1. Unconstrained Isosurface Extraction on Arbitrary Octrees
Michael Kazhdan, Allison Klein, Ketan Dalal, Hugues Hoppe

2. Implicit Representation
In many graphics applications, a 3D model is represented by an implicit function:
Reconstruction
Fluid Dynamics
3D Texturing
Kazhdan 2005
Losasso et al. 2004

3. Octree Representation
To minimize the spatial/temporal complexity, the function is often sampled on an adaptive grid
Popinet 2003
Kazhdan et al. 2006

4. Octree Extraction
Often, in the final processing step, we would like to extract an implicit surface from the function representation.

5. Marching Cubes
If the function is sampled on a regular voxel grid, we can independently triangulate each voxel.

6. Marching Cubes
Although each of the voxels is triangulated independently, the mesh is always water-tight.

7. Marching Cubes
Iso-vertices on an edge are only determined by the values on the corner of the edge:
 Iso-vertices are consistent across voxels.

8. Marching Cubes
Iso-edges on a face are only determined by the values on the face:
 Each iso-edge is shared by two triangles so the mesh is water-tight.

9. Challenges
Extracting a surface by independently triangulating the leaf octants, depth-disparities can cause:
Inconsistent extrapolation to edges
Inconsistent iso-vertex positions

10. Challenges
Extracting a surface by independently triangulating the leaf octants, depth-disparities can cause:
Inconsistent extrapolation to faces
Inconsistent iso-edges

11. Outline
Introduction
Related Work
Approach
Evaluation
Conclusion

12. Related Work [Bloomenthal ’88]
Uses an octree representation of the EDT for adaptive surface polygonization
1,150,915 Vertices
2,301,826 Triangles
139,211 Vertices
106,717 Polygons

13. Related Work [Bloomenthal ’88]
Approach:
Use finer edges to define iso-vertices.

14. Related Work [Bloomenthal ’88]
Approach:
Use finer edges to define iso-vertices.
Use finer faces to define iso-edges

15. Related Work [Bloomenthal ’88]
Properties:
Triangles become polygons
Each iso-edge is shared by two polygons

16. Related Work [Bloomenthal ’88]
Properties:
Triangles become polygons
Each iso-edge is shared by two polygons
Watertight polygon mesh!

17. Related Work [Bloomenthal ’88]
Properties:
Triangles become polygons
Each iso-edge is shared by two polygons
Limitation: There cannot be more than one isovalue-crossing along an edge of a leaf octant.
Watertight polygon mesh!

18. Related Work Although this limitation can be resolved by:
Refining the octree [Bloomenthal 88, Muller et al. 93]
Restricting the topology [Westermann et al. 99]
Modifying corner values [Velasco and Torres 01]
Re-sampling [Ju et al. 02, Schaefer et al. 04]
we want a solution that is true to the input.

19. Outline Introduction Related Work Approach Evaluation Conclusion
Edge-Trees (Polygonization)
Triangulation
Evaluation
Conclusion

20. Nodes in the edge-treesa0
The topology of the octree defines a set of binary trees: a
a10 a1
a11
Nodes in the edge-trees a a0
Edges of octree nodes a1
a10
a11

21. Nodes in the edge-treesa0
The topology of the octree defines a set of binary trees: a
a10
b10 a1 b1
a11
b11
Nodes in the edge-trees a a0
Edges of octree nodes a1 b1
a10
a11
b10
b11

22. Nodes in the edge-treesa0 b0
The topology of the octree defines a set of binary trees: a
a10
b10 a1 b1
a11
b11
Nodes in the edge-trees a a0
Edges of octree nodes a1 b1 b0
a10
a11
b10
b11

23. Edge-Trees The topology of the octree defines a set of binary trees.
Given an isovalue:
Can label nodes in the edge-tree (0/1). a
a10
b10 a1 b1
a11
b11 a a0 a1 b1 b0 1 1
a10
a11
b10
b11 1 1 1

24. Edge-Trees a0 b0
Note:
Parents’ values are determined by values of their children.
Isovalue-crossing nodes/edges have exactly one child that is isovalue-crossing. a
a10
b10 a1 b1
a11
b11 a a0 a1 b1 b0 1 1
a10
a11
b10
b11 1 1 1

25. Outline Introduction Related Work Approach Evaluation Conclusion
Edge-Trees (Polygonization)
Iso-Vertex Consistency
Polygonization
Triangulation
Evaluation
Conclusion

26. Iso-Vertex Consistency
We define the position of an iso-vertex in terms of leaf nodes in the edge-tree. e 1 e’ 1 e 1 1 e’ 1

27. Iso-Vertex Consistency
We define the position of an iso-vertex in terms of leaf nodes in the edge-tree:
An isovalue-crossing edge defines a unique path to a leaf node in an edge-tree. e 1 e e’ 1 1 1 e’ 1

28. Polygonization Observation:
Number of unsealed iso-vertices along an octant edge is even.
Generate polygons by stitching pairs of unsealed iso-vertices.

29. Polygonization
Challenge: With more than two unsealed iso-vertices, which pairs do we stitch together? ? ?

30. Polygonization
Challenge: With more than two unsealed iso-vertices, which pairs do we stitch together?
Sequential pairings can lead to inconsistent (i.e. non watertight) results! ? ?

