Atelier APMEP Octobre 2008

1. Atelier APMEPOctobre 2008 Pr. Eric Andres Université de Poitiers

3. Visualisation Surfacique Visualisation Volumique Quelles sont les différences ?

4. Quelles sont les différences ? Visualisation Surfacique Visualisation Volumique

5. Droite et Hyperplans discrets

6. Discrete Analytical Geometry Discrete Analytical Line definition J.-P. Reveillès (1991) Representation in comprehension Analytical equation :

7. Exercice 1 (**) quelle est la « meilleure » des deux discrétisations ? Introduction

8. Corrigé Exercice 1 quelle est la « meilleure » des deux discrétisations ? Introduction

9. 0  2x+5y+9z < 16 Discrete analytical hyperplane definition J.-P. Reveillès (1991) 0  2x+5y+9z < 9 Representation in comprehension Analytical equation : Arithmetical thickness : w = B - A

10. Application en Géologie ARGILE GRANIT PETROLE ARGILE X SABLE EAU Hyperplans

11. Application en Géologie ARGILE GRANIT PETROLE ARGILE X SABLE EAU Hyperplans

12. Application en Géologie ARGILE GRANIT PETROLE ARGILE X SABLE EAU Hyperplans

13. Application en Géologie ARGILE GRANIT PETROLE ARGILE X SABLE EAU Hyperplans

14. Soit l'hyperplan euclidien Alors si : x1 Application en géologie • Hyperplan discret de localisation : Épaisseur arithmétique : w = a1 Hyperplans

15. Coupe Oblique [Andres 1996]

16. Coupe Oblique Plan de Coupe P : ax+by+cz+d = 0 Examinons le problème en 2D: ax+by+c=0 ax+by+c = -(a+b)/2 ax+by+c = -(a+b)/2 ax+by+c  0 ax+by+c = 0 ax+by+c = +(a+b)/2

17. La valeure R(x,y,z) = ax+by+cz+d+ détermine la coupe. Coupe Oblique Plan de Coupe P : ax+by+cz+d = 0 Voxels coupés : St(P) : -  ax+by+cz+d  < Plan 3D supercouverture standard

18. Coupe Oblique C A B D E

19. Coupe Oblique Discrète P : 0  ax+by+cz+d+ < a+b+c avec 0  a  b  c, m = min(c,a+b), M = max(c,a+b) A(x,y,z) un point de P et pol(A) = vox(A)  P, alors : • si 0< r(A) <a alors pol(A) est de type A(0) • si a < r(A) < b alors pol(A) est de type B(0,3) • si b < r(A) < m alors pol(A) est de type C(0,1,3) • si m < r(A) < M et M=c alors pol(A) est de type D(0,1,2,3) • si m < r(A) < M et m=c alors pol(A) est de typeE(0,1,3,4) Coupe Oblique • si M < r(A) < a+c alors pol(A) est de typeC(4,6,7) • si a+c < r(A) < b+c alors pol(A) est de typeB(4,7) • si b+c < r(A) < a+b+c alors pol(A) est de typeA(6)

20. Coupe Oblique • Avec cette approche arithmétique, nous pouvons : • déterminer comment un voxel est coupé • déterminer comment les arêtes sont coupées • montrer qu’il y a |a|+|b|+|c| polygones de coupes différents • montrer que les coupes parallèles sont similaires

21. Hyperplan pythagoricien Épaisseur arithmétique : • Droite pythagoricienne avec Épaisseur arithmétique : w = |b| + 1 Permet de définir une rotation discrète bijective Différents types d'hyperplans [Andres 1992] Hyperplans

22. Soit l'hyperplan euclidien Alors : Différents types d'hyperplans • Hyperplan supercouverture [Andres 1996] Épaisseur arithmétique : ou Hyperplans

23. Application example: Discrete Analytical Ridgelet Transform Point border Ridgelet domain Domaine de Radon Radon Transform Radon domain Wavelet transform Image Idea : points and lines are linked via the Radon transform The ridgelet transform has been specifically invented to efficiently represent edges (borders of regions) in an image [Candès98] [Carré & Andres 2000-2006]

