1. Isocontour/surface Extractions 2D Isocontour 3D Isosurface

2. Isocontour (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

3. Isocontour (1) Consider a simple case: one cell data set The problem of extracting an isocontour is an inverse of value interpolation. That is: p2 p3 p0 p1 Gieven 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

4. Isocontour (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

5. Isocontouring – 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

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

7. Isocontouring – 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

8. Isocontour cases How many cases can an isocontour intersect a cell? p2 p3 p0 p1 When comparing the value of Pi with the isovalue C, there can be two cases: Vi > = C or Vi < C So there can be 2 * 2 * 2 * 2 = 16 cases!

9. How many cases again? In fact, there are only 4 unique topological cases (1) Complete outside(inside) (2) One inside(outside), 3 outside(inside) (3) Two inside(outside), two outside(inside) (4) Two contours pass through

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

11. Put it all together Divide-and-conquer algorithm 1.Look at one cell at a time 2.Compare the values at 4 vertices with the isovalue C 3.Linear interpolate along the edges 4.Connects the interpolated points together

12. 3D Isocontour (Isosurface)

13. 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 Isosurface Extraction

14. Divide-and-Conquer _ + + + + _ _ _ + + + + _ _ _ _ (2 triangles)

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

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

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

18. Isosurface Cases Total number of cases: 14

19. 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= sovalue) Each cell has an index mapped to a value ranged [0,255] Index = v8 v7 v6 v5 v4 v3 v2 v1

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

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

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