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Discrete Geometry

Discrete Geometry. Tutorial 2. © Maks Ovsjanikov, Alex & Michael Bronstein tosca.cs.technion.ac.il/book. Numerical geometry of non-rigid shapes Stanford University, Winter 2009. Neighborhood. Neighborhood of point in. Discrete equivalent of metric ball. Nearest neighbors.

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Discrete Geometry

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  1. Discrete Geometry Tutorial 2 © Maks Ovsjanikov, Alex & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

  2. Neighborhood Neighborhood of point in Discrete equivalent of metric ball

  3. Nearest neighbors K nearest neighbors of

  4. Connectivity • Neighborhood is a topological definition independent of a metric • Two points are adjacent (directly connected) if they belong to the same neighborhood • The connectivity structure can be represented as an undirected graph with vertices and edges Vertices Edges • Connectivity graph can be represented as a matrix

  5. Shape representation Graph Cloud of points

  6. Connectivity in the plane Four-neighbor Six-neighbor Eight-neighbor

  7. Delaunay tessellation Define connectivity as follows: a pair of points whose Voronoi cells are adjacent are connected The obtained connectivity graph is dual to the Voronoi diagram and is called Delaunay tesselation Boris Delaunay (1890-1980) Voronoi regions Connectivity Delaunay tesselation

  8. Delaunay tessellation For surfaces, the triangles are replaced by geodesic triangles [Leibon & Letscher]: under conditions that guarantee the existence of Voronoi tessellation, Delaunay triangles form a valid tessellation Replacing geodesic triangles by planar ones gives Delaunay triangulation Geodesic triangles Euclidean triangles

  9. Shape representation Graph Cloud of points Triangular mesh

  10. Triangular meshes A structure of the form consisting of • Vertices • Edges • Faces is called a triangular mesh The mesh is a purely topological object and does not contain any geometric properties The faces can be represented as an matrix of indices, where each row is a vector of the form , and

  11. Triangular meshes The geometric realization of the mesh is defined by specifying the coordinates of the vertices for all The coordinates can be represented as an matrix The mesh is a piece-wise planar approximation obtained by gluing the triangular faces together, Triangular face

  12. Example of a triangular mesh Vertices Coordinates Edges Faces Topological Geometric

  13. Barycentric coordinates Any point on the mesh can be represented providing • index of the triangle enclosing it; • coefficients of the convex combination of the triangle vertices Vector is called barycentric coordinates

  14. Manifold meshes • is a manifold • Neighborhood of each interior vertex is homeomorphic to a disc • Neighborhood of each boundary vertex is homeomorphic to a half-disc • Each interior edge belongs to two triangles • Each boundary edge belongs to one triangle

  15. Non-manifold meshes Edge shared by four triangles Non-manifold mesh

  16. Geometry images Surface Geometry image Global parametrization Sampling of parametrization domain on a Cartesian grid

  17. Geometry images Six-neighbor Eight-neighbor Manifold mesh Non-manifold mesh

  18. Geometric validity Topologically valid Geometrically invalid Topological validity (manifold mesh) is insufficient! Geometric validity means that the realization of the triangular mesh does not contain self-intersections

  19. Skeleton For a smooth compact surface , there exists an envelope (open set in containing ) such that every point is continuously mappable to a unique point on The mapping is realized as the closest point on from Problem when is equidistant from two points on (such points are called medial axis or skeleton of ) If the mesh is contained in the envelope (does not intersect the medial axis), it is valid

  20. Skeleton Points equidistant from the boundary form the skeleton of a shape

  21. Local feature size Distance from point on to the medial axis of is called the local feature size, denoted Local feature size related to curvature (not an intrinsic property!) [Amenta&Bern, Leibon&Letscher]: if the surface is sampled such that for every an open ball of radius contains a point of , it is guaranteed that does not intersect the medial axis of Conclusion: there exists sufficiently dense sampling guaranteeing that is geometrically valid

  22. Geometric validity Insufficient density Invalid mesh Sufficient density Valid mesh

  23. Approximation quality How well does the mesh approximate the underlying surface ? • Sampling quality • Topological equivalence (manifold meshes) • Geometric properties such as area, normals, etc. computed on are close to those computed on

  24. Discretization of geometric quantities Given a the mesh evaluate quantities: • Areas, Lengths • Normals • Curvatures (principal, Gauss, mean, etc.) Integral Differential

  25. Discretization of geometric quantities Given a the mesh evaluate quantities: • Areas, Lengths • Normals • Curvatures (principal, Gauss, mean, etc.)

