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: Algorithm:
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