1. Fast Isocontouring For Improved Interactivity Chandrajit L. Bajaj Valerio Pascucci Daniel R. Schikore

2. Introduction ► A preprocessing step selects a subset S of the volume cells which are considered as seed cells. ► Given a particular isovalue, all cells in S which intersect the given isocontour are extracted using a high performance range search. ► Keyword: range search

3. Introduction ► Unstructured volume data ► Isocontours C={ x | F (x)=w }  w: isovalue ► The average number of cells intersected by an isocontour will be for a d-dimensional domain. ► Data structure  Interval tree

4. Algorithm Overview (1/3) ► Three know techniques  The extraction of an isocontour does not require searching all the cells of the mesh.  To improve the efficiency of the cell extraction, it is necessary to define a search structure over the set of cells.  The search space we need to work on is the two dimensional span space.

5. Algorithm Overview (2/3) ► Exploiting above three main ideas we get the following isocontouring algorithm.  Preprocessing A Reduce the set of cells to a subset S that encompasses at least one cell per connected component of each isocontour.  Preprocessing B Construct an efficient search structure over the cells in the set S.

6. Algorithm Overview (3/3)  Step 1 Given the scalar value w, perform a logarithmic search on the set S to find all the cells in S which intersect the isocontour of value w.  Step 2 For each cell c found in step 1, trace the entire connected component of the isocontour intersected by c. (contour propagation)  Time complexity:

7. Contour Propagation ► Advantage  Surfaces are easily transformed into a triangle strip representation for more efficient rendering.

8. Cell Connectivity (1/5) ► Connectivity Graph  interval of cell ► R(c) = [min,max]  connecting interval ► R(f) = [min ’,max ’ ]  R(c1)  R(c2) 304050 60 70 C1 C2 R(C1)=[30,60] R(C2)=[40,70] R(f 12 )=[40,60]

9. Cell Connectivity (2/5) ► Based on the above information, we construct a labeled graph G. 304050 60 70 C1 C2 R(C1)=[30,60] R(C2)=[40,70] R(f 12 )=[40,60] C1C2 f 12

10. Cell Connectivity (3/5) ► All cells which intersect the same connected component of a contour of isovalue w we call w - connected.

11. Cell Connectivity (4/5) ► Definition 1  Consider a scalar value w and two nodes c 1, c 2 of G. c 1 and c 2 are said to be w – connected if one of the two following conditions holds. (a) c 1 and c 2 are connected by an arc f such that w  R( f ). (b) There exists a node c 3 that is w - connected to both c 1 and c 2. C1C2 f 12 R(f 12 )=[40,60] C3 f 23 R(f 23 )=[45,65] W=50

12. Cell Connectivity (5/5) ► Definition 2  Consider a subset S of the nodes of G and a node c  G. The node c is connected to S if. For any w  R(c), there exists a node c ’  S that is w -connected to c. S G c c’c’

13. Seed Sets (1/8) ► ► The seed sets are important because any isocontour of the entire original mesh can be traced by propagating from the cells of any seed set. ► Definition 3  A subset S of the nodes of G is a seed set of G if all the nodes of G are connected to S.

14. Seed Sets (2/8) ► w ► If we wish to determine quickly all the cells of a mesh whose range contains a particular scalar value w, we can proceed as follows:  Search for all the nodes c  S such that w  R(c) ;  Starting from the nodes we have found and using the w -connectivity relation on the graph G, we find the cells of the mesh whose range contains w.

15. Seed Sets (3/8) ► To reduce the search time we need to reduce the cardinality of the seed set S as much as possible. ► Property 1  If S is a seed set and c  S is a cell connected to S - {c}, then S - {c} is a seed set.  Proof: from definition 1 (b)

16. Seed Sets (4/8) ► we wish to find a small seed set ► initial  starting with the entire set of the cells – that is the largest seed set ► repeated  keep removing until we achieve a minimal seed set.

17. Seed Sets (5/8) ► At each step, we remove the selected cell c along with all its incident arcs and add some new arcs between pairs of cells that were connected to c. ► This new arc f needs to be inserted if 30 40 805060 70 C1 C2 90 95

18. Seed Sets (6/8) ► If above condition is true, then the new arc is added with label ► We can remove a cell c of the current seed set if :  where f 1, ……, f k are all the arcs incident to the cell c in the reduced graph of the current seed set.

19. Seed Sets (7/8) ► Example: B A C D E A  E B  E D  E C  E

20. Seed Sets (8/8) The sweep hyperplane is a line parallel to the y direction moving along the x direction.

21. Interval Tree In an interval tree, each node holds a split value s, and each interval is classified as less than (max s), or spanning (min < s < max). With each node, the intervals are sorted into two lists. Store time complexity: O(N) Search time complexity: O(logN)

22. Conclusions (1/3) Result of seed selection technique

23. Conclusions (2/3)

24. Conclusions (3/3) Evident from the plot is that our algorithm provides the greatest speedup when the surface of interest is small compared to the volume.