A reversible and statistical method for discrete surfaces smoothing

1. A reversible and statistical method for discrete surfaces smoothing Bertrand Kerautret Achille Braquelaire Computers & Graphics 30 (2006) 54-61

2. Outline • About Author • Introduction • Related work • Some definitions • Main strategy • conclusions

3. About Author • Bertrand Kerautret Ph. D., Teaching and research assistant in LaBRI , Université Bordeaux I • research: Reconstruction and smoothing of discrete Surfaces

4. About author • Achille Braquelaire a researcher in LaBRI , Université Bordeaux I • Research interest: digital geometry and surface reconstruction

5. Introduction • Processing data sets of Three-dimensional discrete images bring up the problem of extraction and representation of the surface of 3D objects. • use a polygonal representation to represent the boundary of the discrete object.

6. Related work • The first approach is the marching cube algorithm • Marching cubes:a height resolution 3d surface reconstruction algorithm(siggraph 87) • Triangulate the surface by associating centers of voxels to each other • polygon mesh generation for discrete surfaces in 3d space Euro graphics 1997 • Euclidean Nets • Automatic and reversible geometric smoothing of the boundary of a discrete 3d object 2000 DGCI

7. Some definitins • Discrete point& Euclidean point • Discrete plane • Discrete net& Euclidean net

8. Discrete point & Euclidean point • Discrete point: • Euclidean point: let P be a discrete point, then q is a Euclidean point if q belongs to the cell of P.

9. Discrete plane • The discrete plane is the set of points (x,y,z) satisfying the double inequality

10. Discrete net& Euclidean net • Discrete net: a graph associated with a connected component of the polygonal decomposition of the boundary of a 3d object • Euclidean net: the graph obtained by replacing each discrete point by an associated Euclidean point.

11. Main strategy • Find a tangent plane • Project the discrete point to the plane and get the Euclidean point • Reconstruct surface from the Euclidean points.

12. Find a tangent plane • Estimate tangent plane’s normal • Tangent plane’s position

13. Estimate tangent plane’s normal • Considering a random lignels draw in first quadrant ,the probability of type1 and type2 is

14. Estimate tangent plane’s normal Considering a discrete plane in the first 8th space ,the probability to obtain different surfel type is

15. Select surfels around point Random selection Numbers of surfels raws

16. Estimate tangent plane’s normal this method

17. Position of the tangent

18. Surface reconstruction • First we define the real projection plane • The new point resulting from the projection is where

19. Surface reconstruction The Euclidean net after moved the discrete point

20. Experiments

21. Experiments (a) represents the initial surface nets. (b) rendered with flat shading (c) Rendered with gouraud shading.

22. Conclusion • The new geometric and statistical method to smooth discrete surfaces works very well both on the visual and geometric points of view • The resulting surface representation can be used for discrete surface rendering and for geometrical properties extraction

23. The end thank you