mirror of https://github.com/CGAL/cgal
Updated SMP's doc to include Orbifold-Tutte Embeddings
This commit is contained in:
Mael Rouxel-Labbé
parent
521f85bca9
commit
3f87e9d4fb
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@ -2761,6 +2761,16 @@ pages = "207--221"
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organization={IEEE}
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}
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@article{aigerman2015orbifold,
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title={Orbifold Tutte embeddings},
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author={Aigerman, Noam and Lipman, Yaron},
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journal={ACM Transactions on Graphics},
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volume={34},
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number={6},
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pages={190},
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year={2015}
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}
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% ----------------------------------------------------------------------------
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% END OF BIBFILE
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% ----------------------------------------------------------------------------
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@ -20,6 +20,7 @@ Construction and destruction are undefined.
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\cgalHasModel `CGAL::Surface_mesh_parameterization::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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\cgalHasModel `CGAL::Surface_mesh_parameterization::LSCM_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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\cgalHasModel `CGAL::Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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\cgalHasModel `CGAL::Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3<SeamMesh, SparseLinearAlgebraTraits_d>`
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\cgalHasModel `CGAL::Surface_mesh_parameterization::Circular_border_parameterizer_3<TriangleMesh>`
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\cgalHasModel `CGAL::Surface_mesh_parameterization::Square_border_parameterizer_3<TriangleMesh>`
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\cgalHasModel `CGAL::Surface_mesh_parameterization::Two_vertices_parameterizer_3<TriangleMesh>`
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@ -104,6 +104,9 @@ The package performs the next checks:
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/// \defgroup PkgSurfaceParameterizationMainFunction Main Function
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/// \ingroup PkgSurfaceParameterization
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/// \defgroup PkgSurfaceParameterizationEnums Enums
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/// \ingroup PkgSurfaceParameterization
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/// \defgroup PkgSurfaceParameterizationConcepts Concepts
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/// \ingroup PkgSurfaceParameterization
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@ -272,7 +272,7 @@ Least squares conformal maps.
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`Surface_mesh_parameterization::ARAP_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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An as rigid as possible parameterization was introduced by Liu et al. \cgalCite{liu2008local}.
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shape-preserving method based on an iterative energy minimization process.
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It is a shape-preserving method based on an iterative energy minimization process.
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Each step alternates a local optimization technique to find the best local mapping
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and a global stitching technique equivalent to the resolution of a linear system
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to guarantee that the parameterized mesh is a triangulation.
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@ -300,6 +300,27 @@ in the current version of this package.
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</UL>
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\subsection Surface_mesh_parameterizationBorderless Borderless Parameterizations
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\subsubsection Surface_mesh_parameterizationOrbi Orbifold Tutte Embedding
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`Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3<SeamMesh, SparseLinearAlgebraTraits_d>`
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Orbifold-Tutte Planar Embedding was introduced by Aigerman and Lipman \cgalCite{aigerman2015orbifold}
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and is a generalization of Tutte’s embedding to other topologies, and in particular
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spheres. The orbifold-Tutte embedding bijectively maps the original surface to
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a canonical, topologically equivalent, two-dimensional flat surface called
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a Euclidean orbifold. There are 17 Euclidean orbifolds, of which only the 4 sphere
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orbifolds are implemented here.
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The orbifold-Tutte embedding yields a seamless, globally bijective parameterization that,
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similarly to the classic Tutte embedding, only requires solving a sparse linear system
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for its computation.
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The parameterization process intrinsically requires the uses of seams, but whose choice
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have no influence on the result. The `Seam_mesh` structure (see also next Section)
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is used for this purpose.
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\section secCuttingaMesh Cutting a Mesh
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The surface parameterization methods proposed in this package only
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@ -365,11 +386,18 @@ and is therefore non-singular (Gram theorem).
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</UL>
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<LI> Boundaryless
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<UL>
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<LI>One-to-one mapping
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The Orbifold-Tutte embedding is guaranteed to exist and to be computable
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via a sparse linear system.