24. Definition of radial lines passing thru the origin 2D Fourier transform of the image Inverse 1D Fourier transform of the 1D lines Definition of the 2D ridgelet transform The Radon transform Pixels are summed along a direction Classic strategy Fourier strategy

25. Discrete Analytical lines FFT 2D Extraction of the Fourier coefficients Discrete geometry approach Computing strategy for the 2D DART Fourier coefficients Extraction of the Fourier coefficients iFFT Image projection 1D wavelet transform of the 1D vectors Ridgelet [Carré&Andres2002]

26. Discrete Analytical Radon Transform The discrete analytical lines we used for the transform are defined by: with [p,q] the direction of the Radon projection and , function of (p,q), the arithmetical thickness Closed naïve lines (8-connected) Supercover lines (4-connected) Closed pythagorean lines (8-connected)

27. Naïve line Pythagorean line Supercover line 3D discrete analytical Radon transform Definition of 3D discrete analytical lines

28. Discrete planes z, t y x Other 3D line definition Discrete Analytical planes naïve supercover pythagorean

29. Other 3D line definition The 3D planes are seen as a 2D space The projection of the plane is mapped by 2D lines y x z Final definition of the 3D lines:

30. 3D Denoising of a MRI Noisy image Original image Denoising with a wavelet transform DART denoising

31. Color video denoising Original video

32. Color Video denoising Noisy video 3D DART video denoising

33. Color Video denoising Noisy video 3D DART video denoising

34. Offset zone Supercover with arbitrary thickness Discrete Analytical Model: the Supercover model [Andres 2000]

35. S(e1(F)) : 1.5  y + 2z  1.5 S(e2(F)) : -5  3x + 7z  5 S(e3(F)) : -6.5  6x – 7y  6.5 Example : 3D Supercover line Line (0,0,0)-(7,6,-3)

36. Properties

37. Representation in comprehension 12 inequations Example : Supercover 3D line 1.5  x2 + 2 x3 1.5 -5  3 x1 + 7 x3 5 -6.5  6 x1 – 7 x2 6.5 -0.5  x1 7.5 -0.5  x2 6.5 -3.5  x3 0.5

38. Representation in comprehension 17 inequations Example : supercover 3D triangle -1/2  x  19/2 -1/2  y  17/2 -1/2  z  9/2 7x + 6y  151/2 x - 9y  5 -8x + 3y  11/2 x - 9z  5 -4x + 3z  7/2 x + 2z  25/2 y - 2z  3/2 -3y + 7z  9 -y + z  1 -53  4x + 33y - 69z  53

39. - 0.5 x  0.5 -0.5 y  0.5 A - 0.5 z  0.5 2.5 x  3.5 7.5 y  8.5 B 3.5 z  4.5 B(3,8,4) Y 8.5 x  9.5 0.5 y  1.5 C 0.5 z  1.5 Z -5.5 - 8x + 3y  5.5 - 6 - 4y + 8z  6 AB - 3.5 - 3z + 4x  3.5 62.5 7x + 6y  75.5 - 9 3y – 7z  1 BC 28.5 6z + 3x  37.5 - 5 - x + 9y  5 - 1 - y + z  1 AC - 5 - 9z + x  5 C(9,1,1) ABC X - 53  4x + 33y – 69z  53 A(0,0,0) Example : triangle 3D - 53  4x + 33y – 69z  53 (Slide made by M. Dexet)

40. Example in R5

41. Analytical description

42. Discrete modeling Voxel view Analytical view Euclidean view

43. Discrete modeling in grey levels

44. Discrete Analytical Hough Transform “Dual” of a pixel and a voxel Work done by Martine Dexet

45. Discrete Analytical Hough Transform Recognition of a small line segment Work done by Martine Dexet

46. Analytical continuation : 2D contour reconstruction Work done by Rodolphe Breton

47. Pixel level with a segmented image Region level Analytical level After a reconstruction phase Continuous level with pixels Continuous Level Final view Discrete Modeling Software MODELER ARCHITECTURE SPAMOD :Spatial Modeler Work done by Martine Dexet

48. Illustrations Level 0: Original image before segmentation

49. Illustrations Level 1: Region level

50. Illustrations Level 2: Analytical level