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Representation of Objects with Sharp Details in Truncated Distance Fields Pavol Novotný Comenius University, Bratislava, Slovakia Miloš Šrámek Austrian Academy of Sciences, Vienna, Austria
Representation of Objects with Sharp Details in Truncated
Distance Fields
Pavol NovotnýComenius University,
Bratislava, Slovakia
Miloš ŠrámekAustrian Academy of Sciences,
Vienna, Austria
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 2
Outline
• Object representation by truncated distance fields (TDFs)
• CSG operations with voxelized solids (related technique)
• Proposed technique:
Voxelization of implicit solids with sharp details in TDFs
• Results and future work
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 3
Distance Fields
• Distance to the object surface is stored in voxels
• Inside and outside area can be distinguish using different signs
• Surface can be reconstructed by interpolation and thresholding
• Distance estimation for implicid solids defined by function f(X) = 0 can be done as follows:
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 4
Object Representation by TDFs
Volume is divided into three areas:
• Inside• Outside• Transitional:
– In the surface vicinity
– Thickness: 2r
– Stored values:• density (distance from the
surface)• direction of the density
gradient (surface normal)
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 5
Problem of Sharp Details• Edge artifacts
• A problem of representation
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 6
The Object Representability Criterion
• Only solids with smooth surfaces without sharp details are representable in a discrete grid
[Baerentzen 2000]
• The criterion:– It is possible to roll a sphere
of the given radius r from both sides of the surface
– r : • defines thickness of the
transitional area
• determined by the reconstruction filter
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 7
CSG Operations
• The result of CSG operations often contains sharp details
[ Novotný, Dimitrov, Šrámek: CGI’04 ]
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 8
Representable CSG Solids• To avoid artifacts, edges of CSG solids must be
rounded!
CSG solid with artifacts A representable CSG solid
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 9
Our Earlier Results
CSG solids with artifacts
Representable CSG solids
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 10
Voxelization of Implicit Solids:An SDC Method
Problem: • Implicit solids can
contain sharp details (artifacts)
The proposed solution:• Round edges to get
representable objects
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 11
SDC Method – Overview
Stage 1:• Evaluate voxels in a
standard way• Identify critical areas
Stage 2:• Extrapolate values from
non-critical areas (linearly)• Compute final values of
voxels by approximation: CSG intersection of two halfspaces
Stage 1:• Evaluate voxels in a
standard way• Identify critical areas
Stage 2:• Extrapolate values from
non-critical areas (linearly)• Compute final values of
voxels by approximation: CSG intersection of two halfspaces
Stage 1:• Evaluate voxels in a
standard way• Identify critical areas
Stage 2:• Extrapolate values from
non-critical areas (linearly)• Compute final values of
voxels by approximation: CSG intersection of two halfspaces
Stage 1:• Evaluate voxels in a
standard way• Identify critical areas
Stage 2:• Extrapolate values from
non-critical areas (linearly)• Compute final values of
voxels by approximation: CSG intersection of two halfspaces
Stage 1:• Evaluate voxels in a
standard way• Identify critical areas
Stage 2:• Extrapolate values from
non-critical areas (linearly)• Compute final values of
voxels by approximation: CSG intersection of two halfspaces
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 12
Stage 1 – Overview
• Voxelization inside, outside and transitional voxels
• Identification of critical voxels
• Adjustment of the critical area
• Voxelization inside, outside and transitional voxels
• Identification of critical voxels
• Adjustment of the critical area
• Voxelization inside, outside and transitional voxels
• Identification of critical voxels
• Adjustment of the critical area
• Voxelization inside, outside and transitional voxels
• Identification of critical voxels
• Adjustment of the critical area
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 13
Stage 1 – Details
• Voxelization of solids defined by an implicit equation: f(X) = 0
• Distance estimation:
• Identification of the critical area – the normal consistency test:
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 14
Stage 2 – Extrapolation
Transfer density and normal values from faces through the critical area by front propagation:
• Initialization – find critical voxels neighbouring with transitional area
• Fronts may overlap
Transfer density and normal values from faces through the critical area by front propagation:
• Initialization – find critical voxels neighbouring with transitional area
• Fronts may overlap
Transfer density and normal values from faces through the critical area by front propagation:
• Initialization – find critical voxels neighbouring with transitional area
• Fronts may overlap
Transfer density and normal values from faces through the critical area by front propagation:
• Initialization – find critical voxels neighbouring with transitional area
• Fronts may overlap
Active front propagation
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 15
Stage 2 – Final Evaluation
• At the end of the front propagation – each critical voxel stores several values of density and gradient (description of several halfspaces)
• Resulting value: CSG intersection of halfspaces (according to our previous paper)
• More than two faces: sequential calculation
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 16
Results
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 17
Dependency on the Grid Resolution
64 64 64 128 128 128 256 256 256 512 512 512
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 18
Time Complexity
• 30-65% increase of processing time for all the tested objects and grid resolutions
• Important factor – size of the critical area:– sharpness of
edges– length of edges
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 19
Open Problems• Solids with non-convex sharp details proper
combination of CSG intersection and union needed non-trivial analysis necessary
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 20
Conclusion
• SDC method – alias-free voxelization of implicit solids with sharp details
• Main idea: rather smooth edges than jaggy• It works correctly for a number of objects, but
the solution is still not universal
Future work:• Extend the technique also for solids with non-
convex sharp details
Pavol Novotný, Comenius University, Bratislava, Slovakia
VG 2005 21
Thank you for attention.
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