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</UL>
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</UL>
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\section Surface_mesh_parameterizationExtendingthe Implementation History
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@ -111,7 +111,8 @@ namespace Surface_mesh_parameterization {
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/// \sa `CGAL::Surface_mesh_parameterization::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::LSCM_parameterizer_3<TriangleMesh, BorderParameterizer_3>`
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/// \sa `CGAL::Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3<SeamMesh, SparseLinearAlgebraTraits_d>`
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///
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template
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<
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class TriangleMesh,
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@ -66,8 +66,8 @@ namespace Surface_mesh_parameterization {
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\sa `CGAL::Surface_mesh_parameterization::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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\sa `CGAL::Surface_mesh_parameterization::LSCM_parameterizer_3<TriangleMesh, BorderParameterizer_3>`
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\sa `CGAL::Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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\sa `CGAL::Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3<SeamMesh, SparseLinearAlgebraTraits_d>`
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*/
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template
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<
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class TriangleMesh,
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@ -69,6 +69,7 @@ namespace Surface_mesh_parameterization {
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/// \sa `CGAL::Surface_mesh_parameterization::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::LSCM_parameterizer_3<TriangleMesh, BorderParameterizer_3>`
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/// \sa `CGAL::Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3<SeamMesh, SparseLinearAlgebraTraits_d>`
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///
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template
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<
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@ -69,6 +69,7 @@ namespace Surface_mesh_parameterization {
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/// \sa `CGAL::Surface_mesh_parameterization::Discrete_authalic_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::LSCM_parameterizer_3<TriangleMesh, BorderParameterizer_3>`
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/// \sa `CGAL::Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3<SeamMesh, SparseLinearAlgebraTraits_d>`
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///
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template
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<
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@ -27,6 +27,8 @@ namespace CGAL {
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namespace Surface_mesh_parameterization {
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/// \ingroup PkgSurfaceParameterizationEnums
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///
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/// List of errors detected by this package
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enum Error_code
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{
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@ -83,7 +83,6 @@ namespace Surface_mesh_parameterization {
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/// \sa `CGAL::Surface_mesh_parameterization::Barycentric_mapping_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Discrete_authalic_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::LSCM_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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///
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template
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@ -56,7 +56,7 @@ namespace Surface_mesh_parameterization {
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/// \ingroup PkgSurfaceParameterizationMethods
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///
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/// The class `LSCM_parameterizer_3` implements the
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/// *Least Squares Conformal Maps (LSCM)* parameterization \cgalCite{cgal:lprm-lscm-02}.
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/// *Least Squares Conformal Maps (LSCM)* parameterization \cgalCite{cgal:lprm-lscm-02}.
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///
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/// This is a conformal parameterization, i.e. it attempts to preserve angles.
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///
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@ -74,13 +74,14 @@ namespace Surface_mesh_parameterization {
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/// Note: We may use a symmetric definite positive solver because LSCM
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/// solves the system in the least squares sense.
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///
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/// \sa `CGAL::Surface_mesh_parameterization::Fixed_border_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Two_vertices_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::ARAP_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Barycentric_mapping_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Discrete_authalic_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3<SeamMesh, SparseLinearAlgebraTraits_d>`
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///
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template
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<
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class TriangleMesh,
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@ -69,6 +69,7 @@ namespace Surface_mesh_parameterization {
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/// \sa `CGAL::Surface_mesh_parameterization::Discrete_authalic_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::LSCM_parameterizer_3<TriangleMesh, BorderParameterizer_3>`
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/// \sa `CGAL::Surface_mesh_parameterization::Orbifold_Tutte_parameterizer_3<SeamMesh, SparseLinearAlgebraTraits_d>`
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///
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template
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<
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@ -57,8 +57,6 @@
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/// \file Orbifold_Tutte_parameterizer_3.h
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#define CGAL_SMP_OUTPUT_ORBIFOLD_MATRICES
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// @todo checks that cones are different, are on seams, seam is one connected
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// component
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// @todo Should the order of cones provided in entry matter ? Map the first cone
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@ -68,12 +66,40 @@ namespace CGAL {
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namespace Surface_mesh_parameterization {
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/// \ingroup PkgSurfaceParameterizationEnums
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///
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/// Weight type used in the parameterization computation.
