtained from scanning, AQFC provides robust curvature estimation.Comment: 14 page