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Stereological Techniques for Solid Textures Rob Jagnow MIT Julie Dorsey Yale University Holly Rushmeier Yale University Objective Given a 2D slice through an aggregate material, create a 3D volume with a comparable appearance. Real-World Materials • • • • Concrete Asphalt Terrazzo Igneous minerals • Porous materials Independently Recover… • Particle distribution • Color • Residual noise In Our Toolbox… e Stereology (ster'e-ol' -je) The study of 3D properties based on 2D observations. Prior Work – Texture Synthesis • 2D 2D • 3D 3D • Procedural Textures Efros & Leung ’99 • 2D 3D – Heeger & Bergen 1995 – Dischler et al. 1998 – Wei 2003 Heeger & Bergen ’95 Wei 2003 Prior Work – Texture Synthesis Input Heeger & 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 • d max: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 N A = Profile density (number of circles per unit area) NV = Particle density (number of spheres per unit volume) H = Mean caliper particle diameter The fundamental relationship of stereology: N A H NV Recovering Sphere Distributions Group profiles and particles into n bins according to diameter Particle densities = N A (i), {1 i n} Profile densities = NV (i), {1 i n} Densities NV , K ij N A are related by the values K ij Relative probabilities: j n -a sphere in the j th histogram bin with diameter -a profile in the i th histogram bin with diameter (i 1) i d n n 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? = N A (4) K ij K 44 NV (4) = Probability that particle NV(j) exhibits profile NA(i) Recovering Sphere Distributions How many profiles of the smallest size? = + N A (1) K11NV (1) K ij + K12 NV (2) + K13 NV (3) K14 NV (4) = Probability that particle NV(j) exhibits profile NA(i) Recovering Sphere Distributions Putting it all together… = NA K NV Recovering Sphere Distributions Some minor rearrangements… N A = d max K NV d max = Maximum diameter Normalize probabilities for each column j: n K i 1 ij j/n Recovering Sphere Distributions N A d max KNV K is upper-triangular and invertible For spheres, we can solve for K analytically: 1 / n K ij j 2 (i 1) 2 0 j2 i2 for j i otherwise Solving for particle densities: NV 1 d max K 1 N A Other Particle Types We cannot classify arbitrary particles by d/dmax Instead, we choose to use A / Amax Algorithm inputs: + Approach: Collect statistics for 2D profiles and 3D particles Profile Statistics Segment input image to obtain profile densities NA. Input Segmentation Bin profiles according to their area, A / Amax Particle Statistics • Polygon mesh:random orientation • Render Particle Statistics Look at thousands of random slices to obtain H and K 0.45 sphere cube long ellipsoid flat ellipsoid 0.4 0.35 probability 0.3 Example probabilities of A / Amax for simple particles 0.25 0.2 0.15 probability 0.1 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 A/Amax 0.7 0.8 0.9 1 Scale Factor • Scale factor s :to relate the size of particle P to the size of the particles in input image s Aimg /APmax – profile maximum area • Aimg :input image • APmax:particle P • Mean caliper diameter H sH P Recovering Particle Distributions Just like before, N A H KNV Solving for the particle densities, NV 1 1 H K NA Use NV to populate a synthetic volume. Managing Multiple Particle Types • particle type:i • • • • mean caliper diameter:H i representative matrix:K i distribution: NVi probability that a particle is type i :P( i ) N A ( H i K i NVi ) i N A ( H i K i P(i ) NV ) i ( H i K i P(i ))NV i • total particle density: NV NVi 1 NV ( H i K i P(i)) N A i 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 Putting it all together Input Synthetic volume without noise Prior Work – Revisited Input Heeger & Bergen ’95 Our 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 Input Result 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