1. Marching Cubes A High Resolution 3D Surface Construction Algorithm

2. Slice Data to Volumetric Data(1)

3. Slice Data to Volumetric Data(2)

4. Marching Cube Create cells (cubes) Classify each vertex Build an index Get edge list Based on table look-up Interpolate triangle vertices Obtain polygon list and do shading in image space

5. Cube Consider a cube defined by 8 data values, 4 from slice k, and another 4 from slice k+1

6. Classify each vertex Label 1 or 0 as to whether it lies inside or outside the surface Match!!! 0 1 0 1 1 0 0 0

7. Build an index Create an index of 8 bits from the binary labeling of each vertex.

8. Get edge list Give an index, store a list of edges. Because symmetry ： 256/2=128 rotation ： 128/8=16 256 cases are reduced to 14 cases.

9. Interpolate triangle vertices X = i + ii+1X i X = 20 = 10 iso_value=18iso_value=14 (iso_value - D(i)) (D(i+1) - D(i))

10. Problems about MC Empty cells 30-70% of isosurface generation time was spent in examining empty cells. Speed Ambiguity

11. The Asymptotic Decider Resolving the Ambiguity in Marching Cubes

12. Ambiguity Problem (1) Ambiguous Face : a face that has two diagonally oppsed points with the same sign + +

13. Ambiguity Problem (2) Certain Marching Cubes cases have more than one possible triangulation. Case 6 Case 3 Mismatch!!! + + + + Hole!

14. Ambiguity Problem (3) To fix it … Case 6 Case 3 B Match!!! + + + + The goal is to come up with a consistent triangulation

15. Asymptotic Decider (1) Based on bilinear interpolation over faces B01 B00B10 B11 (s,t) B(s,t) = (1-s, s) B00 B01 B10 B11 1-t t The contour curves of B: {(s,t) | B(s,t) =  } are hyperbolas = B00(1- s)(1- t) + B10(s)(1- t) + B01(1- s)(t) + B11(s)(t)

16. Asymptotic Decider (2) (0,0) (1,1) Asymptote (S  T  If B(S  T  >=  (S  T  Not Separated

17. Asymptotic Decider (3) (1,1) Asymptote (S  T  (0,0) If B(S  T  <  (S  T  Separated

18. Asymptotic Decider (4) (S 1, 1) (S  T  (S 0, 0) S  B00 - B01 B00 + B11 – B01 – B10 T  B00 – B10 B00 + B11 – B01 – B10 B(S  T  B00 B11 + B10 B01 B00 + B11 – B01 – B10 (0, T 0 ) (1, T 1 ) B( S  ) = B( S , 1) B( 0, T  ) = B( 1, T  )

19. Asymptotic Decider (5) case 3, 6, 12, 10, 7, 13 (These are the cases with at least one ambiguious faces)