Representation of Objects with
Sharp Details in Truncated
Distance Fields
Pavol Novotný
Comenius University,
Bratislava, Slovakia
Miloš Šrámek
Austrian Academy of Sciences,
Vienna, Austria
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
2
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
3
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
4
Problem of Sharp Details
• Edge artifacts
• A problem of
representation
Pavol Novotný, Comenius University,
Bratislava, Slovakia
VG 2005
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The Object Representability
Criterion [Baerentzen 2000]
• Only solids with smooth surfaces without sharp details
are representable in a discrete grid
• 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
6
CSG Operations
[ Novotný, Dimitrov, Šrámek: CGI’04 ]
• The result of
CSG operations
often contains
sharp details
Pavol Novotný, Comenius University,
Bratislava, Slovakia
VG 2005
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Representable CSG Solids
• To avoid artifacts, edges of CSG solids must be
rounded!
CSG solid with artifacts
Pavol Novotný, Comenius University,
Bratislava, Slovakia
A representable CSG solid
VG 2005
8
Our Earlier Results
CSG solids with
artifacts
Representable
CSG solids
Pavol Novotný, Comenius University,
Bratislava, Slovakia
VG 2005
9
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
10
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
Pavol Novotný, Comenius University,
Bratislava, Slovakia
VG 2005
11
Stage 1 – Overview
• Voxelization  inside,
outside and transitional
voxels
• Identification of critical
voxels
• Adjustment of the
critical area
Pavol Novotný, Comenius University,
Bratislava, Slovakia
VG 2005
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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
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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
Active front propagation
Pavol Novotný, Comenius University,
Bratislava, Slovakia
VG 2005
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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
15
Results
Pavol Novotný, Comenius University,
Bratislava, Slovakia
VG 2005
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Dependency on the Grid Resolution
64  64  64
128  128  128
Pavol Novotný, Comenius University,
Bratislava, Slovakia
256  256  256
VG 2005
512  512  512
17
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
18
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
19
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 nonconvex sharp details
Pavol Novotný, Comenius University,
Bratislava, Slovakia
VG 2005
20
Thank you for attention.
Emails:
• novotny@sccg.sk
• milos.sramek@oeaw.ac.at
Pavol Novotný, Comenius University,
Bratislava, Slovakia
VG 2005
21