Integrating Surface Normal Vectors Using Fast Marching Method

Abstract. Integration of surface normal vectors is a vital component in many shape reconstruction algorithms that require integrating surface normals to produce their final outputs, the depth values. In this paper, we introduce a fast and efficient method for computing the depth val-ues from surface normal vectors. The method is based on solving the Eikonal equation using Fast Marching Method. We introduce two ideas. First, while it is not possible to solve for the depths Z directly using Fast Marching Method, we solve the Eikonal equation for a function W of the form W = Z+λf. With appropriately chosen values for λ, we can ensure that the Eikonal equation for W can be solved using Fast March-ing Method. Second, we solve for W in two stages with two different λ values, first in a small neighborhood of the given initial point with large λ, and then for the rest of the domain with a smaller λ. This step is needed because of the finite machine precision and rounding-off errors. The proposed method is very easy to implement, and we demonstrate experimentally that, with insignificant loss in precision, our method is considerably faster than the usual optimization method that uses conju-gate gradient to minimize an error function.

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

A discrete contact model for crowd motion

The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: We first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people; The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e. velocities which do not violate the non-overlapping constraint). We describe here the underlying mathematical framework, and we explain how recent results by J.F. Edmond and L. Thibault on the sweeping process by uniformly prox-regular sets can be adapted to handle this situation in terms of well-posedness. We propose a numerical scheme for this contact dynamics model, based on a prediction-correction algorithm. Numerical illustrations are finally presented and discussed.Comment: 22 page

Physics Of Eclipsing Binaries. II. Towards the Increased Model Fidelity

The precision of photometric and spectroscopic observations has been systematically improved in the last decade, mostly thanks to space-borne photometric missions and ground-based spectrographs dedicated to finding exoplanets. The field of eclipsing binary stars strongly benefited from this development. Eclipsing binaries serve as critical tools for determining fundamental stellar properties (masses, radii, temperatures and luminosities), yet the models are not capable of reproducing observed data well either because of the missing physics or because of insufficient precision. This led to a predicament where radiative and dynamical effects, insofar buried in noise, started showing up routinely in the data, but were not accounted for in the models. PHOEBE (PHysics Of Eclipsing BinariEs; http://phoebe-project.org) is an open source modeling code for computing theoretical light and radial velocity curves that addresses both problems by incorporating missing physics and by increasing the computational fidelity. In particular, we discuss triangulation as a superior surface discretization algorithm, meshing of rotating single stars, light time travel effect, advanced phase computation, volume conservation in eccentric orbits, and improved computation of local intensity across the stellar surfaces that includes photon-weighted mode, enhanced limb darkening treatment, better reflection treatment and Doppler boosting. Here we present the concepts on which PHOEBE is built on and proofs of concept that demonstrate the increased model fidelity.Comment: 60 pages, 15 figures, published in ApJS; accompanied by the release of PHOEBE 2.0 on http://phoebe-project.or