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##### Discrete geometry

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**Discrete geometry**© Alexander & Michael Bronstein, 2006-2009 © Michael Bronstein, 2010 tosca.cs.technion.ac.il/book 048921 Advanced topics in vision Processing and Analysis of Geometric Shapes EE Technion, Spring 2010**‘**‘ ‘ ‘ The world is continuous, but the mind is discrete David Mumford**Discretization**Continuous world Discrete world • Surface • Metric • Topology • Sampling • Discrete metric (matrix of distances) • Discrete topology (connectivity)**Sampling density**• How to quantify density of sampling? • is an -covering ofif Alternatively: for all , where is the point-to-set distance.**Sampling efficiency**• Are all points necessary? • An -covering may be unnecessarily dense (may even not be a discrete set). • Quantify how well the samples are separated. • is -separated if for all . • For , an -separated set is finite if is compact. Also an r-covering!**Farthest point sampling**• Good sampling has to be dense and efficient at the same time. • Find and -separated -covering of . • Achieved using farthest point sampling. • We defer the discussion on • How to select ? • How to compute ?**Farthest point sampling**Farthest point**Farthest point sampling**• Start with some . • Determine sampling radius • If stop. • Find the farthest point from • Add to**Farthest point sampling**• Outcome: -separated -covering of . • Produces sampling with progressively increasing density. • A greedy algorithm: previously added points remain in . • There might be another -separated -covering containing less points. • In practice used to sub-sample a densely sampled shape. • Straightforward time complexity: number of points in dense sampling, number of points in . • Using efficient data structures can be reduced to .**Sampling as representation**• Sampling represents a region on as a single point . • Region of points on closer to than to any other • Voronoi region (a.k.a. Dirichlet or Voronoi-Dirichlet region, Thiessen polytope or polygon, Wigner-Seitz zone, domain of action). • To avoid degenerate cases, assume points in in general position: • No three points lie on the same geodesic. (Euclidean case: no three collinear points). • No four points lie on the boundary of the samemetric ball. (Euclidean case: no four cocircular points).**Voronoi decomposition**Voronoi region Voronoi edge Voronoi vertex • A point can belong to one of the following • Voronoi region ( is closer to than to any other ). • Voronoi edge ( is equidistant from and ). • Voronoi vertex ( is equidistant from three points ).**Voronoi decomposition**• Voronoi regions are disjoint. • Their closure covers the entire . • Cutting along Voronoi edges produces a collection of tiles . • In the Euclidean case, the tiles are convex polygons. • Hence, the tiles are topological disks (are homeomorphic to a disk).**Voronoi tesellation**• Tessellation of (a.k.a. cell complex): a finite collection of disjoint open topological disks, whose closure cover the entire . • In the Euclidean case, Voronoi decomposition is always a tessellation. • In the general case, Voronoi regions might not be topological disks. • A valid tessellation is obtained is the sampling is sufficiently dense.**Non-Euclidean Voronoi tessellations**• Convexity radius at a point is the largest for which the closed ball is convex in , i.e., minimal geodesics between every lie in . • Convexity radius of = infimum of convexity radii over all . • Theorem (Leibon & Letscher, 2000): An -separated -covering of with convexity radius of is guaranteed to produce a valid Voronoi tessellation. • Gives sufficient sampling density conditions.**Sufficient sampling density conditions**Invalid tessellation Valid tessellation**MATLAB® intermezzo**Farthest point sampling and Voronoi decomposition**Representation error**• Voronoi decomposition replaces with the closest point . • Mapping copying each into . • Quantify the representation error introduced by . • Let be picked randomly with uniform distribution on . • Representation error = variance of**Optimal sampling**• In the Euclidean case: (mean squared error). • Optimal sampling: given a fixed sampling size , minimize error Alternatively: Given a fixed representation error , minimize sampling size**Centroidal Voronoi tessellation**• In a sampling minimizing , each has to satisfy (intrinsic centroid) • In the Euclidean case – center of mass • In general case: intrinsic centroid of . • Centroidal Voronoi tessellation (CVT): Voronoi tessellation generated by in which each is the intrinsic centroid of .**Lloyd-Max algorithm**• Start with some sampling (e.g., produced by FPS) • Construct Voronoi decomposition • For each , compute intrinsic centroids • Update • In the limit , approaches the hexagonal honeycomb shape – the densest possible tessellation. • Lloyd-Max algorithms is known under many other names: vector quantization, k-means, etc.**Sampling as clustering**Partition the space into clusters with centers to minimize some cost function • Maximum cluster radius • Average cluster radius In the discrete setting, both problems are NP-hard Lloyd-Max algorithm, a.k.a. k-means is a heuristic, sometimes minimizing average cluster radius (if converges globally – not guaranteed)**Farthest point sampling encore**• Start with some , • For • Find the farthest point • Compute the sampling radius Lemma • .**Proof**For any**Proof (cont)**Since , we have**Almost optimal sampling**Theorem (Hochbaum & Shmoys, 1985) Let be the result of the FPS algorithm. Then In other words: FPS is worse than optimal sampling by at most 2.**Idea of the proof**Let denote the optimal clusters, with centers Distinguish between two cases One of the clusters contains two or more of the points Each cluster contains exactly one of the points**Proof (first case)**Assume one of the clusters contains two or more of the points , e.g. triangle inequality Hence:**Proof (second case)**Assume each of the clusters contains exactly one of the points , e.g. Then, for any point triangle inequality**Proof (second case, cont)**We have: for any , for any point In particular, for**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**Shape representation**Graph Cloud of points**Connectivity in the plane**Four-neighbor Six-neighbor Eight-neighbor**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**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**Shape representation**Graph Cloud of points Triangular mesh**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**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**Example of a triangular mesh**Vertices Coordinates Edges Faces Topological Geometric**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**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**Non-manifold meshes**Edge shared by four triangles Non-manifold mesh**Geometry images**Surface Geometry image Global parametrization Sampling of parametrization domain on a Cartesian grid**Geometry images**Six-neighbor Eight-neighbor Manifold mesh Non-manifold mesh**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**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**Skeleton**Points equidistant from the boundary form the skeleton of a shape**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