Representation of Objects with Sharp Details in Truncated Distance Fields

1
Representation of Objects with Sharp Details in
Truncated Distance Fields

• Pavol Novotný
• Comenius University,
• Bratislava, Slovakia
• Milo rámek
• Austrian Academy of Sciences, Vienna, Austria

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

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

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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)

5
Problem of Sharp Details

• Edge artifacts
• A problem of representation

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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

7
CSG Operations
Novotný, Dimitrov, rámek CGI04

• The result of CSG operations often contains sharp
  details

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Representable CSG Solids

• To avoid artifacts, edges of CSG solids must be
  rounded!

CSG solid with artifacts
A representable CSG solid
9
Our Earlier Results
CSG solids with artifacts Representable CSG
solids
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Voxelization of Implicit SolidsAn SDC Method

• Problem
• Implicit solids can contain sharp details
  (artifacts)
• The proposed solution
• Round edges to get representable objects

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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

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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

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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

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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

• 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
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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

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Results
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Dependency on the Grid Resolution
64 ? 64 ? 64
128 ? 128 ? 128
256 ? 256 ? 256
512 ? 512 ? 512
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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

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Open Problems

• Solids with non-convex sharp details ? proper
  combination of CSG intersection and union needed
  ? non-trivial analysis necessary

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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

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Thank you for attention.

• Emails
• novotny_at_sccg.sk
• milos.sramek_at_oeaw.ac.at