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

Unconstrained Isosurface Extraction on

Arbitrary Octrees

Michael Kazhdan,Allison Klein,Ketan Dalal,Hugues Hoppe

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

Octree Representation

To minimize the spatial/temporal complexity, the function is often sampled on an adaptive grid

Kazhdan et al. 2006Popinet 2003

Octree Extraction

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

Marching Cubes

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

Marching Cubes

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

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.

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.

Challenges

Extracting a surface by independently triangulating the leaf octants, depth-disparities can cause:

Inconsistent extrapolation to edges Inconsistent iso-vertex positions

Challenges

Extracting a surface by independently triangulating the leaf octants, depth-disparities can cause:

Inconsistent extrapolation to faces Inconsistent iso-edges

Outline

Introduction Related Work Approach Evaluation Conclusion

Related Work [Bloomenthal ’88]

1,150,915 Vertices2,301,826 Triangles

139,211 Vertices106,717 Polygons

Uses an octree representation of the EDT for adaptive surface polygonization


Approach: Use finer edges to define iso-vertices.


Approach: Use finer edges to define iso-vertices. Use finer faces to define iso-edges


Properties: Triangles become polygons Each iso-edge is shared by two polygons


Properties: Triangles become polygons Each iso-edge is shared by two polygonsWatertight polygon mesh!


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!

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.

Outline

Introduction Related Work Approach

Edge-Trees (Polygonization) Triangulation

Evaluation Conclusion

Edge-Trees

a

a1a0

a11a10

a

a1

a0

a11

a10

The topology of the octree defines a set of binary trees:

Nodes in the edge-trees

Edges of octree nodes

Edge-Trees

b1

b11b10

a

a1a0

a11a10

a

a1

a0

a11

a10

b1

b11

b10




Edge-Trees

b0b1

b11b10

a

a1a0

a11a10

a

a1

a0

a11

a10

b1

b11

b10

b0




Edge-Trees

0

1 1

0b0b1

b11b10

0

1 1

a

a1a0

0 1a11a10

a

a1

a0

a11

a10

b1

b11

b10

b0

The topology of the octree defines a set of binary trees.

Given an isovalue:

Can label nodes in the edge-tree (0/1).

Edge-Trees

Note: Parents’ values are

determined by values of their children.

Isovalue-crossing nodes/edges have exactly one child that is isovalue-crossing.

a

a1

a0

a11

a10

b1

b11

b10

b0

0

1 1

a

a1a0

0 1a11a10

0

1 1

0b0b1

b11b10

Outline


Edge-Trees (Polygonization) Iso-Vertex Consistency Polygonization

Triangulation Evaluation Conclusion

Iso-Vertex Consistency

We define the position of an iso-vertex in terms of leaf nodes in the edge-tree.

1

00 1

e

e 1

1

11 0e’

e’

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.

1

00 1

e

1

1

11 0e’

e

e’

Polygonization

Observation:

Number of unsealed iso-vertices along an octant edge is even.

Generate polygons by stitching pairs of unsealed iso-vertices.

Polygonization

?? ??

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

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

Polygonization

?? ??Sequential pairings can lead to inconsistent (i.e. non watertight) results!

Polygonization

0

?? ?

e

0

1

1

1

11

11

We stitch unsealed iso-edges in a canonical manner by defining the twin of an iso-vertex:

e

v

vv

Polygonization

0

?? ?

e

0

1

1

1

11

11 v


An iso-vertex ve is unsealed if the path from vto e passes through a 0-labeled node.

e

vv

Polygonization

0

?? ?

e

0

1

1

1

11

11 v



The other child of the 0-labeled nodealso defines an unsealed iso-vertex.

v’

e

vv

v’v’

Polygonization

0

?? ?

e

0

1

1

1

11

11 v



The other child of the 0-labeled nodealso defines an unsealed iso-vertex.

v’

e

vv

v’v’

For octrees, this type of pairing always leads to consistent (i.e. watertight) polygonization!

Outline


Edge-Trees (Polygonization) Triangulation

Evaluation Conclusion

Triangulation

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

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.

Triangulation

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

Triangulation


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

Triangulation


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].

Outline


Evaluation

To evaluate the extraction method, we used an octree to adaptively sample the EDT of a mesh and thenextracted thezero-crossingisosurface.

1,150,915 Vertices2,301,826 Triangles

139,211 Vertices106,717 Polygons

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


68,572 Vertices 58,876 Polygons

64,639 Vertices 53,304 Polygons

Original Simplified(Restricted)

Simplified(Unrestricted)

173,974 Vertices345,944 Triangles


68,572 Vertices 58,876 Polygons181,161 Nodes












145,829 Vertices125,858 Polygons380,681 Nodes




543,652 Vertices1,087,716 Triangles






543,652 Vertices1,087,716 Triangles

Outline


Contribution

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

Contribution


We walk down the tree to defineiso-vertex positions consistently 1

00 1

e

1

1

11 0

e

e’

Contribution


We walk up an then down thetree to stitch up iso-polygons

0

?? ?

e

0

1

1

1

11

11 v

v’

e

vv

v’v’

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

Thank You!

http://www.cs.jhu.edu/~misha/Code/IsoOctree/

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

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