Processing a tesselation
The disclosure relates to a computer-implemented method for tessellation processing. The tessellation forms a surface representation of a real-world 3D object. The method comprises providing a constrained tetrahedral Delaunay mesh comprising the tessellation. The mesh is conformal and comprises one or more sets of tetrahedrons, each set representing a gap between portions of the tessellation, each gap having a size lower than a predefined threshold. The method further comprises determining a set of one or more tetrahedra faces of the mesh to be added to the tessellation, includes minimizing an objective function that includes a term penalizing surface creation by face addition to the tessellation. The minimization is under the constraint that a given set of tetrahedra faces of the Delaunay mesh is to be added to the tessellation. The given set of tetrahedra faces includes, for each gap of one or more gaps, the tetrahedra faces meshing the gap.
1. A computer-implemented method for processing a tessellation forming a surface representation of a real-world 3D object, the method comprising:
obtaining a constrained tetrahedral Delaunay mesh including the tessellation, the Delaunay mesh being conformal and including one or more sets of tetrahedrons each representing a respective gap between portions of the tessellation, each respective gap having a size lower than a predefined threshold; and
determining a set of one or more tetrahedra faces of the Delaunay mesh to be added to the tessellation, the determining including minimizing an objective function that includes a term penalizing surface creation by face addition to the tessellation, the minimization being under a constraint that a given set of tetrahedra faces of the Delaunay mesh is to be added to the tessellation, the given set of tetrahedra faces including, for each gap of one or more gaps, the tetrahedra faces meshing the gap,
wherein the minimization includes obtaining an initial classification of the tetrahedrons of the Delaunay mesh into a first group and a second group, the first group including tetrahedrons that are inside a bounding box of the tessellation, the minimization having as free variable a classification of the tetrahedrons into the first and second groups, the constraint being that tetrahedrons meshing said one or more gaps belong to the first group, and
wherein the minimization outputs a classification of the tetrahedrons into the first and second groups that defines an interface between the first group and the second group, and the determining of a set of one or more tetrahedra faces of the Delaunay mesh to be added to the tessellation further includes another minimization to further minimize the objective function by modifying the interface between the first group and the second group.
2. The computer-implemented method of claim 1 , wherein said another minimization comprises:
traversing points of the Delaunay mesh which are on the interface but not on the tessellation, and assessing for each point whether moving the point, by bringing the point closer to another point of the interface along a direction of an edge having the point as vertex, further minimizes the objective function; and
determining one or more angles, each determined angle being respective to a traversed point and being an angle formed by two triangles of the Delaunay mesh that both belong to the interface and that intersect at the traversed point, each determined angle having a value smaller than a predefined threshold, and for each determined angle, inserting a point that is to be part of the interface, the point being either inserted on an existing edge, thereby dividing the existing edge, or inserted on no existing edge in which case new edges are created for connecting the point to the interface, the created edges being also part of the interface, the insertion of the point being performed to further optimize the objective function and thereby to flatten the angle.
3. The computer-implemented method of claim 1 , wherein the term is of a type
Area(∂ D (ω)− S ),
where ω∈Ω and Ω is the set of all tetrahedra classifications in the first and second groups, ∂D(ω) is the interface between the first and second groups in the classification ω, S is the tessellation.
4. The computer-implemented method of claim 3 , wherein the objective function further comprises a term of a type:
αArea(∂ D (ω)∩ S )
where α is a positive real number.
5. The computer-implemented method of claim 3 , wherein the objective function further comprises a term of a type:
Area(π D 0(ω)∩ S−∂D (ω)),
where πD0(ω) is the set of faces for all tetrahedra classified in the second group D0 in classification ω.
6. The computer-implemented method of claim 1 , wherein the obtaining of the constrained tetrahedral Delaunay mesh further comprises:
obtaining the tessellation, and
making the tessellation conformal and inserting the tessellation in a tetrahedral Delaunay mesh by determining a tetrahedral Delaunay mesh that includes, as part of the mesh, the tessellation, the determined tetrahedral Delauney mesh forming the obtained constraint tetrahedral Delaunay mesh.
