1. Stereological Techniques for Solid Textures Rob Jagnow MIT Julie Dorsey Yale University Holly Rushmeier Yale University

2. Given a 2D slice through an aggregate material, create a 3D volume with a comparable appearance. Objective

3. Real-World Materials Concrete Asphalt Terrazzo Igneous minerals Porous materials

4. Independently Recover… Particle distribution Color Residual noise

5. Stereology (ster'e-ol' -je) e The study of 3D properties based on 2D observations. In Our Toolbox…

6. Prior Work – Texture Synthesis 2D 3D Efros & Leung ’99 2D 3D –Heeger & Bergen 1995 –Dischler et al. 1998 –Wei 2003 Heeger & Bergen ’95 Wei 2003 Procedural Textures

7. Prior Work – Texture Synthesis InputHeeger & Bergen, ’95

8. Prior Work – Stereology Saltikov 1967 Particle size distributions from section measurements Underwood 1970 Quantitative Stereology Howard and Reed 1998 Unbiased Stereology Wojnar 2002 Stereology from one of all the possible angles

9. Recovering Sphere Distributions = Profile density (number of circles per unit area) = Mean caliper particle diameter = Particle density (number of spheres per unit volume) The fundamental relationship of stereology:

10. Recovering Sphere Distributions Group profiles and particles into n bins according to diameter Particle densities = Profile densities = For the following examples, n = 4

11. Recovering Sphere Distributions Note that the profile source is ambiguous

12. Recovering Sphere Distributions How many profiles of the largest size? = = Probability that particle N V (j) exhibits profile N A (i)

13. Recovering Sphere Distributions How many profiles of the smallest size? = ++ + = Probability that particle N V (j) exhibits profile N A (i)

14. Recovering Sphere Distributions Putting it all together… =

15. Recovering Sphere Distributions Some minor rearrangements… = Normalize probabilities for each column j : = Maximum diameter

16. Recovering Sphere Distributions For spheres, we can solve for K analytically: K is upper-triangular and invertible for otherwise Solving for particle densities:

17. Testing precision Input distribution Estimated distribution

18. Other Particle Types We cannot classify arbitrary particles by d/d max Instead, we choose to use Approach: Collect statistics for 2D profiles and 3D particles Algorithm inputs: +

19. Profile Statistics Segment input image to obtain profile densities N A. Bin profiles according to their area, Input Segmentation

20. Particle Statistics Look at thousands of random slices to obtain H and K Example probabilities of for simple particles probability 0.10.20.30.40.50.60.70.80.91 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 sphere cube long ellipsoid flat ellipsoid A/A max probability

21. Recovering Particle Distributions Just like before, Use N V to populate a synthetic volume. Solving for the particle densities,

22. Recovering Color Select mean particle colors from segmented regions in the input image Input Mean Colors Synthetic Volume

23. Recovering Noise How can we replicate the noisy appearance of the input? - = Input Mean Colors Residual The noise residual is less structured and responds well to Heeger & Bergen’s method Synthesized Residual

24. without noise Putting it all together Input Synthetic volume with noise

25. Prior Work – Revisited InputHeeger & Bergen ’95Our result

26. Results – Physical Data Physical Model Heeger & Bergen ’95 Our Method

27. Results Input Result

28. Results InputResult

29. Summary Particle distribution –Stereological techniques Color –Mean colors of segmented profiles Residual noise –Replicated using Heeger & Bergen ’95

30. Future Work Automated particle construction Extend technique to other domains and anisotropic appearances Perceptual analysis of results

31. Thanks to… Maxwell Planck, undergraduate assistant Virginia Bernhardt Bob Sumner John Alex