Stereological Techniques for Solid Textures

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

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

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 In Our Toolbox… The study of 3D properties based on 2D observations.

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

7. Prior Work – Texture Synthesis Input Heeger & 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. The fundamental relationship of stereology: Recovering Sphere Distributions = Profile density (number of circles per unit area) = Particle density (number of spheres per unit volume) = Mean caliper particle diameter

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 particleNV(j)exhibits profileNA(i)

13. Recovering Sphere Distributions How many profiles of the smallest size? + + + = = Probability that particleNV(j)exhibits profileNA(i)

14. Recovering Sphere Distributions Putting it all together… =

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

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

17. Testing precision Input distribution Estimated distribution

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

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

20. 0.45 sphere 0.4 cube 0.35 long ellipsoid flat ellipsoid 0.3 0.25 probability 0.2 0.15 0.1 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A / A max Particle Statistics Look at thousands of random slices to obtain H and K Example probabilities of for simple particles probability

21. Solving for the particle densities, Recovering Particle Distributions Just like before, UseNVto populate a synthetic volume.

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

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

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

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

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

27. Results Input Result

28. Results Input Result

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