31. Polygonization
We stitch unsealed iso-edges in a canonical manner by defining the twin of an iso-vertex: e ? ? e v 1 1 1 1 1 v

32. Polygonization
We stitch unsealed iso-edges in a canonical manner by defining the twin of an iso-vertex:
An iso-vertex ve is unsealed if the path from v to e passes through a 0-labeled node. e ? ? e v 1 1 1 1 1 v

33. Polygonization
We stitch unsealed iso-edges in a canonical manner by defining the twin of an iso-vertex:
An iso-vertex ve is unsealed if the path from v to e passes through a 0-labeled node.
The other child of the 0-labeled node also defines an unsealed iso-vertex. e ? ? e v v’ 1 1 1 1 v’ 1 v

34. Polygonization
We stitch unsealed iso-edges in a canonical manner by defining the twin of an iso-vertex:
An iso-vertex ve is unsealed if the path from v to e passes through a 0-labeled node.
The other child of the 0-labeled node also defines an unsealed iso-vertex. e
For octrees, this type of pairing always leads to consistent (i.e. watertight) polygonization! ? ? e v v’ 1 1 1 1 v’ 1 v

35. Outline Introduction Related Work Approach Evaluation Conclusion
Edge-Trees (Polygonization)
Triangulation
Evaluation
Conclusion

36. Triangulation
To obtain a triangulated surface, we need to triangulate the leaf octants’ iso-polygons.

37. Triangulation
Challenge: In general, triangulating a 3D polygon is both open and hard [Barequet et al. ’98]:
Open: Determining if it can be triangulated requires determining if it’s knotted.
Hard: Even if it can be, this may require introducing exponentially many new vertices.

38. Triangulation
Observation: The polygon is on the surface of a convex solid.

39. Triangulation
Observation: The polygon is on the surface of a convex solid.
Minimal Surfaces [Meeks and Yau ’80]: A minimal area surface of a simple closed curve on the surface of a convex solid is embedded.

40. Triangulation
Observation: The polygon is on the surface of a convex solid.
Minimal Surfaces [Meeks and Yau ’80]: A minimal area surface of a simple closed curve on the surface of a convex solid is embedded.
Approach: Triangulate the polygonization by computing the minimal area triangulation [Barequet et al. ’95].

41. Outline
Introduction
Related Work
Approach
Evaluation
Conclusion

42. Evaluation
To evaluate the extraction method, we used an octree to adaptively sample the EDT of a mesh and then extracted the zero-crossing isosurface.
1,150,915 Vertices
2,301,826 Triangles
139,211 Vertices
106,717 Polygons

43. Evaluation (Restricted Octrees)
Previous work addresses the problem by restricting the depth disparity between adjacent leaf octants [Westermann et al. ’99].
Unrestricted Tree
Restricted Tree

44. Evaluation (Restricted Octrees)
Original
Simplified (Restricted)
Simplified (Unrestricted)
173,974 Vertices
345,944 Triangles
68,572 Vertices
58,876 Polygons
64,639 Vertices
53,304 Polygons

45. Evaluation (Restricted Octrees)
Original
Simplified (Restricted)
Simplified (Unrestricted)
173,974 Vertices
345,944 Triangles
68,572 Vertices
58,876 Polygons
181,161 Nodes
64,639 Vertices
53,304 Polygons
106,745 Nodes

46. Evaluation (Restricted Octrees)
Original
Simplified (Restricted)
Simplified (Unrestricted)
173,974 Vertices
345,944 Triangles
68,572 Vertices
58,876 Polygons
181,161 Nodes
64,639 Vertices
53,304 Polygons
106,745 Nodes

47. Evaluation (Restricted Octrees)
Original
Simplified (Restricted)
Simplified (Unrestricted)
543,652 Vertices
1,087,716 Triangles
145,829 Vertices
125,858 Polygons
380,681 Nodes
128,146 Vertices
104,868 Polygons
205,617 Nodes

48. Evaluation (Restricted Octrees)
Original
Simplified (Restricted)
Simplified (Unrestricted)
543,652 Vertices
1,087,716 Triangles
145,829 Vertices
125,858 Polygons
380,681 Nodes
128,146 Vertices
104,868 Polygons
205,617 Nodes

49. Outline
Introduction
Related Work
Approach
Evaluation
Conclusion

50. Contribution
We have shown that an octree defines a set of binary edge-trees that provide a solution to the surface extraction problem:

51. Contribution
We have shown that an octree defines a set of binary edge-trees that provide a solution to the surface extraction problem:
We walk down the tree to define iso-vertex positions consistently e 1 e e’ 1 1 1 1

52. Contribution
We have shown that an octree defines a set of binary edge-trees that provide a solution to the surface extraction problem:
We walk up an then down the tree to stitch up iso-polygons e ? ? v 1 1 e 1 1 v’ v’ 1 v

53. Conclusion
Using the edge-trees, we can extract a watertight isosurface in a local manner:
For arbitrary tree topology/values
Without knowing the original implicit function
Independent of the isovalue
871,414 Triangles
73,164 Polygons
181,052 Triangles

54. Thank You!