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

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

Real-World Materials Concrete Asphalt Terrazzo Igneous minerals Porous materials

Independently Recover… Particle distribution Color Residual noise

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

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

Prior Work – Texture Synthesis InputHeeger & Bergen, ’95

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

Estimating 3D Distributions Macroscopic statistics of a 2D image are related to, but not equal to the statistics of a 3D volume –Distributions of Spheres –Distributions for Other Particles –Managing Multiple Particle Types

Distributions of Spheres ： maximum diameter Establish a relationship between –the size distribution of 2D circles (as the number of circles per unit area) –the size distribution of 3D spheres (as the number of spheres per unit volume)

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:

Recovering Sphere Distributions Group profiles and particles into n bins according to diameter Particle densities = Profile densities = Densities, are related by the values Relative probabilities ： － a sphere in the j th histogram bin with diameter － a profile in the i th histogram bin with diameter

Recovering Sphere Distributions Note that the profile source is ambiguous For the following examples, n = 4

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

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

Recovering Sphere Distributions Putting it all together… =

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

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

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: +

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

Particle Statistics Polygon mesh ： random orientation Render

Particle Statistics Look at thousands of random slices to obtain H and K Example probabilities of for simple particles probability sphere cube long ellipsoid flat ellipsoid A/A max probability

Scale Factor Scale factor s ： to relate the size of particle P to the size of the particles in input image –profile maximum area ： input image ： particle P Mean caliper diameter

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

Managing Multiple Particle Types particle type ： i mean caliper diameter ： representative matrix ： distribution ： probability that a particle is type i ： P( i ) total particle density ：

Reconstructing the Volume Particle Positions Color Adding Fine Detail

Particle Position - Annealing Populate the volume with all of the particles, ignoring overlap Perform simulated annealing to resolve collision –Repeatedly searches for all collision (in the x, y, z directions) –Relaxes particle positions to reduce interpenetration

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

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

without noise Putting it all together Input Synthetic volume

Prior Work – Revisited InputHeeger & Bergen ’95Our result

Results- Testing Precision Input distribution Estimated distribution

Result- Comparison

Collection of Particle Shapes Can’t predict exact particle shapes Unable to count small profiles Limited to fewer profile observation Calculations error

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

Results Input Result

Results InputResult

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

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