Download - Well-composed cell complexes
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Rocio Gonzalez-DiazMaria-Jose Jimenez
Belen Medrano
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Our ultimate purpose:
To extract (co)homological information of a 3D model.
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For this aim, first we need to compute representative (co)cycles of (co)homology generators of dimension 1 in the model.
Extracted from Dey et al., Computing Geometry-aware Handle and Tunnel Loops in 3D Models. ACM Transactions on Graphics, Vol. 27, No. 3, Article 45, (2008).
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T. Dey et al:
Computing loops on the surface that wraps around their ‘handles’ and ‘tunnels’. Applications:
Feature detection Topological simplification…
Dey T. K., Li K., Sun J.: On computing handle and tunnel loops. IEEE Proceedings of the international conference on cyberworlds; 2007.p.357-66.
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All the computations are carried out over a connected closed surface in R3.
Can we start from a similar scenario if we consider the cubical complex associated to a 3D digital picture?
Our answer: Well-composed cell complexes
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Well-composed images enjoy important topological and
geometric properties:
1. There is only one type of connected component.
2. Some algorithms used in computer vision,
computer graphics and image processing are
simpler.
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Well-composed images enjoy important topological and
geometric properties:
3. Thinning algorithms can be simplified and naturally
made parallel if the input image is well-composed.
4. Some algorithms for computing surface curvature
or extracting adaptive triangulated surfaces assume
that the input image is well-composed.
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There are several methods for turning binary digital
images that are not well-composed into well-composed
ones:
…but these methods “destroy the topology”.
Original image Well-composed Differences in gray
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To “transform’’ the cubical complex induced by a 3D binary digital picture into a homotopy equivalent cell complex, whose boundary is made up by 2-manifolds:
WELL –COMPOSED CELL COMPLEX.
Our goal in this paper:
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I = (Z3, B): set of unit cubes (voxels) centered at the
points of B together with all the faces.
Bc = Z3\B = background
Example: I = (Z3, B), B = {(0,0,0), (0,1,0), (0,0,1), (1,0,1)}
B = foreground
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{ Voxels } { 3D cubes in R3 }
Combinatorial struture: CUBICAL COMPLEX
Voxel
0-cells = vertices
1-cells = edges
2-cells = squared faces
3-cells = cubes
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{ Voxels } { 3D cubes in R3 }
Combinatorial struture: CUBICAL COMPLEX
)2
1,2
1,2
1( kjipl
0-cells notation:
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{ Voxels } { 3D cubes in R3 }
Combinatorial struture: CUBICAL COMPLEX
),2
1,2
1(
)2
1,,
2
1(
)2
1,2
1,(
kjia
kjia
kjia
l
l
l
1-cells notation:
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{ Voxels } { 3D cubes in R3 }
Combinatorial struture: CUBICAL COMPLEX
2-cells notation:
)2
1,,(
),2
1,(
),,2
1(
kjic
kjic
kjic
l
l
l
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{ Voxels } { 3D cubes in R3 }
Combinatorial struture: CUBICAL COMPLEX
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I = (Z3, B) is a well-composed image if the boundary of
the cubical complex associated, ∂Q(I), is a 2D-manifold.
[Latecki97] A 3D binary digital image is well-composed iff
the configuractions C1, C2 and C3 do not occur in Q(I).
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This point has not a neighborhood
homeomorphic to R2.
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Critical
configurations
within a block of
eight cubes
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A 3D digital image is not generally a well-composed image
Cubical complex Q(I)
Cell complex K(I) such that ∂K(I) is composed by 2D-manifolds: a well-composed cell complex.
Homotopy equivalent
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Key idea: to create a true face adyacency to avoid the critical configurations
Cubical complex Q(I)
Cell complex K(I) such that ∂K(I) is composed by 2D-manifolds: a well-composed cell complex.
Homotopy equivalent
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INPUT OUTPUTCubical complex Q(I) Well-composed cell complex K(I)
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So, we need “more space” to add new cells:
Adjacency = face relation between cells.
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Step 1: Label critical edges and critical vertices of Q(I).
Cubical complex Q(I)
Cell complex K(I) such that ∂K(I) is composed by 2D-manifolds: a well-composed cell complex.
Homotopy equivalent
Step 2: Repair critical edges of Q(I).
Step 3: Repair critical vertices of Q(I).
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Step 1: Label critical edges and critical vertices of Q(I).
A := set of critical edges of Q(I)Vi := set of critical vertices of Q(I)
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Step 2: Repair critical edges of Q(I).
Step 1: Label critical edges and critical vertices of Q(I).
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Step 2: Repair critical edges of Q(I).
Step 1: Label critical edges and vertices.
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Step 2: Repair critical edges of Q(I).
Step 1: Label critical edges and vertices.
Adjacency = face relation between cells.
υ
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Step 2: Repair critical edges of Q(I).
Step 1: Label critical edges and critical vertices of Q(I).
Step 3: Repair critical vertices of Q(I).
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Step 2: Repair critical edges of Q(I).
Step 1: Label critical edges and critical vertices of Q(I).
Step 3: Repair critical vertices of Q(I).
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Step 2: Repair critical edges of Q(I).
Step 1: Label critical edges and critical vertices of Q(I).
Step 3: Repair critical vertices of Q(I).
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Step 2: Repair critical edges of Q(I).
Step 3: Repair critical vertices of Q(I).
Step 1: Label critical edges and critical vertices of Q(I).
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Step 2: Repair critical edges of Q(I).
Step 1: Label critical edges and critical vertices of Q(I).
Step 3: Repair critical vertices of Q(I).
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Step 2: Repair critical edges of Q(I).
Step 1: Label critical edges and critical vertices of Q(I).
Step 3: Repair critical vertices of Q(I).
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Aims: To compute the homology of the foreground image as well as the background by computing the homology of the boundary surface;
Geometrically control the representative (co)-cycles of homology generators;
Deal with other 3D digital images: (6,26), (18,6) or (6,18) 3D images.
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Thanks for your attention!Thanks for your attention!
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