1. Automatic Isosurface Propagation Using an Extrema Graph and Sorted Boundary Cell Lists

2. Introduction Visiting non-intersected cells is very time-consuming in huge volumes. Elimination of non-intersected cells outside the process is more effective. The number of visited cells is less than O(n)

3. Propagation Algorithm (1)

4. Propagation Algorithm (2) 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 Basic Idea: Given an starting cell that contains isosurface, the remainder of the isosurface can be found by propagation

5. Propagation Algorithm (3)

6. Challenges Need to know the starting cells! For any given isovalue C, finding the starting cells to start the propagation. You could do a global search, but … Solution ： Extrema Graph & Boundary Cell Lists

7. Extremum Point Extremum points ard defined as grid-points whose scalar values are higher or lower than the values of all adjacent grid-points.

8. Generating an Extrema Graph Searching for starting cells: Find all the local minimum and maximum points, and connect them together by straight lines (Arcs). The closed isosurface is intersected by at least one of the arcs.

9. Extrema Graph Extreme Graph: { E, A: E: extrema points A: Arcs conneccts E } Problem ： Holes!! Hole

10. Generating Boundary Cell Lists Hole The open isosurface is intersected by visiting boundary cells in order.

11. Generating Isosurfaces Searching for starting cells

12. Outline of the Algorithm Void main(){ /* Preprocess */ ExtremumPointExtraction(); GenerateGraph(); GenerateBoundList(); /* Isosurfacing process*/ while(1){ Specify an isovalue C; GenerateSurface(C); }

13. About Arc

14. Image of Isosurface

15. Volume Thinning for Automatic Isosurface Propagation

16. Topology of an extrema graph and an isosurface Boundary cells are not necessary if there is a cycle around a through-hole

17. Image thinning method (a)(b)(c)(d)(e) (f)(g) ppppp 1 23 4 5 67 8

18. Volume thinning method We initially assumes that a seed set of a volume contains all cells in the volume. The extremum point will never be eliminated from the seed set during the process. Finally, the seed set form a one-cell-wide skeleton. Bubble-like layer Extremum point

19. Elimination of Bubble-Like Layers of Cells

20. Visited cells in the isosurfacing process Preprocess

21. Outline of the Algorithm Void main(){ /* Preprocess */ ExtremumPointExtraction(); VolumeThinning(); /* Isosurfacing process */ while(1){ Specify an isovalue; Extract isosurface cells from the extrema skeleton; IsosurfacePropagation(); }

22. Image of Volume Thinning Process

24. Image of Isosurface(1)

25. Image of Isosurface(2)