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. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 20052 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. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 20053 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:

4. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 20054 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. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 20055 Problem of Sharp Details Edge artifacts A problem of representation

6. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 20056 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

7. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 20057 CSG Operations The result of CSG operations often contains sharp details [ Novotný, Dimitrov, Šrámek: CGI’04 ]

8. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 20058 Representable CSG Solids To avoid artifacts, edges of CSG solids must be rounded! CSG solid with artifacts A representable CSG solid

9. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 20059 Our Earlier Results CSG solids with artifacts Representable CSG solids

10. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200510 Voxelization of Implicit Solids: An SDC Method Problem: Implicit solids can contain sharp details (artifacts) The proposed solution: Round edges to get representable objects

11. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200511 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

12. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200512 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

13. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200513 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:

14. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200514 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

15. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200515 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

16. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200516 Results

17. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200517 Dependency on the Grid Resolution 64  64  64 128  128  128256  256  256512  512  512

18. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200518 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

19. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200519 Open Problems Solids with non-convex sharp details  proper combination of CSG intersection and union needed  non-trivial analysis necessary

20. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200520 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

21. Pavol Novotný, Comenius University, Bratislava, Slovakia VG 200521 Thank you for attention. Emails: novotny@sccg.sk milos.sramek@oeaw.ac.at