Lecture 3 : Isosurface Extraction

1. Lecture 3 : Isosurface Extraction Bong-Soo Sohn School of Computer Science and Engineering Chung-Ang University

2. Isosurface Definition • Constant Density Surface from a 3D array of data • I={X|F(X)=w} • I : isosurface • F : 3D volume • w : isovalue • Ideal isosurface vs Discretized (triangularized) isosurface

3. Introduction • Idea: • create a triangular mesh that wil approximate the iso-surface • calculate the normals to the surface at each vertex of the triangle • Algorithm: • locate the surface in a cube of eight pixels • calculate normals • march to the next cube

4. Marching Cubes • Marching cubes • method for approximating surface • defined by isovalue α, given by grid data • Input: • Grid data (set of 2D images) • Threshold value (isovalue) α • Output: • Triangulated surface that matches isovalue surface of α

5. Surface Intersection in a Cube • assign ZERO to vertex outside the surface • assign ONE to vertex inside the surface • Note: • Surface intersects those cube edges where one vertex is outside and the other inside the surface

6. Surface Intersection in a Cube • There are 28=256 ways the surface may intersect the cube • Triangulate each case

7. Patterns • Note: • using the symmetries reduces those 256 cases to 15 patterns

8. 15 Cases

9. Surface intersection in a cube • Create an index for each case: • Interpolate surface intersection along each edge

10. Calculating normals • Calculate normal for each cube vertex: • Interpolate the normals at the vertices of the triangles:

11. Summary • Read four slices into memory • Create a cube from four neighbors on one slice and four neighbors on the next slice • Calculate an index for the cube • Look up the list of edges from a pre-created table • Find the surface intersection via linear interpolation • Calculate a unit normal at each cube vertex and interpolate a normal to each triangle vertex • Output the triangle vertices and vertex normals

12. Ambiguity Problem

13. Trilinear Function • Trilinear Function • Saddle point • Face saddle • Body saddle

14. Trilinear Isosurface Topology

15. Triangulation