1. Outline ● Introduction – What is the problem ● Generate stochastic textures ● Improve realism ● High level approach - Don't just jump into details – Why is it important – High level approach to solution ● Stochastic geometry overview – Main point - introducing stochastic geometry to CG – Stochastic textures with controllable properties ● Stochastic reconstruction – Input info / values -> ideal geometry -> distribution – Reconstruction algorithm ● Bin packing (makes easier to understand) – Figure 7, on separate slides ● "Animation" ● Point things out on the slides ● Talk through the algorithm as I go – Description of offset parameters ● Offset function ● Show sqrt vs regular – Approaches to usage ● Sample parameter space ● User interface ● Results ● Same distribution on different geometry ● Different distribution on same geometry ● Pictures ● Another tool for creating stochastic textures

2. Stochastic Geometry for Displacement Mapping Craig Schroeder, David Breen, Christopher Cera, William Regli June 16, 2005

3. Problem ● Given – Ideal, smooth object with little detail ● Want – Object with stochastic, textured microdetails

4. Motivation ● Improve realism ● Add stochastic detail ● Creative tool for producing interesting models

5. Overview of Approach ● Input – Feature distribution – Parameters – Polygonal mesh ● Output – Polygonal mesh with stochastic microgeometry

6. Overview of Approach ● Input preprocessing ● Label assignment ● Compute displacement

7. Stochastic Geometry ● Distinct branch of mathematics* – Study of random processes whose outcomes are geometrical objects or spatial patterns – Statistical analysis of geometry * In constrast to the generic computer graphics usage

8. Stochastic Geometry ● What is most important to us – “Isotropic” ● Invariant under rotation – “Spherical Contact Distribution” ● For all points outside object – Distance r to object – Values r form a distribution ● We use this plus its complement

9. Algorithm Input ● Ideal geometry ● Stochastic properties – Spherical contact distribution ● Algorithm parameters

10. Offset Parameters ● Independent for negative and positive regions – Scale factor ● Convert labels into units of geometry – Offset inwards or outwards – Offset function

11. Algorithm Setup ● Mesh is composed of triangles – Subdivide as needed ● Convert stochastic information into a histogram – Positive, Negative regions – Normalize to number of triangles

12. Algorithm ● Each mesh triangle is a bin – Assign one label per bin ● For each bar of the histogram – Outside inwards, ending with bar zero – Assign to bins where forced to by constraint – Assign remaining uniformly

13. Algorithm ● Compute vertex displacement – Average surrounding bin labels – Apply a function ● Displace the mesh geometry

14. Offset Function ● Changes “shape” of bumps – Eg: f(x) = x or f(x) = sqrt(x)

15. Sample Run

23. Parameter Space Exploration ● Sample space of surface characteristics ● Look for good candidates

24. User Interface ● Fine tune parameters ● Interactive

25. Results – Same Distribution

26. Results – Same Geometry

27. Results

28. Overview of Approach ● Setup – Convert feature distribution into a histogram – Mesh is composed of triangles ● Each triangle is a “bin” ● Bins must be filled with labels, one per bin ● Label determines magnitiude of displacement

29. Overview of Approach ● Label Assignment – Use the histogram to fill in labels for each triangle – Labels must satisfy a constraint ● Displacement Mapping – Use labels to determine displacement ● Apply a function to the label ● Displace vertices by the computed value