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 Miloš Šrámek 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 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: 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) VG 2005

5. Problem of Sharp Details • Edge artifacts • A problem of representation VG 2005

6. 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 rfrom both sides of the surface • r : • defines thickness of the transitional area • determined by the reconstruction filter VG 2005

7. CSG Operations [ Novotný, Dimitrov, Šrámek: CGI’04 ] • The result of CSG operations often contains sharp details VG 2005

8. Representable CSG Solids • To avoid artifacts, edges of CSG solids must be rounded! CSG solid with artifacts A representable CSG solid VG 2005

9. Our Earlier Results CSG solids with artifacts Representable CSG solids VG 2005

10. Voxelization of Implicit Solids:An SDC Method Problem: • Implicit solids can contain sharp details (artifacts) The proposed solution: • Round edgesto get representable objects 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 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 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: 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 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 VG 2005

16. Results VG 2005

17. Dependency on the Grid Resolution 64  64  64 128  128  128 256  256  256 512  512  512 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 VG 2005

19. Open Problems • Solids with non-convex sharp details  proper combination of CSG intersection and union needed  non-trivial analysis necessary 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 VG 2005

21. Thank you for attention. • Emails: • novotny@sccg.sk • milos.sramek@oeaw.ac.at VG 2005