About Digital Level Layers

1. About Digital Level Layers GT Géométrie Discrète, 03/12/2010 Yan Gerard & Laurent Provot ISIT, Clermont Universités gerard.research@gmail.com provot.research@gmail.com

2. Outline I Linear Primitives II Unlinear Primitives III Some Applications of DLL IV Algorithms

3. I Linear Primitives

4. digital straight line

5. digital plane and more generally digital hyperplanes of Zd

6. Digital hyperplanes of Zd have at least 3 definitions Algebra Topology Morphology The boundary of the lattice points in the half-space of equation a.x<h

7. Digital hyperplanes of Zd have at least 3 definitions Algebra Topology Morphology Structuring element The track on Zd of a Minskowski sum H+Structuring Element

8. Digital hyperplanes of Zd have at least 3 definitions Algebra Topology Morphology Structuring element ball N0 ball N1 ball N2 The track on Zd of a Minskowski sum H+Structuring Element segments

9. Digital hyperplanes of Zd have at least 3 definitions Algebra Topology Morphology The lattice points in an affine strip of double equation h< a.x <h’

10. Digital hyperplanes of Zd have at least 3 definitions Algebra Topology Morphology Neighborhood Structuring element value h’-h Parameters

11. Digital hyperplanes of Zd have at least 3 definitions Algebra Topology Morphology Neighborhood Structuring element value h’-h h’-h=N (a) Ball N Ball N1 8 8 h’-h=N1 (a) Ball N Ball N1 8 More generally h’-h=N* (a) Ball N Ball N ? The three definitions collapse But what about unlinear primitives ?

12. II Unlinear Primitives

13. Let S be a continuous level set of equation f(x)=0 Problem: define a digital primitive for S.

14. Three approaches Problem: define a digital primitive for S.

15. Three approaches Algebra Topology Morphology

16. Three approaches Algebra Topology Morphology Structuring element

17. Three approaches Algebra Topology Morphology We consider the lattice points between two ellipses f(x)=h et f(x)=h’

18. Three approaches Algebra Topology Morphology Advantages and drawbacks ? The three approaches are equivalent for linear structure but not for unlinear shapes

19. Three approaches Topology Morphology Algebra Properties Advantages and drawbacks ? Topology Morphology Algebraic characterization Recognition algorithm

20. Three approaches Topology Morphology Algebra Properties Topology Morphology Algebraic characterization Recognition algorithm SVM

21. Algebra Topology Morphology

22. Algebra Topology This kind of primitives is not a surface!!!!!! Morphology Definition: The lattice set characterized by a double-inequality h<f(x)<h’ is called a Digital Level Layer (DLL for short).

23. III Some Applications of DLL

24. Estimation of the kth derivative of a digital function Previous works : Error Bounding A. Vialard, J-O Lachaud, F De Vieilleville O(h1/3) for k=1 An approximation based on maximal straight segments S. Fourey, F. Brunet, A. Esbelin, R. Malgouyres k O(h(2/3) ) for k An approximation based on convolutions L. Provot, Y. G O(h(1/(k+1)) ) for k An approximation based on DLL Recognition

25. Estimation of the kth derivative of a digital function Principle : Input: Points

26. Estimation of the kth derivative of a digital function Principle : Input: Points + Vertical thickness (or maximal roughness)>1

27. Estimation of the kth derivative of a digital function Principle : Polynomial of degree ≤ k Input: Points + order k + Vertical thickness (or maximal roughness)>1

28. Estimation of the kth derivative of a digital function Principle : Polynomial of degree ≤ k Output: DLL of double-inequation -roughness ≤ y-P(x) ≤ +roughness containing S the derivative of P(x) as digital derivative

29. Estimation of the kth derivative of a digital function Previous works : Error Bounding A. Vialard, J-O Lachaud, F De Vieilleville O(h1/3) for k=1 An approximation based on maximal straight segments Increase the degree Relax the maximal vertical S. Fourey, F. Brunet, A. Esbelin, R. Malgouyres k O(h(2/3) ) for k An approximation based on convolutions thickness L. Provot, Y. G O(h(1/(k+1)) ) for k An approximation based on DLL Recognition Different general algorithms (chords or GJK)…

33. Second derivative

34. Second derivative

35. Vectorization of Digital Shapes Principle : Alternative ? Digitization Recognition Input: Lattice set S DLL containing S Undesired neighbors

36. Vectorization of Digital Shapes Principle : Digitization Recognition Input: Lattice set S DLL containing S Undesired neighbors + Recognition DLL between the inliers and outliers Forbidden neighbors

37. IV Algorithms

38. Recognition of topological surfaces well-known in the framework of Support Vector Machine (Kernel trick: Aizerman et al. 1964) or Computational Geometry Problem of separation by a level set f(x)=0 with f in a given linear space Problem of linear separability in a descriptive space GJK computes the closest pair of points from the two convex hulls

39. Recognition of DLL with forbidden points Thank you for your attention Problem of separation by two level sets f(x)=h and f(x)=h’ with f in a given linear space Problem of linear separability by two parallel hyperplanes We introduce a variant of GJK in nD