1. 1 www.geometrie.tuwien.ac.at GEOMETRIE Geometrie in der Technik H. Pottmann TU Wien SS 2007

2. 2 www.geometrie.tuwien.ac.at GEOMETRIE Distance function Given: geometric object F (curve, surface, solid, …) Assigns to each point the shortest distance from F F p

3. 3 www.geometrie.tuwien.ac.at GEOMETRIE Distance function Level sets of the distance function are trimmed offsets Not smooth at the cut locus

4. 4 www.geometrie.tuwien.ac.at GEOMETRIE Classical Geometry Studied the graph surfaces of distance functions: developable surfaces of constant slope Relation to circle and sphere geometry, cyclographic mapping

5. 5 www.geometrie.tuwien.ac.at GEOMETRIE PDEs distance function solves the eikonal equation Efficient numerical solvers on a grid (in R 2 and R 3 ) Fast marching (Sethian, Kimmel,…) Fast sweeping (Danielson, Osher, Tsai, Zhao,…) Distance functions in manifolds (Hamilton-Jacobi equations) Signed distance function as level set function in the level set method

6. 6 www.geometrie.tuwien.ac.at GEOMETRIE PDEs: Comparison of algorithms ZhaoTsai

7. 7 www.geometrie.tuwien.ac.at GEOMETRIE Computer-Aided Design Level sets of distance function: offsets Offsets and generalized offsets important for NC machining

8. 8 www.geometrie.tuwien.ac.at GEOMETRIE Computer Vision and Image Processing Central role in Math. Morphology (dilation, erosion, skeleton,…) normalization of level set function in level set evolution for segmentation

9. 9 www.geometrie.tuwien.ac.at GEOMETRIE Robotics Distance functions on manifolds (configuration space) and derived geodesics or splines for motion planning, also in the presence of obstacles Collision avoidance

10. 10 www.geometrie.tuwien.ac.at GEOMETRIE Distance functions in a special manifold: feature sensitive metric Euclidean Feature sensitive ECCV ´04

11. 11 www.geometrie.tuwien.ac.at GEOMETRIE Further application areas Pattern Classification: distance fields in high dimensions; separation of clusters relation to methods from Computational Geometry Computer Graphics: unifying implicit representation adaptively sampled distance fields point cloud processing Scientific Visualization

12. 12 www.geometrie.tuwien.ac.at GEOMETRIE Distance functions d(x) is a distance function if it solves the eikonal equation For a signed distance function, we admit a sign change at the set S to which the distance is computed; unlike d it is smooth at S Geometric meaning of the eikonal equation in R 2 : all tangent planes of the graph surface have slope 1.

13. 13 www.geometrie.tuwien.ac.at GEOMETRIE Fast sweeping Compute distance function on a grid Fast sweeping algorithm (Tsai, Zhao…) in R 2 : Grid points (i,j), i=0:Nx-1,j=0:Ny-1 Compute accurate distance values at grid points close to S Propagate this informtation by sweeping through the grid.

14. 14 www.geometrie.tuwien.ac.at GEOMETRIE Fast sweeping (x+,y+) sweeping: for j=0:Ny-1 for i=0:Nx-1 update d(i,j) Correctly propagates distance information in directions to the first quadrant y x x

15. 15 www.geometrie.tuwien.ac.at GEOMETRIE Fast sweeping (x-,y+) sweeping: for j=0:Ny-1 for i=Nx-1:0 update d(i,j) Correctly propagates distance information in directions to the second quadrant y x x

16. 16 www.geometrie.tuwien.ac.at GEOMETRIE Tsai‘s closest point solver Computes the distance function to a set S on a grid. Uses 4-neighborhood of (i,j) 1. Initialization: For grid points g close to S compute and store the exact distance d(g) and a closest point g* on S. These grid points are marked and not updated anymore. The other grid points get distance value ∞

17. 17 www.geometrie.tuwien.ac.at GEOMETRIE Tsai‘s closest point solver 2. Sweeping: in each of the four sweeps, visit each grid point e that can be updated: A) For each neighbor p l of e compute B) If set (enforces monotonicity) C) distance of current grid point e is set to d 2 (e) =min l d l tmp =:d m tmp and the closest point e* to e is set to e*=p m *.

18. 18 www.geometrie.tuwien.ac.at GEOMETRIE Zhao‘s fast sweeping Computes the distance value at a grid point e only from the distance values of the 4 neighbors (no closest points used) Key idea: use only two neighbors p 1,p 2 (depending on the sweep) and estimate a distance value d(e) from their distances d 1,d 2 by a local approximation of the distance function by the distance function of a straight line L.

19. 19 www.geometrie.tuwien.ac.at GEOMETRIE Zhao‘s fast sweeping For a (x+,y+) sweep: e p1p1 p2p2 L

20. 20 www.geometrie.tuwien.ac.at GEOMETRIE Zhao‘s fast sweeping Updating formulae (h=gridsize) (a) if, set (b) if, set

21. 21 www.geometrie.tuwien.ac.at GEOMETRIE Comparison of algorithms Zhao Tsai

22. 22 www.geometrie.tuwien.ac.at GEOMETRIE Distance fields in the presence of obstacles The fast sweeping algorithm of Zhao can compute the distance function, considering given obstacles Set a flag to grid points inside obstacles and do not use them for updating

23. 23 www.geometrie.tuwien.ac.at GEOMETRIE L. 11: Geodesics Computation Applications in civil engineering image processing 3D vision Robotics