Fundamental Algorithms and Advanced Data Representations

Transcription

1 Fundamental Algorithms and Advanced Data Representations Anders Hast

2 Outline Isosurfaces (Volume Data) Cuberille Contouring Marching Squares Linear Interpolation methods Marching Cubes Non Linear Interpolation

3 Iso-Lines A line going through data having the same value (iso = same) Examples

4 Iso-Surface A surface where the volume data has the same value

5 Isosurfaces in VTK from vtk import * # Read the volume reader = vtkstructuredpointsreader() reader.setfilename("liver.vtk") # Isosurface isosurface = vtkcontourfilter() isosurface.setinputconnection(reader.getoutputport()) isosurface.setvalue(0, 160)

6 VTK... vtkcontourfilter vtkmarchingcontourfilter vtkmarchingcubes vtkslicecubes vtkmarchingsquares vtkimagemarchingcubes

7 Countouring Volumes are often Cut Sections from the scan Each slice can be segmented into a surface using an iso-value Hence we get an inside and an outside The iso-lines in neighbouring planes can be connected by triangulation

8 Example Source:

9 Cuberille Each voxel is a cube Source: Ideal Iso-Surface Cuberille Surface

10 Contour by Segmentation Threshold >=

11 Contouring (Iso-Lines) Just thresholding gives a contour similar to Cuberille We can do better by Marching Squares Let the template march through the slice Cut the surface with the lines in the template

12 Marching Squares Threshold >=

13 Marching Squares Threshold >=

14 Marching Squares Threshold >=

15 Marching Squares Threshold >=

16 Marching Squares Threshold >=

17 Marching Squares Threshold >=

18 Marching Squares Threshold >=

19 Marching Squares Threshold >=

20 Marching Squares Threshold >=

21 Marching Squares Threshold >=

22 Marching Squares Threshold >=

23 Marching Squares Threshold >=

24 Marching Squares Threshold >=

25 Marching Squares Threshold >=

26 Marching Squares Threshold >=

27 Marching Squares Threshold >=

28 Marching Squares Threshold >=

29 Result A Better outline (iso-line) of the object! But we can do even better

30 Linear Interpolation Can be used to do better line cutting to obtain a better fit of the iso-surface. Linear Interpolation: u varies from 0 to 1. Expand The line equation!!!

31 Parametric Lines Linear Interpolation p(u)=p0(1-u)+p1(u) or p(u)=vu+p0 v=p1- p0 p0 p(u) p1 v p1 p0

32 Bilinear Interpolation Linear Interpolation pa(u)=p0(1-u)+p1(u) pb(u)=p3(1-u)+p2(u) p(v)=pa(1-v)+pb(v) v p0 p u pa u pb p1 v p2 p3 u u

33 Bilinear Interpolation over a Polygon Interpolate between a and b to get ab between a and c to get ac upper part: between ab and ac to get d lower part: between bc and ac to get d a b ab d ac c

34 Iso-Surface Interpolation Between two values a and b using the threshold t How long between a and b? u=(a-t)/(a-b) use u in the linear interpolation equation Example t=5, u=(7-5)/(7-4)=2/3 (a=7, b=4)

35 Marching Squares Interpolated t=5, no cut

36 Marching Squares t=5, u=(5-5)/(5-4)=

37 Marching Squares t=5, u=(7-5)/(7-4)=2/

38 Marching Squares t=5, u=(7-5)/(7-2)=2/

39 Marching Squares t=5, u=(5-5)/(5-3)=

40 Marching Squares t=5, u=(5-5)/(5-4)=

41 Marching Squares t=5, no cut

42 Marching Squares t=5, u=(8-5)/(8-3)=3/

43 Marching Squares t=5, u=(6-5)/(6-2)=1/

44 Marching Squares t=5, no cut

45 Marching Squares t=5, no cut

46 Marching Squares t=5, u=(9-5)/(9-4)=4/

47 Marching Squares t=5, u=(6-5)/(6-4)=1/

48 Marching Squares t=5, u=(6-5)/(6-4)=1/

49 Marching Squares t=5, u=(9-5)/(9-3)=2/

50 Marching Squares t=5, u=(9-5)/(9-4)=4/

51 Marching Squares Threshold >=

52 No Interpolation vs. Linear Interpolation The Iso-contour is more exact

53 Concluisions Cuberille gives a lego -like appearance Marching Squares can produce well cut segments in each slice The quality of the iso-surface can be increased by linear interpolation Then each slice must be attached to each neighbour by triangulation BUT, there are ambiguities we have not discussed yet...

54 Ambiguity Threshold >= 5 Is this one or two objects?

55 Ambiguity Threshold >= 5 According to the template there are two

56 Ambiguity Threshold >= 5 If we change the template it is one Note!

57 Ambiguities Both cases are possible Which one to choose depends... Choose one of them and stick to that choice! Better would be to make the choice depending on the neigbouring slices. Don t split the object if it is one object in both neigbouring slices etc...

58 Increased Quality Interpolation of the data while marching This have already been shown Supersampling Magnification and Interpolation Then do the contour tracking This can be combined with the already shown interpolation

59 Segmentation Threshold 5.0

60 Supersampling Finer Mesh magnify 4x4 Bilinear Interpolation

61 Segmentation Threshold 5.0

62 The Object changes with the Threshold Threshold 4.5

63 The Object changes with the Threshold Threshold 5.5

64 Triangulation Which contour vertex in slice n shall be connected to the vertices of slice n+1? Constrained Delaunay? etc... Slice: n Slice: n+1

65 Bifurcation How do we handle such a case? Slice: n Slice: n+1

66 Bifurcation The triangulation must be able to handle cases when the number of objects differ from slice to slice Leads to triangle intersections Not a big problem... The 3D Version of Marching Squares does not lead to triangle intersections: Marching Cubes

67 Bifurcation Source:

68 Marching Cubes Lorensen et al Ambiguities Ugly meshes but very popular Similar techniques Marching Tetrahedrons No ambiguities More triangles Meshes can be improved by edge collapse/contraction

69 Marching Cubes 8 neighbouring voxels can be intersected in several ways.

70 Marching Cubes Produces more triangles than the triangulation of cut slices However leads to complicated ambiguities Cannot be solved by choosing a strategy Marching Tetrahedra is the remedy for this Leads to even more triangles...

71 Result The ugly triangles can be corrected Marching Cubes Edge Contracted

72 Marching Tetrahedron 4 points in the cell Only 8 Cases!

73 Conclusion Nowadays Marching Cubes is often used or Marching Tetrahedra! The latter is straightforward to implement! Bifurcation is handled automatically

74 Non Linear Interpolation Splines are often used to interpolate data They are polynomials often second or third degree Polynomials Two Points: Linear Interpolation Three points: Quadratic Interpoaltion Four points: Cubic Interpolation and so forth Larger degree does not necessarily give better interpolation!

75 Quadratic interpolation p(u)= au 2 +bu+c Solve the system of equations to obtain the coefficients a p0 1/4 1/2 1 b = p c p2 This curve is defined between p0 and p1 for u=[0..1] p0 p1 p2

76 Quadratic Interpolation Can be used on triangles Must have 6 points of data Quads needs 8 points of data

77 Conclusions Iso-surface triangulations can be done by: Marching squares and triangualtion of slices Marching Cubes; gives full triangulation but generally more triangles Marching Tetrahedron Even more triangles, but simpler and no ambiguities Quality can be enhanced by interpolation supersampling