Global surface parametrization often requires the use of cuts or charts due to non-trivial topology. In recent years a focus has been on so-called seamless parametrizations, where the transition functions across the cuts are rigid transformations with a rotation about some multiple of 90 degrees. Of particular interest, e.g. for quadrilateral meshing, paneling, or texturing, are those instances where in addition the translational part of these transitions is integral (or more generally: quantized). We show that finding not even the optimal, but just an arbitrary valid quantization (one that does not imply parametric degeneracies), is a complex combinatorial problem. We present a novel method that allows us to solve it, i.e. to find valid as well as good quality quantizations. It is based on an original approach to quickly construct solutions to linear Diophantine equation systems, exploiting the specific geometric nature of the parametrization problem. We thereby largely outperform the state-of-the-art, sometimes by several orders of magnitude.

Quadrilateral remeshing approaches based on global parametrization enable many desirable mesh properties. Two of the most important ones are (1) high regularity due to explicit control over irregular vertices and (2) smooth distribution of distortion achieved by convex variational formulations. Apart from these strengths, state-of-the-art techniques suffer from limited reliability on real-world input data, i.e. the determined map might have degeneracies like (local) non-injectivities and consequently often cannot be used directly to generate a quadrilateral mesh. In this paper we propose a novel convex Mixed-Integer Quadratic Programming (MIQP) formulation which ensures by construction that the resulting map is within the class of so called Integer-Grid Maps that are guaranteed to imply a quad mesh. In order to overcome the NP-hardness of MIQP and to be able to remesh typical input geometries in acceptable time we propose two additional problem specific optimizations: a complexity reduction algorithm and singularity separating conditions. While the former decouples the dimension of the MIQP search space from the input complexity of the triangle mesh and thus is able to dramatically speed up the computation without inducing inaccuracies, the latter improves the continuous relaxation, which is crucial for the success of modern MIQP optimizers. Our experiments show that the reliability of the resulting algorithm does not only annihilate the main drawback of parametrization based quad-remeshing but moreover enables the global search for high-quality coarse quad layouts – a difficult task solely tackled by greedy methodologies before.

The most popular and actively researched class of quad remeshing techniques is
the family of *parametrization based quad meshing methods*. They all strive
to generate an *integer-grid map*, i.e. a parametrization of the input surface
into R^{2} such that the canonical grid of integer iso-lines forms a
quad mesh when mapped back onto the surface in R^{3}. An essential,
albeit broadly neglected aspect of these methods is the *quad extraction*
step, i.e. the materialization of an actual quad mesh from the mere “quad
texture”. Quad (mesh) extraction is often believed to be a trivial matter but
quite the opposite is true: Numerous special cases, ambiguities induced by
numerical inaccuracies and limited solver precision, as well as imperfections
in the maps produced by most methods (unless costly countermeasures are taken)
pose significant challenges to the quad extractor. We present a method to
sanitize a provided parametrization such that it becomes numerically
consistent even in a limited precision floating point representation. Based
on this we are able to provide a comprehensive and sound description of how to
perform quad extraction robustly and without the need for any complex
tolerance thresholds or disambiguation rules. On top of that we develop a
novel strategy to cope with common local fold-overs in the parametrization.
This allows our method, dubbed *QEx*, to generate all-quadrilateral meshes
where otherwise holes, non-quad polygons or no output at all would have been
produced. We thus enable the practical use of an entire class of maps that was
previously considered defective. Since state of the art quad meshing methods
spend a significant share of their run time solely to prevent local
fold-overs, using our method it is now possible to obtain quad meshes
significantly quicker than before. We also provide `libQEx`

, an open source
C++ reference implementation of our method and thus significantly lower the
bar to enter the field of quad meshing.

We present a theoretical framework and practical method for the automatic construction of simple, all-quadrilateral patch layouts on manifold surfaces. The resulting layouts are coarse, surface-embedded cell complexes well adapted to the geometric structure, hence they are ideally suited as domains and base complexes for surface parameterization, spline fitting, or subdivision surfaces and can be used to generate quad meshes with a high-level patch structure that are advantageous in many application scenarios. Our approach is based on the careful construction of the layout graph's combinatorial dual. In contrast to the primal this dual perspective provides direct control over the globally interdependent structural constraints inherent to quad layouts. The dual layout is built from curvature-guided, crossing loops on the surface. A novel method to construct these efficiently in a geometry- and structure-aware manner constitutes the core of our approach.

We introduce a fully automatic algorithm which optimizes the high-level structure of a given quadrilateral mesh to achieve a coarser quadrangular base complex. Such a topological optimization is highly desirable, since state-of-the-art quadrangulation techniques lead to meshes which have an appropriate singularity distribution and an anisotropic element alignment, but usually they are still far away from the high-level structure which is typical for carefully designed meshes manually created by specialists and used e.g. in animation or simulation. In this paper we show that the quality of the high-level structure is negatively affected by helical configurations within the quadrilateral mesh. Consequently we present an algorithm which detects helices and is able to remove most of them by applying a novel grid preserving simplification operator (GP-operator) which is guaranteed to maintain an all-quadrilateral mesh. Additionally it preserves the given singularity distribution and in particular does not introduce new singularities. For each helix we construct a directed graph in which cycles through the start vertex encode operations to remove the corresponding helix. Therefore a simple graph search algorithm can be performed iteratively to remove as many helices as possible and thus improve the high-level structure in a greedy fashion. We demonstrate the usefulness of our automatic structure optimization technique by showing several examples with varying complexity.

Proceedings of the 2009 SIGGRAPH Conference

We present a novel method for quadrangulating a given triangle mesh. After constructing an as smooth as possible symmetric cross field satisfying a sparse set of directional constraints (to capture the geometric structure of the surface), the mesh is cut open in order to enable a low distortion unfolding. Then a seamless globally smooth parametrization is computed whose iso-parameter lines follow the cross field directions. In contrast to previous methods, sparsely distributed directional constraints are sufficient to automatically determine the appropriate number, type and position of singularities in the quadrangulation. Both steps of the algorithm (cross field and parametrization) can be formulated as a mixed-integer problem which we solve very efficiently by an adaptive greedy solver. We show several complex examples where high quality quad meshes are generated in a fully automatic manner.

The Constrained Mixed-Integer Solver used in this project has been released under GPL and can be found on its projects page.

The aim of Reverse Engineering is to convert an unstructured representation of a geometric object, emerging e.g. from laser scanners, into a natural, structured representation in the spirit of CAD models, which is suitable for numerical computations. Therefore we present a user-controlled, as isometric as possible parameterization technique which is able to prescribe geometric features of the input and produces high-quality quadmeshes with low distortion. Starting with a coarse, user-prescribed layout this is achieved by using affine functions for the transition between non-orthogonal quadrangular charts of a global parameterization. The shape of each chart is optimized non-linearly for isometry of the underlying parameterization to produce meshes with low edge-length distortion. To provide full control over the meshing alignment the user can additionally tag an arbitrary subset of the layout edges which are guaranteed to be represented by enforcing them to lie on iso-lines of the parameterization but still allowing the global parameterization to relax in the direction of the iso-lines.