7. The computer-implemented method of claim 1 , further comprising, prior to the determining of a set of one or more tetrahedra faces of the Delaunay mesh to be added to the tessellation:
locally refining the tetrahedral mesh of each gap.
8. The computer-implemented method of claim 1 , wherein the determining of a set of one or more tetrahedra faces of the Delaunay mesh to be added to the tessellation is iterated, each iteration comprising, before the minimization, modifying the given set of tetrahedra faces.
9. The computer-implemented method of claim 1 , wherein the surface represents an outer surface of a car body.
10. A non-transitory computer-readable data storage medium having recorded thereon a computer program comprising instructions for performing a method for processing a tessellation forming a surface representation of a real-world 3D object, the method comprising:
obtaining a constrained tetrahedral Delaunay mesh including the tessellation, the Delaunay mesh being conformal and comprising one or more sets of tetrahedrons each representing a respective gap between portions of the tessellation, each respective gap having a size lower than a predefined threshold; and
determining a set of one or more tetrahedra faces of the Delaunay mesh to be added to the tessellation, the determining including minimizing an objective function that includes a term penalizing surface creation by face addition to the tessellation, the minimization being under a constraint that a given set of tetrahedra faces of the Delaunay mesh is to be added to the tessellation, the given set of tetrahedra faces including, for each gap of one or more gaps, the tetrahedra faces meshing the gap,
wherein the minimization includes obtaining an initial classification of the tetrahedrons of the Delaunay mesh into a first group and a second group, the first group including tetrahedrons that are inside a bounding box of the tessellation, the minimization having as free variable a classification of the tetrahedrons into the first and second groups, the constraint being that tetrahedrons meshing said one or more gaps belong to the first group, and
wherein the minimization outputs a classification of the tetrahedrons into the first and second groups that defines an interface between the first group and the second group, and the determining of a set of one or more tetrahedra faces of the Delaunay mesh to be added to the tessellation further includes another minimization to further minimize the objective function by modifying the interface between the first group and the second group.
11. The non-transitory computer-readable data storage medium of claim 10 , wherein said another minimization comprises:
traversing points of the Delaunay mesh which are on the interface but not on the tessellation, and assessing for each point whether moving the point further minimizes the objective function;
determining one or more angles formed at the traversed points by edges of the mesh and having a value lower than a predefined threshold, and for each determined angle, inserting a point that is to be part of the interface to flatten the angle.
12. A computer system comprising:
a processor coupled to a memory, the memory having recorded thereon a computer program including instructions for processing a tessellation forming a surface representation of a real-world 3D object that when executed by the processor causes the processor to be configured to:
obtain a constrained tetrahedral Delaunay mesh comprising the tessellation, the Delaunay mesh being conformal and comprising one or more sets of tetrahedrons each representing a respective gap between portions of the tessellation, each respective gap having a size lower than a predefined threshold, and
determine a set of one or more tetrahedra faces of the Delaunay mesh to be added to the tessellation, the determining including minimizing an objective function that includes a term penalizing surface creation by face addition to the tessellation, the minimization being under a constraint that a given set of tetrahedra faces of the Delaunay mesh is to be added to the tessellation, the given set of tetrahedra faces including, for each gap of one or more gaps, the tetrahedra faces meshing the gap,
wherein the minimization includes obtaining an initial classification of the tetrahedrons of the Delaunay mesh into a first group and a second group, the first group including tetrahedrons that are inside a bounding box of the tessellation, the minimization having as free variable a classification of the tetrahedrons into the first and second groups, the constraint being that tetrahedrons meshing said one or more gaps belong to the first group, and
wherein the minimization outputs a classification of the tetrahedrons into the first and second groups that defines an interface between the first group and the second group, and the determining of a set of one or more tetrahedra faces of the Delaunay mesh to be added to the tessellation further includes another minimization to further minimize the objective function by modifying the interface between the first group and the second group.
13. The computer system of claim 12 , wherein said another minimization includes:
traversing points of the Delaunay mesh which are on the interface but not on the tessellation, and assessing for each point whether moving the point further minimizes the objective function; and
determining one or more angles formed at the traversed points by edges of the mesh and having a value lower than a predefined threshold, and for each determined angle, inserting a point that is to be part of the interface to flatten the angle.