We present a new globally smooth parameterization method for triangulated surfaces of arbitrary topology. Given two orthogonal piecewise linear vector elds dened over the input mesh (typically the estimated principal curvature directions), our method computes two piecewise linear periodic functions, aligned with the input vector elds, by minimizing an objective function. The bivariate function they dene is a smooth parameterization almost everywhere on the surface, except in the vicinity of singular vertices, edges and triangles, where the derivatives of the parameterization vanish. We extract a quadrilateral chart layout from the parameterization function and propose an automatic procedure to detect the singularities, and x them by splitting and re-parameterizing the containing charts. Our method can construct both quasi-conformal (angle preserving) and quasi-isometric (angle and area preserving) parameterizations. The more restrictive class of quasi-isometric parameterizations is constructed at the expense of introducing more singularities. The constructed parameterizations can be used for a variety of geometry processing applications. Since we can align the parameterization with the principal curvature directions, our result is particularly suitable for surface tting and remeshing.

Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction elds (unit tangent vector elds) dened over a surface. For instance, these direction elds are used in texture synthesis, in geometry processing or in non-photorealistic rendering to distribute and orient elements on the surface. Such direction elds can be designed in fundamentally dierent ways, according to the symmetry requested: inverting a direction or swapping two directions may be allowed or not.Despite the advances realized in the last few years in the domain of geometry processing, a unied formalism is still lacking for the mathematical object that characterizes these generalized direction elds. As a consequence, existing direction eld design algorithms are limited to use non-optimum local relaxation procedures.In this paper, we formalize N-symmetry direction elds, a generalization of classical direction elds. We give a new denition of their singularities to explain how they relate with the topology of the surface. Namely, we provide an accessible demonstration of the Poincaré-Hopf theorem in the case of N-symmetry direction elds on 2-manifolds. Based on this theorem, we explain how to control the topology of N-symmetry direction elds on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly ecient algorithm to design a smooth eld interpolating user dened singularities and directions.

A Texture Atlas is an efficient color representation for 3D Paint Systems. The model to be textured is decomposed into charts homeomorphic to discs, each chart is parameterized, and the unfolded charts are packed in texture space. Existing texture atlas methods for triangulated surfaces suffer from several limitations, requiring them to generate a large number of small charts with simple borders. The discontinuities between the charts cause artifacts, and make it difficult to paint large areas with regular patterns.In this paper, our main contribution is a new quasi-conformal parameterization method, based on a least-squares approximation of the Cauchy-Riemann equations. The so-defined objective function minimizes angle deformations, and we prove the following properties: the minimum is unique, independent of a similarity in texture space, independent of the resolution of the mesh and cannot generate triangle flips. The function is numerically well behaved and can therefore be very efficiently minimized. Our approach is robust, and can parameterize large charts with complex borders.We also introduce segmentation methods to decompose the model into charts with natural shapes, and a new packing algorithm to gather them in texture space. We demonstrate our approach applied to paint both scanned and modeled data sets.The remainder of this section presents the existing methods for these three steps, and their limitations with respect to the requirements mentioned above. We then introduce a new texture atlas generation method, meeting these requirements by creating charts with natural shapes, thus reducing texture artifacts.

Many algorithms in texture synthesis, nonphotorealistic rendering (hatching), or remeshing require to define the orientation of some features (texture, hatches, or edges) at each point of a surface. In early works, tangent vector (or tensor) fields were used to define the orientation of these features. Extrapolating and smoothing such fields is usually performed by minimizing an energy composed of a smoothness term and of a data fitting term. More recently, dedicated structures (N -RoSy and N -symmetry direction fields ) were introduced in order to unify the manipulation of these fields, and provide control over the field's topology (singularities). On the one hand, controlling the topology makes it possible to have few singularities, even in the presence of high frequencies (fine details) in the surface geometry. On the other hand, the user has to explicitly specify all singularities, which can be a tedious task. It would be better to let them emerge naturally from the direction extrapolation and smoothing.This article introduces an intermediate representation that still allows the intuitive design operations such as smoothing and directional constraints, but restates the objective function in a way that avoids the singularities yielded by smaller geometric details. The resulting design tool is intuitive, simple, and allows to create fields with simple topology, even in the presence of high geometric frequencies. The generated field can be used to steer global parameterization methods (e.g., QuadCover).