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Surface Reconstruction From Unorganized Point Sets. piyush@cs. Example Reconstruction. Surface Reconstruction Algorithm. Organization of Talk. Problem Statement Some Basic Definitions Hoppe’s Algorithm : The Beginning Alpha Shapes : Generalizations of the convex hull

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example reconstruction
Example Reconstruction

Surface Reconstruction


organization of talk
Organization of Talk
  • Problem Statement
  • Some Basic Definitions
  • Hoppe’s Algorithm : The Beginning
  • Alpha Shapes : Generalizations of the convex hull
  • Crust : A Narrow problem specification
  • New Algorithms on the Horizon
  • Open Problems
problem statement
Problem Statement
  • Given a set of Sample Points in three dimensions produce a simplicial surface that captures the “most reasonable shape” the points were sampled from.
  • Applications: CG, Medical Imaging, Cartography, Compression, Reverse Engineering, etc.

Delaunay Triangulation, Voronoi Diagram[2D,3D]

hoppe et al s algorithm
Hoppe et al’s Algorithm
  • Estimate Signed Distance Function



  • Use Iso-Surface Extraction after this step
hoppe et al s algorithm7
Hoppe et al’s Algorithm
  • Tangent Plane Estimation(PCA)
  • Consistent tangent plane orientation(MST)
  • Signed distance function
  • Contour tracing (Marching Cubes)
problems with hoppe s algorithm
Problems with Hoppe’s Algorithm
  • All k-nearest neighbors might lie on a single line in some cases hence creating a problem for the normal computation.
  • The Sampling criterion required is uniform. Sampling ideally should be proportional to curvature in some sense.
  • Curless Levoy gave a modification which is one of the best results in Practice.
alpha shapes
Alpha Shapes

The space generated by point pairs that can be touched by an

empty disc of radius alpha.

alpha shapes13
Alpha Shapes

Alpha Controls the desired level of detail.

problems with alpha shapes
Problems with Alpha Shapes
  • Global Value of Alpha may not exist for a correct reconstruction in some cases
  • Experimentation is required to find the appropriate values of alpha
  • A Delaunay computation is done on the point set which is costly in 3D
crust a narrow problem specification
Crust: A Narrow problem Specification
  • Aims to reconstruct only smooth closed Manifolds in 2D and 3D

Medial Axis: of surface F is the closure of points that have more

than one closest point in F.


Local Feature Size f(p) at a point p on F is the least distance of p to

the medial axis.

S is called an r-sample of F if every point has a sample

within a distance rf(p).




the intuition behind crust
The Intuition behind Crust

The Voronoi Cells of a dense sampling are thin and long.

The Medial Axis is the extension of Voronoi Diagram for

continuous surfaces in the sense that the Voronoi Diagram

of S Can be defined as the set of points with more than one

closest point in S. (S = Sample Point Set)

The Sampling criterion of

the Crust breaks down in case

of non-smooth curves and


the crust in 2d
The Crust in 2D

Input : P = Set of sample points in the plane

Output: E = Set of edges connecting points in P

The Algorithm

Compute the Voronoi vertices of P = V

Calculate the Delaunay of (P U V)

Pick the edges (p,q) where both p,q are in P

3d crust definitions
3D Crust: Definitions

Restricted Voronoi Cells:

Restricted Delaunay Triangles: A triangle

is nonempty.

Here S is a surface and p is a sample point.

(What is a correct connection in 3D??)

the problem with 3d
The Problem with 3D
  • Slivers : Even very dense sampling does not guarantee that the Voronoi Vertices approximate the Medial axis which is true in case of 2D Crust. Actually the Voronoi Vertices in 3D can be arbitrarily far from the Medial Axis.
crust in 3d
Crust in 3D
  • Compute the 3D Voronoi diagram of the sample points.
  • For each sample point s, pick the farthest vertex v of its Voronoi cell, and the farthest vertex v' such that angle vsv' exceeds 90 degrees.
  • Compute the Voronoi diagram of the sample points and the "poles", the Voronoi vertices chosen in the second step.
  • Add a triangle on each triple of sample points with neighboring cells in the second Voronoi diagram.
problems with crust
Problems with Crust
  • Very Slow in 3D for large pointsets
  • Cannot handle Boundaries
  • Cannot deal with Sharp turns and Corners
  • Needs computation of Delaunay Twice
  • Theoretically requires very high density sampling in 3D (r <= .06)
the hot solutions
The Hot Solutions
  • Spiralling Edge (P.Crossno & E.Angel, Visualization 99)*
  • Interactive Approach (L.P.Kobelt and M.Botsch, EuroGraphics 2k)
  • Surface Reconstruction based on Lower Dimensional Localized Delaunay Triangulation (Gopi et al., Eurographics 2000)
  • Single Pass Crust(Amenta et al, SoCG 2k)*
  • Graph-Based Surface Reconstruction Using Structures in Scattered Point Sets (R. MenclandH. Müller in CGI 98)*
graph based reconstruction
Graph Based Reconstruction
  • EMST Computation
  • SDG Computation(in successive stages)
  • Fill SDG(final) with triangles
single pass crust
Single Pass Crust

Single Pole Computation

Cross section of the

Voronoi cell of p in

3 Dimensions

If Edge e has three such

supporting points, the

dual triangle to e is in

the crust

Edge e



Near pi/2 angle

spiralling edge
Spiralling Edge
  • Advancing Front Flavour: Uses Edge Rings.
  • Gabriel Graph Computation in k-nearest neighbours.
  • Starts with normal information.
  • Handles Boundary and Corner points as special cases.
  • No Theoretical Guarantees.

open problems
Open Problems
  • Provable reconstruction for Manifolds with boundary and sharp turns.
  • Can Provable algorithms be given using subset of Delaunay instead of global Delaunay computation?
  • Can Unorganised points be decimated before reconstruction?