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123 maart 2006
Surface construction techniques for volumetric objects
How to maintain convex and concave features?
Eddy Loke and Erik Jansen
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Introduction
Discretized Marching Cubes builds a surface• Halfway inbetween object and background• Rounding of edges and corners • Can we maintain sharp features?
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Surface construction from voxel data
• Data defined on grid• Surface construction with
MC (linear approximation)• Binary data• Surface construction on
binary data• Discrete Marching Cubes
(DMC)• Discrete version of MC• Less triangles• Manifold• Rounded volumes• Ambiguous cases
4-boundary 8-boundary 8+
Discrete Marching CubesDiscrete Marching Cubes – alt.Marching cubes
8-connectivity4-connectivity
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Ambiguous cases
X-cell:• MC: asymptotic decider• 4-connected
“background priority”• 8-connected
“object priority”• Subgrid
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Boundary construction cell-by-cell
• Apply configurations• We can also apply it on
subgrid• But then we have some
ambiguous cases
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Ambiguous cases
X-cell:• 4-connected
“background priority”
• 8-connected “object priority”
Boundary connectivity and node connectivity are two different things!
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3D
Connectivity:• 6-neighbors• 18 neighbors• 26 neighbors
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3D “Object priority”
P4a P4b
P4d P4e P4g
P4cP3a
P7a
P5a
P5c P6bP6a
P5b
P6c
Kenmochi et al. 99
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Object vs background priority
Object priority Background priority
P6b P6b
Node connectivity 18/26 Node connectivity 6
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Connectivity
P5a - OP P5a - BPP4g - OP P4g - BP
P6c - OP P6c - BP
Some other cases
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Background priority
P3a
P7a
P4a P4b P4c
P4d P4e P4g P5a P5b
P5c P6b P6cP6a
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Object versus background priority
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Convex vs. concave
Object priority: convex edges Background priority: concave edges
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Object versus background priority
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Discrete Marching Cubes
DMC- n DMC- o
DMC- e DMC- k DMC- l
DMC-p
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• Build a DMC configuration by using subgrid triangulation
Discrete Marching Cubes
DMC- p DMC- p DMC- p
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BP versionDMC-p OP-version
DMC triangulation
DMC has background priority triangulation
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• DMC configurations are background priority
• and triangulation is BP as well!
Discrete Marching Cubes
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Results
DMC standard DMC refined
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Concave
P7a – OP/BP P7a – SBPP6a P6a - SBP
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Concave variants
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Convex extrapolation
• DMC rounded corners
• Feature detection• Subgrid
construction• OP-triangulation
Similar for concave edges
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Results
DMC standard DMC refined
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Results
DMC standard DMC refined
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Conclusions
• Flexible triangulation scheme for convex and concave edges
• DMC extended/refined• Discrete models may be of interest for
high resolution models
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Triangulation
Configurations(cell 2x2x2 nodes)
Other possibilities?Black versus white
diagonals
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Two P6b cells with object priorityTwo P6b cells with background priority
Non-manifold
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DMC- e DMC- l
DMC- n DMC- pDMC- o
DMC- k
DMC configurations with BP-triangulation
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Concave variants
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Filters feature detection
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?
?
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positive triangulations negative triangulations
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Surface construction from voxel data
• Data defined on grid• Surface construction
with MC (continue)• Surface construction
on binary data
Marching cubes Discrete Marching Cubes4-boundary 8-boundary
