Presentation of the ACM Tog paper at Siggraph 2010.

2. SIGGRAPH 2010 Accurate Multi-DimensionalPoisson-Disk Sampling Manuel Gamito and Steve Maddock Lightwork Design Ltd – The University of Sheffield

3. Introduction Definition of Poisson-Disk sampling Previous Poisson-Disk sampling methods Use of a spatial subdivision tree for sampling Description of the sampling algorithm Results and Conclusions Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling

4. Poisson-Disk Sampling Definition: Each sample is placed with uniform probability density No two samples are closer than 2𝑟, where 𝑟 is some chosendistribution radius A distribution is maximal if no more samples can be inserted Poisson-Disk sampling is useful for: Distributed ray tracing [Cook 1986; Hachisuka et al. 2008] Object placement and texturing[Lagae and Dutré 2006; Cline et al. 2009] Stippling and dithering[Deussen et al. 2000; Secord et al. 2002] Global Illumination [Lehtinen et al. 2008] Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling

5. Previous Methods Approximate Methods Relax at least one of the sampling conditions Accurate Methods Brute force Dart Throwing [Dippé and Wold 1985] Assisted by a spatial data structure Voronoi diagram [Jones 2006] Scalloped sectors [Dunbar and Humphreys 2006] Uniform grid [Bridson 2007] Simplified subdivision tree and uniform grid [White et al. 2007] Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling

6. Sampling with Subdivision Trees Quadtree in 2D, Octree in 3D,... Signals the regions of space where sample insertion is allowed Node A has already been removed from the tree – no new samples allowed there A new sample randomly inserted in node B is always accepted A new sample randomly inserted in node C may have to be rejected Each leaf node keeps a list of the samples that intersect with it Allows efficient lookup of neighbour samples A B 2r 2r C Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling

7. The Main Algorithm While tree not empty Randomly traverse tree towards a leaf node Generate a sample randomly in the leaf node If sample is accepted Insert sample in the tree Else Subdivide leaf node An empty tree signals that a maximal distribution has been reached The relative areas of the child nodes are used as the probabilities of choosing a child for descent Sample is accepted or rejected by looking at the leaf node’s sample list Tree is recursively updated to reflect the disk occupied by the new sample See example in later slide Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling

8. First subdivision level Child probabilities: Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling Tree Descent We choose the right-top child

9. Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling Tree Descent Second subdivision level Child probabilities: The right-top child is inevitably chosen

10. Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling Tree Descent Third subdivision level Child probabilities: The left-top leaf node is finally chosen for sample insertion

11. A valid sample has been placed in the chosen leaf node A recursive descent of the tree with the new sample finds: Node A fully inside the new disk Itself and its descendants are deleted Node B fully outside the new disk Itself and its descendants are left untouched Any remaining leaf node that straddles the boundary of the new disk is subdivided Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling Tree Updating A B

12. Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling Leaf Node Subdivision Original Leaf Node Is intersected by the disks of samples s1 and s2 After subdivision we have: Node A is inside the disk of s1 It is deleted Node B is a new leaf node It has s1 and s2 as its samples Nodes C and D are new leaf nodes They both have s1 as their only sample A B s2 C D s1

13. A distribution can be specifiedby supplying either The distribution radius 𝑟 The desired number of samples 𝑁 When the number of samples is specified The algorithm uses a radius 𝑟𝑁,𝛾 based on 𝑁 and on the measured packing density 𝛾of sample disks (see paper for the maths) The packing density was obtained by averaging the packing densities measured from 100 distributions generated by our algorithm The number of samples of the resulting maximal distribution is approximately equal to the desired number 𝑁 (𝑒𝑟𝑟𝑜𝑟<5%) Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling Radius vs. Number of Samples

14. Number of samples Sampling time Samples per second Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling Results

15. Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling Results

16. A Poisson-Disk Sampling Algorithm that Is statistically correct (see proof in paper) Is efficient through the use of a subdivision tree Works in any number of dimensions Subject to available physical memory Generates maximal distributions Allows approximate control over the number of samples Can enforce periodic or wall boundary conditions on the boundaries of the domain Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling Conclusions

17. Make it multi-threaded Distant parts of the domain can be sampled inparallel with different threads Some synchronisation between threads is still required Generate non-uniform distributions Have the distribution radius 𝑟(𝐱) be a function of theposition 𝐱 in the domain Work over irregular domains Discard subdivided tree nodes that fall outside the domain Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling Future Work

18. Source code available at http://www.dcs.shef.ac.uk/~mag/poisson.html Gamito and Maddock – Accurate Multi-dimensional Poisson-Disc Sampling Thank you!