1. DEpthLAUNAY1 DEpthLAUNAY www.dma.fi.upm.es/mabellanas/delonedepth/ Manuel Abellanas Alfredo de las Vegas Facultad de Informática - UPM

2. DEpthLAUNAY2 Starting point Article: “Point set stratification and Delaunay depth” Written by Manuel Abellanas, Mercè Claverol & Ferrán Hurtado Available at: http://arxiv.org/abs/cs.CG/0505017 Notions of depth in 2D data analysis Convex and Delaunay depth

3. DEpthLAUNAY3 Convex hull The smallest convex region enclosing a specified group of points.

4. DEpthLAUNAY4 Convex depth Recursively eliminates points of convex hull and recalculate it.

5. DEpthLAUNAY5 Delaunay property Every triangle has a circumscribed circle. The Delaunay property forces that all of those circles must be empty of points.

6. DEpthLAUNAY6 Delaunay Triangulation The Delaunay triangulation is a collection of edges satisfying an "empty circle“ or Delaunay property. Also maximizes the angles.

7. DEpthLAUNAY7 Delaunay depth Delaunay depth of a point is the minimum number of edges used to reach the convex hull.

8. DEpthLAUNAY8 Delaunay layers Delaunay layers are the subgraphs of DT(S) induced by the points of S with same Delaunay depth

9. DEpthLAUNAY9 Delaunay levels Delaunay levels are regions of plane formed with points with a determined depth

10. DEpthLAUNAY10 Voronoi diagram Voronoi cells are the set of points closer to a vertex than any other vertex. Voronoi diagram is the partition induced by Voronoi cells. Dual of Delaunay triangulation

11. DEpthLAUNAY11 Voronoi diagram Voronoi cells are the set of points closer to a vertex than any other vertex. Voronoi diagram is the partition induced by Voronoi cells. Dual of Delaunay triangulation

12. DEpthLAUNAY12 Voronoi diagram by depth A Voronoi diagram can be coloured based of Delaunay depth of generator points

13. DEpthLAUNAY13 Goals of DEpthLAUNAY Teaching tool Help in research Interactivity Use of C.G.A.L. Efficient tool to compute depths ~ O(n log n) Runs in Windows and Linux (with wine)

14. DEpthLAUNAY14 Computational Geometry Algorithms Library www.cgal.org

15. DEpthLAUNAY15 CGAL The goal of the CGAL Open Source Project is to provide easy access to efficient and reliable geometric algorithms to users in industry and academia in the form of a C++ library.

16. DEpthLAUNAY16 CGAL CGAL, the Computational Geometry Algorithms Library, offers data structures and algorithms like triangulations (2D constrained triangulations and Delaunay triangulations in 2D and 3D), Voronoi diagrams (for 2D and 3D points, 2D additively weighted Voronoi diagrams, and segment Voronoi diagrams), Boolean operations on polygons and polyhedral, arrangements of curves, mesh algorithms (2D Delaunay mesh generation and 3D surface mesh generation, surface mesh subdivision and parameterization), alpha shapes, convex hull algorithms (in 2D, 3D and dD), operations on polygons (straight skeleton and offset polygon), search structures (kd trees for nearest neighbour search, and range and segment trees), interpolation (natural neighbour interpolation and placement of streamlines), optimization algorithms (smallest enclosing sphere of points or spheres, smallest enclosing ellipsoid of points, principal component analysis), and kinetic data structures. All these data structures and algorithms operate on geometric objects like points and segments, and perform geometric tests on them. These objects and predicates are regrouped in CGAL Kernels.

17. DEpthLAUNAY17 Algorithm of DT CGAL uses “incremental insertion algorithm” to triangulate a set of points. When a point is inserted, triangulates it and test the Delaunay property. If fails, flip and propagate.

18. DEpthLAUNAY18 DEpthLAUNAY Three layers design Only Presentation layer is Operative System dependant CNube class is a wrapper of CGAL, so the programmer doesn’t need to know CGAL.

19. DEpthLAUNAY19 DEpthLAUNAY Divided in three parts from user’s point of view: –Entering data (mouse, keyboard, images, automatic generation) –Manipulating data. Various types of diagrams (real-time depending of number of points) (Use of clipboard) –Exporting data (four different file formats, image)

20. DEpthLAUNAY20 Using DEpthLAUNAY

21. DEpthLAUNAY21 Parts of window Menus Tool bar Status bar Main view

22. DEpthLAUNAY22 Entering data Mouse (add, delete, move) Keyboard (absolute, relative) File (various formats) Automatic generation (regular, random) (quick, slow) Image.

23. DEpthLAUNAY23 Data manipulation Clipboard (cut, copy, paste) Shake vertex Compute structures (alone and together) Cloud (scroll, zoom, Canvas, cloud move, cloud adjust) Show paths (Crtl +Right mouse) Preferences

24. DEpthLAUNAY24 Export data Image (with or without scroll bars) File formats: –OFF (Geomview) –VTC (CGAL) –PNT (Simple) –NIV (All info) View data in 3D with external program Spring program (muelles) Thanks in movement