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///
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/// MVC weights are guaranteed to generate positive edge weights, and the parameterization
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/// is guaranteed to be injective.
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///
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/// In case the cotangent weights are used, the orbifold-Tutte embedding globally
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/// minimizes the Dirichlet energy and approximates conformal mappings.
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enum Weight_type
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{
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Cotangent = 0,
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Mean_value
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};
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/// \ingroup PkgSurfaceParameterizationMethods
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///
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/// The class `Orbifold_Tutte_parameterizer_3` implements <em>Orbifold Tutte Planar
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/// Embeddings</em> \cgalCite{aigerman2015orbifold}.
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///
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/// This is a borderless parameterization. A one-to-one mapping is guaranteed.
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///
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/// \cgalModels `Parameterizer_3`
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///
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/// \tparam SeamMesh must be a `Seam_mesh`, with underlying mesh any model of `FaceGraph`
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/// \tparam SparseLinearAlgebraTraits_d Traits class to solve a sparse linear system. <br>
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///
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/// \sa `CGAL::Surface_mesh_parameterization::ARAP_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Barycentric_mapping_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Discrete_authalic_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::Discrete_conformal_map_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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/// \sa `CGAL::Surface_mesh_parameterization::LSCM_parameterizer_3<TriangleMesh, BorderParameterizer_3>`
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/// \sa `CGAL::Surface_mesh_parameterization::Mean_value_coordinates_parameterizer_3<TriangleMesh, BorderParameterizer_3, SparseLinearAlgebraTraits_d>`
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///
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template
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<
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typename SeamMesh,
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@ -100,8 +126,8 @@ private:
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typedef typename Sparse_LA::Matrix Matrix;
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// Kernel subtypes
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typedef typename internal::Kernel_traits<SeamMesh>::Kernel Kernel;
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typedef typename internal::Kernel_traits<SeamMesh>::PPM PPM;
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typedef typename internal::Kernel_traits<SeamMesh>::Kernel Kernel;
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typedef typename internal::Kernel_traits<SeamMesh>::PPM PPM;
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typedef typename Kernel::FT NT;
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typedef typename Kernel::Vector_2 Vector_2;
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typedef typename Kernel::Vector_3 Vector_3;
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@ -112,7 +138,7 @@ private:
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const Weight_type weight_type;
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private:
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/// check input's correctness.
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/// Check input's correctness.
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template<typename ConeMap>
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Error_code check_cones(ConeMap cmap) const
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{
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@ -203,6 +229,7 @@ private:
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++id_r; // current line index in A is increased
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}
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/// Add the constraints from a seam segment to the linear system.
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void constrain_seam_segment(const std::vector<std::pair<int, int> >& seam_segment,
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NT ang, int& current_line_id_in_A,
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Matrix& A, Vector& B) const
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@ -427,6 +454,7 @@ private:
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}
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}
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/// Compute the system weights using a Cotangent Laplacian.
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template<typename VertexIndexMap>
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void cotangent_laplacien(SeamMesh& mesh,
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VertexIndexMap vimap,
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@ -594,22 +622,29 @@ private:
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}
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public:
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/// Flattens the mesh to one of the orbifolds. In the end, the
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/// position of each vertex is stored in the property map `uvmap`.
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/// Compute a one-to-one mapping from a triangular 3D surface mesh
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/// to a piece of the 2D space.
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/// The mapping is piecewise linear (linear in each triangle).
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/// The result is the (u,v) pair image of each vertex of the 3D surface.
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///
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///
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/// \tparam VertexUVmap must be a model of `ReadWritePropertyMap` with
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/// `boost::graph_traits<TM>::%vertex_descriptor` as key type and
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/// %Point_2 (type deduced from `TriangleMesh` using `Kernel_traits`)
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/// as value type.
