1. Advanced Iso-Surfacing Algorithms Jian Huang, CS594, Spring 2002 This set of slides are developed and used by Prof. Han-Wei Shen at Ohio State University.

2. Iso-contour/surface Extractions 2D Iso-contour 3D Iso-surface

3. Iso-contour (0) Remember bi-linear interpolation p2 p3 p0 p1 P =? p4 p5 To know the value of P, we can first compute p4 and P5 and then linearly interpolate P

4. Iso-contour (1) Consider a simple case: one cell data set The problem of extracting an iso-contour is an inverse of value interpolation. That is: p2 p3 p0 p1 Given f(p0)=v0, f(p1)=v1, f(p2)=v2, f(p3)=v3 Find the point(s) P within the cell that have values F(p) = C

5. Iso-contour (2) p2 p3 p0 p1 We can solve the problem based on linear interpolation (1) Identify edges that contain points P that have value f(P) = C (2) Calculate the positions of P (3) Connect the points with lines

6. Iso-contouring – Step 1 (1) Identify edges that contain points P that have value f(P) = C v1v2 If v1 < C < v2 then the edge contains such a point

7. Iso-contouring – Step 2 (2) Calculate the position of P Use linear interpolation: P = P1 + (C-v1)/(v2-v1) * (P2 – P1) v1v2 Pp1 p2 C

8. Iso-contouring – Step 3 p2 p3 p0 p1 Connect the points with line(s) Based on the principle of linear variation, all the points on the line have values equal C

9. Inside or Outside? Just a naming convention 1.If a value is smaller than the iso-value, we call it “Inside” 2.If a value is greater than the iso-value, we call it “Outside” p2 p3 p0 p1 - + outside cell p2 p3 p0 p1 - inside cell

10. Extend the same divide-and-conquer algorithm to three dimension 3D cells Look at one cell at a time Let’s only focus on voxel Iso-surface Extraction

11. Divide and Conquer _ + + + + _ _ _ + + + + _ _ _ _ (2 triangles)

12. How many cases? Now we have 8 vertices So it is: 2 = 256 8 How many unique topological cases?

13. Case Reduction (1) Value Symmetry + + __ _ _ _ _ + + _ _ + + + +

14. Case Reduction (2) Rotation Symmetry + + __ _ _ _ _ _ _ + + _ _ _ _ By inspection, we can reduce 256 14

15. Iso-surface Cases Total number of cases: 14 + 3

16. Marching Cubes Algorithm A Divide-and-Conquer Algorithm v1 v2 v3v4 v5 v6 v7v8 Vi is ‘1’ or ‘0’ (one bit) 1: > C; 0: <C (C= iso-value) Each cell has an index mapped to a value ranged [0,255] Index = v8 v7 v6 v5 v4 v3 v2 v1

17. Marching Cubes (2) Given the index for each cell, a table lookup is performed to identify the edges that has intersections with the iso-surface 0 1 2 3 14 e1, e3, e5 … Index intersection edges e1 e2 e3 e4 e5 e6 e7 e8 e9e10 e11 e12

18. Marching Cubes (3) + + + + _ _ _ _ Perform linear interpolations at the edges to calculate the intersection points Connect the points

19. Why is it called marching cubes? Linear search through cells Row by row, layer by layer Reuse the interpolated points for adjacent cells

20. Iso-surface cells: cells that contain iso- surface. min < iso-value < max Marching cubes algorithm performs a linear search to locate the iso-surface cells – not very efficient for large-scale data sets. Iso-surface cell search

21. Iso-surface Cells For a given iso-value, only a smaller portion of cells are iso-surface cell. For a volume with n x n x n cells, the average number of the iso-surface cells is O(n x n) (ratio of surface v.s. volume) n n n

22. Efficient iso-surface cell search Problem statement: Given a scalar field with N cells, c 1, c 2, …, c n, with min-max ranges (a 1,b 1 ), (a 2,b 2 ), …, (a n, b n ) Find {C k | a k < C < b k ; C=iso-value}

23. Efficient search methods 1.Spatial subdivision (domain search) 2.Value subdivision (range search) 3.Contour propagation

24. Domain search Subdivide the space into several sub-domains, check the min/max values for each sub-domain If the min/max values (extreme values) do not contain the iso-value, we skip the entire region Min/max Complexity = O(Klog(n/k))

25. Range Search (1) Subdivide the cells based on their min/max ranges Global minimumGlobal maximum Isovalue Hierarchically subdivide the cells based on their min/max ranges

26. Range Search (2) Within each subinterval, there are more than one cells To further improve the search speed, we sort them. Sort by what ? Min and Max values Max Min M5 M2 M6 M4 M1 M3 M7 M8 M11 M10 M9 m5 m1 m6 m3 m8 m7 m2 m9 m11 m4 m10 G1 G2 Isosurface cells = G1 G2

27. Range Search (3) ? A clean range subdivision is difficult … Difficult to get an optimal speed

28. Range Search: Span Space Span Space: Instead of treating each cell as a range, we can treat it as a 2D point at (min, max) This space consists of min and max axes is called span space Any problem here?