  26. Discretization of geometric quantities Given a the mesh evaluate quantities: • Areas, Lengths • Normals • Curvatures (principal, Gauss, mean, etc.)

  27. Discretization of geometric quantities Given a the mesh evaluate quantities: • Areas, Lengths • Normals • Curvatures (principal, Gauss, mean, etc.)

  28. Discretization of geometric quantities Given a the mesh evaluate quantities: • Areas, Lengths • Normals • Curvatures (principal, Gauss, mean, etc.)

  29. Discretization of geometric quantities Given a the mesh evaluate quantities: • Areas, Lengths • Normals • Curvatures (principal, Gauss, mean, etc.)

  30. Discretization of geometric quantities Given a the mesh evaluate quantities: • Areas, Lengths • Normals • Curvatures (principal, Gauss, mean, etc.)

  31. Discretization of geometric quantities Given a the mesh evaluate quantities: • Areas, Lengths • Normals • Curvatures (principal, Gauss, mean, etc.) • A whole envelope of possibilities. Which one is the right one?

  32. Discretization of geometric quantities Hundreds of methods for estimating differential information. • Case specific. • Try to preserve essential properties of these quantities. • Express them in terms of integral ones. Averaging. Local Regions smooth surface mesh

  33. Discretization of geometric quantities Estimating Normals. • Well defined on the triangles • On edges and vertices can be defined by averaging: • Scale each normal by the area of corresponding triangle Integral: mesh

  34. Discretization of geometric quantities Estimating Normals. • Well defined on the triangles • On edges and vertices can be defined by averaging: • Important: scale each normal by the area of corresponding triangle mesh Integral:

  35. Discretization of geometric quantities Estimating Curvature. • Equals to 0 on triangles/edges. • Cannot average. • Limits are often dropped in the discrete case.

  36. Discretization of geometric quantities Estimating Curvature. • Equals to 0 on triangles/edges. • Cannot average. • Limits are often dropped in the discrete case.

  37. Discretization of geometric quantities Estimating Curvature. • Equals to 0 on triangles/edges. • Cannot average. • Limits are often dropped in the discrete case. Can be extended to 3D as well. [Taubin ’95]

  38. Discretization of geometric quantities Estimating Gaussian Curvature. • Equals to 0 on triangles/edges. • Cannot average. • Limits are often dropped in the discrete case.

  39. Discretization of geometric quantities Estimating Gaussian Curvature. • Equals to 0 on triangles/edges. • Cannot average. • Limits are often dropped in the discrete case. • Task: Compute area of spherical polygon.

  40. Discretization of geometric quantities Estimating Gaussian Curvature. • Equals to 0 on triangles/edges. • Cannot average. • Limits are often dropped in the discrete case. • Task: Compute area of spherical polygon.

  41. Approximation quality How well does the mesh approximate the underlying surface ? • Sampling quality • Topological equivalence (manifold meshes) • Geometric properties such as area, normals, etc. computed on are close to those computed on

  42. Schwarz lantern M=10, N=10 M=20, N=20 M=20, N=10

  43. Schwarz lantern Set . Then, in the limit Set . Then, in the limit Reason: triangles become infinitely thin [Morvan & Thibert 2002]: if the triangulation is sufficiently fat, the area approximation error is bounded Hermann Schwarz (1843-1921)

  44. Conclusion Sampling Voronoi tessellation Connectivity Farthest point sampling Delaunay tessellation Triangular meshes Geometric validity Topological validity Sufficiently dense sampling Manifold meshes Schwarz lantern

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