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/// \tparam VertexIndexMap must be a model of `ReadablePropertyMap` with
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/// `boost::graph_traits<TM>::%vertex_descriptor` as key type and
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/// a unique integer as value type.
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/// \tparam VertexParameterizedMap must be a model of `ReadWritePropertyMap` with
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/// `boost::graph_traits<TM>::%vertex_descriptor` as key type and
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/// a Boolean as value type.
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///
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/// \param mesh a model of the `FaceGraph` concept
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/// \param bhd a halfedge on the border of the seam mesh
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/// \param cmap
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/// \param cmap a mapping of 4 (or 6) `vertex_descriptor`s that are cones
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/// to their respective `Cone_type`.
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/// \param uvmap an instanciation of the class `VertexUVmap`.
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/// \param vimap an instanciation of the class `VertexIndexMap`.
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/// \param orb the type of orbifold mapping
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/// \param convexBoundary - omitted or false for free boundary, of one of
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/// the 4 sphere orbifolds as specified in the constructor; true
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/// for the classic Tutte embedding with fixed boundary into a disk;
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/// 'square' for "classic" Tutte on a square with prefixed boundary
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/// map; 'freesquare' for the square disk orbifold; 'freetri' for
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/// the triangle disk orbifold.
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/// \param wt weight type (cotan weights or MVC weights).
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///
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/// \pre cones and seams must be valid.
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template<typename ConeMap,
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return OK;
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}
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/// Constructor
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public:
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/// Constructor.
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Orbifold_Tutte_parameterizer_3(const Orbifold_type orb_type = Square,
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const Weight_type weight_type = Cotangent)
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:
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@ -37,10 +37,19 @@
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#include <utility>
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#include <vector>
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/// \file Orbifold_cone_helper.h
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namespace CGAL {
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namespace Surface_mesh_parameterization {
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/// \ingroup PkgSurfaceParameterizationEnums
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///
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/// The types of cones used in Orbifold Tutte parameterization.
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///
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/// `Unique_cones` are found at the beginning and the end of the seam. All other
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/// cones are duplicated in the sense that when the seam is `opened`, the vertex
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/// is duplicated at two different positions.
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enum Cone_type
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{
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First_unique_cone = 0,
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@ -48,6 +57,9 @@ enum Cone_type
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Duplicated_cone
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};
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/// \ingroup PkgSurfaceParameterizationEnums
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///
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/// The four Orbifold types available in the Orbifold Tutte parameterization.
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enum Orbifold_type
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{
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Square = 0,
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@ -56,6 +68,9 @@ enum Orbifold_type
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Parallelogram
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};
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/// Get message corresponding to an error code
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/// \param orb_type The integer value in the enum
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/// \return The string describing the Orbifold type
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const char* get_orbifold_type(int orb_type)
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{
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// Messages corresponding to Error_code list above. Must be kept in sync!
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@ -94,10 +109,10 @@ Point_container get_cones_parameterized_coordinates(const Orbifold_type orb_type
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return tcoords;
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}
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/// Angles are minus as we go around the seam border in a counterclockwise manner
|
||||
template<typename NT_container>
|
||||
NT_container get_angles_at_cones(const Orbifold_type orb_type)
|
||||
{
|
||||
// Note that angles are minus as we go around the seam border in a counterclockwise manner
|
||||
NT_container angs;
|
||||
if(orb_type == Square) {
|
||||
angs.push_back(4.);
|
||||
|
|
@ -183,6 +198,7 @@ void find_start_cone(const ConeMap& cmap,
|
|||
CGAL_postcondition(false);
|
||||
}
|
||||
|
||||
/// Check the validity of the input cones in the `Seam_mesh` mesh.
|
||||
template<typename SeamMesh,
|
||||
typename Cones_in_Seam_mesh_map,
|
||||
typename Cones_in_Base_mesh_container>
|
||||
|
|
|
|||
Loading…
Reference in New Issue