29. Span Space min max What are the iso-surface cells? C How to search them?

30. Span Space Search (1) With the point representation, subdividing the space is much easier now. Search method 1: K-D tree subdivision (NOISE algorithm) K-d tree: A multi-dimensional version of binary tree Partition the data by alternating between each each of the dimensions at each level of the tree

31. NOISE Algorithm (K-d tree) left right up down ……… Construction min max * One node per cell Min Max ? Median point

32. NOISE Algorithm (Query) Complexity = O( N + k) left right up down ……… Min Max ? Median point If ( iso-value < root.min ) check the ?? Sub-tree If (iso-value > root.min) Check the ?? Sub-tree Don’t forget to check the root‘s interval as well.

33. Span Space Search (2) Search Method (2): ISSUE, discretized span space O(log(N/L)) O(1) ? Complexity = ?

34. Range Search: Interval Tree Interval Tree: II I left I right Sort all the data points (x1,x2,x3,x4,…., xn) Let  = x  mid point) n/2 We use  to divide the cells into three sets I  I left, and I right I  cells that have min <  max I left: cells that have max <  I right: cells that have min >  … …

35. Interval Tree II I left I right …… I  cells that have min <  max I left: cells that have max <  I right: cells that have min >  Now, given an isovalue C 1)If C <  2)If C >  3) If C =  Complexity = O(log(n)+k) Optimal!!

36. Range Search Methods In general, range search methods all are super fast – two orders of magnitude faster than the marching cubes algorithm in terms of cell search But they all suffer a common problem … Excessive extra memory requirement!!!

37. Basic Idea: Given an initial cell that contains iso-surface, the remainder of the iso-surface can be found by propagation Contour Propagation A B D C E Initial cell: A Enqueue: B, C Dequeue: B Enqueue: D … FIFO Queue A B C C C D …. Breadth-First Search

38. Challenges Need to know the initial cells! For any given iso-value C, finding the initial cells to start the propagation is almost as hard as finding the iso-surface cells. You could do a global search, but …

39. Solutions (1)Extrema Graph (Itoh vis’95) (2)Seed Sets (Bajaj volvis’96) Problem Statement: Given a scalar field with a cell set G, find a subset S G, such that for any given iso-value C, the set S contains initial cells to start the propagation. We need search through S, but S is usually (hopefully) much smaller than G. We will only talk about extrema graph due to time constraint

40. Extrema Graph (1) Basic Idea: If we find all the local minimum and maximum points (Extrema), and connect them together by straight lines (Arcs), then any closed Iso-contour is intersect by at least one of the arcs.

41. Extrema Graph (2)

42. Extrema Graph (3) E1 E2 E3 E4 E7 E5 E6E8 a2 a3 a4 a5 a6 a7 a1 Extreme Graph: { E, A: E: extrema points A: Arcs conneccts E } An ‘arc’ consists of cells that connect extrema points (we only store min/max of the arc though)

43. Extrema Graph (4) Algorithm: Given an iso-value 1)Search the arcs of the extrema graph (to find the arcs that have min/max contains the iso-value 2)Walk through the cells along each of the arcs to find the seed cells 3)Start to propagate from the seed cells 4)…. There is something more needs to be done…

44. We are not done yet … What ?! We just mentioned that all the closed iso-contours will intersect with the arcs connecting the extrema points How about non-closed iso-contours? (or called open iso- contours)

45. Extrema Graph (5) Contours missed These open iso-contours will intersect with ?? cells Boundary Cells!!

46. Extrema Graph (6) Algorithm (continued) Given an iso-value 1)Search the arcs of the extrema graph (to find the arcs that have min/max contains the iso-value 2)Walk through the cells along each of the arcs to find the seed cells 3)Start to propagate from the seed cells 4)Search the cells along the boundary and find seed cells from there 5)Propagate open iso-contours

47. Extrema Graph Efficiency - Number of cells visisted: –extrema graph - N 0.33 –boundary - N 0.66 –Iso-surface - N 0.66 based on tetrahedra - will create more surface triangles... should extract the same number of cells/ triangulation as Marching Cubes

48. Ambiguity Problem Certain Marching Cube cases have more than one possible triangulation Case 6 Case 3 Mismatch!!! + + + + Hole!

49. The Problem Ambiguous Face: a face that has two diagonally opposing points with the same sign + + Connecting either way is possible

50. To fix it … Case 6 Case 3 B Match!!! + + + + The goal is to come up with a consistent triangulation

51. Solutions There are many solutions available – we present a method called: Asymptotic Decider by Nielson and Hamann (IEEE Vis’91)

52. Asymptotic Decider 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

53. Asymptotic Decider (2) (0,0) (1,1) Where the hyperbolas go through the cell depends on the values at the corners, I.e., B00, B01, B10, B11

54. Asymptotic Decider (3) (0,0) (1,1) Asymptote (S  T  If  B(S  T 

55. Asymptotic Decider (4) (1,1) Asymptote (S  T  (0,0) If  B(S  T 

56. Asymptotic Decider (5) (1,1) (S  T  (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

57. Asymptotic Decider (6) Based on the result of asymptotic decider, we expand the marching cube case 3, 6, 12, 10, 7, 13 (These are the cases with at least one ambiguious faces).