1. Feature Sensitive Surface Extraction from Volume Data Leif P. Kobbelt Mario Botsch Ulrich Schwanecke Hans-Peter Seidel Computer Graphics Group, RWTH-Aachen Computer Graphics Group, MPI Saarbrucken Proc. Of ACM SIGGRAPH 2001, page 57 66

2. Abstract A new technique to extract high quality triangle meshes from volume data. The main two contributions are: –Enhanced distance field representation –Extended Marching Cubes (EMC) About Standard Marching Cubes (MC)

3. Abstract The above figures show reconstructions of the well-known “fandisk” dataset. Standard MC Standard MC + Enhanced distance field Extended MCExtended MC + Enhanced distance field uniform 65×65×65 grid

4. About Standard Marching Cubes

5. Introduction The volume data is usually sampled on a regular grid with a given step width. We often observe severe alias artifacts at sharp features on the extracted surfaces. –Reduce these alias effect –Keep the simple algorithmic structure of the standard MC algorithm

6. Alias artifacts The Marching-Cubes-type algorithm process discrete volume data. The sampling of the implicit surface f(x,y,z)=0 is performed on the basis of a uniform spatial grid.

7. Parametric surfaces v.s. Implicit surfaces Parametric surfaces –A mapping from R 2 (u,v) to R 3 (x,y,z) –Parametrized by u and v. x=f(u,v) y=g(u,v) z=h(u,v) –Allows easy enumeration of points. Just plug in values for u and v.

8. Parametric surfaces v.s. Implicit surfaces Implicit surfaces –Defined by f(x,y,z)=0 Advantages –Easy to check whether a point is “inside” and “outside” –Inside: f(x,y,z) < 0 Disadvantages –One cannot easily enumerate points on the surface. x 2 +y 2 +z 2 -R 2 =0

9. Introduction The central contributions of this paper are: –Enhanced representation of the distance field This allows us to find more accurate surface. Store directed distance in x, y, and z directions. –Extended Marching Cubes algorithm Reduce alias (converge to the original surface’s normals.)

10. Distance field representation For a given surface, a volume representation consists of a scalar valued function such that Signed distance field function > 0 outside the surface < 0 inside the surface = 0 on the surface

11. Operation A B U A - B

12. Distance field representation The standard way to sample f on a uniform spatial grid g i,j,k = [ ih, jh, kh ]. The sampled distances d i,j,k = f ( ih, jh, kh ) can be interpolated on each grid cell. C i,j,k (h) = [ ih, (i+1)h ] × [ jh, (j+1)h ] × [ kh, (k+1)h ]

13. Distance field representation The major limitation of this technique is that the samples on S * are not necessarily close to S in the vicinity of sharp features. S S*S*

14. Distance field representation To improve the approximation one could refine the discretization grid h  h’ < h or switch to higher order polynomial interpolants within each cell C i,j,k (h). First case: –output a larger number of triangles. Second case: –local computations are getting more complicated.

15. Distance field representation Therefore we suggest a third alternative to avoid these difficulties by using the directed distance field. We store at each grid point g i,j,k three directed distances in x, y, and z direction. > 0 outside the surface < 0 inside the surface = 0 on the surface The three directed distances at one grid point always have the same sign. ( inside / outside status )

16. Distance field representation The intersection point (Interpolation) It is valid if and have opposite signs. g i,j,k g i+,j,k S h

17. Distance field representation Standard MCStandard MC + Enhanced distance field

18. Extended marching cubes Marching cubes in general cannot reconstruct very sharp features and result in aliasing artifacts. –Problem : normals don’t converge –Alias errors in surfaces generated by the MC algorithm. –By decreasing the grid size, the effect becomes less and less visible.

19. Extended marching cubes What we can do –By using point and normal information on both sides of the sharp feature one can find a good estimate for the feature point at the intersection of the tangent elements.

20. Extended marching cubes D2Y > 0 X Y D1X < 0 D3X > 0 D1Y < 0 Surface Exact intersection point

21. Extended marching cubes normal tangent element normal tangent element Reconstructed surface

22. Extended marching cubes This works only if there is at most one sharp feature. Just like the standard MC, the extended algorithm processes each cell C i,j,k (h) separately. If the cell does not contain a sharp feature –by using the standard Marching Cubes table. If a feature is present –We compute one new sample point close to the expected feature. (generate a triangle fan)

23. Extended marching cubes For each cell we first have to check if a feature is present and if yes, which type of feature.

24. Extended marching cubes Postprocessing step –Left : the cells/patches that contain a feature are identified. –Center : one new sample is included per cell. –Right : by using edge flipping to reconstruct the feature edges.

25. Extended marching cubes Extended MCExtended MC + Enhanced distance field

26. Result –The execution times include only the running times for standard and extended MC, respectively. –The (directed) distance fields and gradients have been generated in a pre-process.

27. Result - CSG

28. Result – Fan Disk Standard MC Standard MC + Enhanced distance field Extended MCExtended MC + Enhanced distance field

29. Result – Max Planck Low pass filter

30. Result – CAD 129  129  129

31. Application – Milling simulation

32. Application – Surface reconstruction –The original dataset consists 200K scattered points

33. Application - Remeshing Polygonal meshes that are generated at some intermediate stage of an industrial CAD process often have a bad quality. –Convert a CAD model into a volume representation by sampling its distance field on a uniform grid. –Apply the extended Marching Cubes algorithm to this volume gives a remeshed version.

34. Conclusions and future directions Adaptively refined octrees. –The problem is to fix the gaps (different refinement levels meet) Parallelization –The algorithm processes each cell individually. (like the standard Marching Cubes)