Approximating Normals for Marching Cubes applied to Locally Supported Isosurfaces

1. Approximating Normals for Marching Cubes applied to Locally Supported Isosurfaces Gregory M. Nielson H. Adam Huang (Speaker) Steve Sylvester Computer Science and Engineering Arizona State University

2. Contents • I. Introduction • Isosurface Approximation - Marching Cubes • Normal Estimation - Gradient • Motivation - Problems with Central Difference Approx. • II. Alternative Methods • Triangular Mesh Topology • New Method - 4 Star • III. Results • IV. Conclusions 3DK - November 9, 2014 (2)

3. Isosurfaces • Isosurface ST for a field function F (x, y, z) is ST  { (x, y, z) : F (x, y, z) = 0 } • Usually samples or measurements are known at regular lattice • Marching Cubes-an effective method to compute isosurface approximation 3DK - November 9, 2014 (3)

4. Marching Cubes (MC) • Samples F (iΔx, jΔy, kΔz) are knownDomain can be divided into small cubes as lattice building blocks 3DK - November 9, 2014 (4)

5. V V 2 3 V V 6 7 +y V +x V 0 1 +z V V 4 5 Marching Cubes (MC) • Isosurface approximation within an individual cube if F(i,j,k) ≥ α 3DK - November 9, 2014 (5)

6. V V 2 3 V V 6 7 +y V +x V 0 1 +z V V 4 5 Marching Cubes (MC) • Isosurface approximation within another individual cube 3DK - November 9, 2014 (6)

7. Example • Approximation surfaces comprised of triangles using MC algorithm 3DK - November 9, 2014 (7)

8. Normal Estimates-Gradient • Gradient as the normal vector • Apply numerical differentiation to obtain normals at the lattice points N(i,j+1,k) N(i,j,k) 3DK - November 9, 2014 (8)

9. Central Difference Approx. • Calculate normals at the lattice points using standard central difference approx. • Formulas for the lattice points on the boundaries 3DK - November 9, 2014 (9)

10. N(i,j+1,k) N(i,j+t,k) N(i,j,k) Normal Estimates-Gradient • Gradient as the normal vector • Apply numerical differentiation to obtain normals at the lattice points • Linear interpolation along edges to obtain normals at the triangular mesh vertices 3DK - November 9, 2014 (10)

11. Bunny Example Point Cloud Data Gradient from Central Difference Approximation 4-Star Method 3DK - November 9, 2014 (11)

12. Fungus Example Confocal microscopic fungus data Gradient from Central Difference Approximation 4-Star Method 3DK - November 9, 2014 (12)

13. (i+1, j+1) (i, j+1) (i+1, j) (i, j) (i-1, j) (i+2,j) (i, j-1) (i+1,j-1) Motivation • Problems with gradient estimate methods (x, j) Not specified or default value 3DK - November 9, 2014 (13)

14. II. Alternative Methods • Normal Vector is Perpendicular to Local Planar Surface Approximation • Computing average normals from the triangular mesh topology built from MC • 4-Star average normal estimates without triangular mesh topology 3DK - November 9, 2014 (14)

15. 7 5 3 2 8 4 6 1 9 7 4 8 5 1 6 3 2 Triangular Mesh Topology Weighted average normal from local topology 3DK - November 9, 2014 (15)

16. Py+ Px- i,j,k P Px+ i,j,k+1 Py- New Method 4-Star Observation: Except on the boundary voxels, there will always be exactly 4 faces that share an edge +Y +Z +X 3DK - November 9, 2014 (16)

17. Py+ Px- i,j,k P Px+ i,j,k+1 Py- 4-Star Method Observation: Except on the boundary voxels, there will always be exactly 4 vertices that connect to P with an edge on a voxel face N++=Normal of Triangle: (P,Py+,Px+) N-+=Normal of Triangle: (P,Px-,Py+) N--=Normal of Triangle: (P,py-,px-) N+-=Normal of Triangle: (P,Px+,Py-) +Y +Z +X 3DK - November 9, 2014 (17)

18. Py+ Px- i,j,k P Px+ i,j,k+1 Py- 4-Star Method if Fi-1,j,k < αthen Px- [i,j,k to i-1,j,k] else ( ifFi-1,j,k+1 < αthen Px- [i-1,j,k+1 to i-1,j,k] else Px- [i-1,j,k+1 to i,j,k+1]) if Fi+1,j,k < αthen Px+ [i,j,k to i+1,j,k] else ( ifFi+1,j,k+1 < αthen Px+ [i+1,j,k+1 to i+1,j,k] else Px+ [i+1,j,k+1 to i,j,k+1] ) 3DK - November 9, 2014 (18)

19. III. Results-Stanford Bunny Gradient from Central Difference Approximation Average Normal from 4-Star Average Normal from Local Topology 3DK - November 9, 2014 (19)

20. Results-Pollen Gradient from Central Difference Approximation Average Normal from 4-Star Average Normal from Local Grid Topology 3DK - November 9, 2014 (20)

21. Results-Fungus 4_Star Central Difference Approximation Local Grid Topology 3DK - November 9, 2014 (21)

22. Results-Bust Differene Approximation 4-Star Local Grid Topology 3DK - November 9, 2014 (22)

23. Results-Bust Differene Approximation 4-Star Local Grid Topology 3DK - November 9, 2014 (23)

24. III. Results † RMS differences of normal compared to the triangular mesh topology method in degrees * Difference Ratio of 4-Star and Gradient ~ 1/6 -2/3 3DK - November 9, 2014 (24)

25. IV. Conclusions • Gradient approximation methods can give poor results where the field function is defined only at lattice points close to isosurfaces. • 4-Star methods produce results that are close to topology methods. • They are efficient and easy to implement. • If topology is not needed, or quick rendering is used to find proper isovalues, 4-Star methods are recommended. 3DK - November 9, 2014 (25)