Surfaces, meshes, and topology

Transcription

1 Surfaces from Point Samples Surfaces, meshes, and topology A surface is a 2-manifold embedded in 3- dimensional Euclidean space Such surfaces are often approximated by triangle meshes 2 1

2 Triangle mesh A piecewise linear surface consisting of triangular faces which are pasted together along their edges Constructed from points (vertices), edges, triangles These building blocks are referred to as simplices: a vertex is a 0-simplex an edge a 1-simplex a triangle a 2-simplex. Edges of a triangle, or vertices of an edge are called faces. This can be generalized by introducing the notion of simplicial complexes. 3 Topology Two surfaces S and S' are said to be topologically equivalent (or homeomorphic) when one can transform S into S' by an elastic deformation D, that is, a mapping that does not tear the surface apart 1. The transformation D:S S' is called a topological mapping (or homeomorphism) and D is said to preserve the topology of the surface S. 1 More precisely: continuous bijection with continuous inverse 4 2

3 Topologically equivalent surfaces 5 Manifolds A surface is called a manifold when a region around each point is topologically equivalent to a disk (in 2D) or a ball (in 3D). That is, a small disk or ball can be placed on the surface without tearing or overlapping. Topology that is not manifold is called nonmanifold. 6 3

4 Non-manifold topology: examples triangle meshes where an edge is used by more than two triangles triangles connected to each other only at their vertices, i.e., not sharing an edge 7 Point-Based Surfaces Many applications need a definition of surface based on point samples Reduction Up-sampling Interrogation (e.g., ray tracing) Desirable surface properties Manifold Smooth Local (efficient computation) 8 4

5 Introduction & Basics Notation, Terms - Regular/Irregular, Approximation/Interpolation, Global/Local Standard interpolation/approximation techniques Global: Triangulation, Voronoi-Interpolation, Least Squares (LS), Radial Basis Functions (RBF) Local: Shepard/Partition of Unity Methods, Moving LS Problems Sharp edges, feature size/noise Functional (height field) Manifold 9 Notation Consider functional (height) data for now Data points are represented as: Location in parameter space p i With certain height f i = f(p i ) Goal is to approximate f from {f i, p i } 10 5

6 Terms: Regular/Irregular Regular (on a grid) or irregular (scattered) Neighborhood (topology) is unclear for irregular data 11 Terms: Approximation/Interpolation Noisy data -> Approximation Perfect data -> Interpolation 12 6

7 Terms: Global/Local Global approximation Local approximation Locality comes at the expense of smoothness 13 Triangulation Exploit the topology in a triangulation (e.g., Delaunay) of the data Interpolate the data points on the triangles Piecewise linear C 0 Piecewise quadratic C 1?

8 Triangulation: piecewise linear Barycentric interpolation on simplices (triangles) Given n+1 points x i with values f i and a point x inside the simplex defined by x i Compute α i from Then 15 Least Squares Fits a primitive to the data Minimizes squared distances between the p i 's and primitive g 19 8

9 Least Squares: Example Primitive is a polynomial of degree M Minimization: Put derivative of sum w.r.t. c j to zero for each j=0,1,..., M: Linear system of equations for unknowns c 0,c 1,...,c M 20 Radial Basis Functions Represent interpolant as - Sum of radial functions r( x ) - Centered at the data points p i, i=0,1,,n 21 9

10 Radial Basis Functions Solve j=0,1,,n to compute weights w i, i=0,1,,n Linear system of equations 22 Radial Basis Functions Solvability depends on radial function Several choices assure solvability - thin plate spline: - Gaussian: * h is a data parameter * h reflects the feature size or anticipated spacing among points 23 10

11 Function Spaces Problems - Many points lead to large linear systems - Evaluation requires global solutions Solutions - RBF with compact support * Matrix is sparse * Still: solution depends on every data point though drop-off is exponential with distance - Local approximation approaches 24 Moving Least Squares Compute a local LS approximation at t Data points weighted based on distance to t 29 11

12 Moving Least Squares The interpolated set is a smooth curve, if and only if the kernel θ(.) is smooth 30 Moving Least Squares Typical choices: Note: is fixed For each t - Standard weighted LS problem - Linear iff corresponding LS is linear 31 12

13 Typical Problems Sharp corners/edges Noise vs. feature size 32 Functional Manifold Standard techniques are applicable if data represents a function (e.g., height field) Manifolds are more general No parameter domain No knowledge about neighbors, Delaunay triangulation connects non-neighbors 33 13

14 Implicits Each orientable n-manifold can be embedded in (n+1) space Idea: Represent n-manifold as zero-set of a scalar function in (n+1) space Inside: f (x)< 0 On the manifold: f (x) = 0 Outside: f (x) > 0 34 Implicits - Illustration Courtesy: Greg Turk 35 14

15 Implicits from point samples Function should be zero at data points Use standard approximation techniques to find Trivial solution: Additional constraints are needed 36 Implicits from point samples Constraints define inside and outside Simple approach (Turk, O Brien) 37 15

16 Implicits from point samples Use normal information Normals could be computed from scan Or, normals have to be estimated 38 Implicits from point samples Compute non-zero anchors in the distance field Use normal information directly as constraints 39 16

17 Computing Implicits Given N points and normals and constraints Let An RBF approximation leads to a system of linear equations 40 Computing Implicits Practical problems: N > Matrix solution becomes difficult Two solutions Sparse matrices allow iterative solution (use compactly supported RBFs) Smaller number of RBFs (only where quality is not sufficient) 41 17

18 RBF Implicits - Results 42 RBF Implicits - Results 43 18