Summer Seminar

1. Summer Seminar Ruizhen Hu

2. Sampling • Spectral Sampling of Manifolds (Siggraph Asia 2010) • Accurate Multidimensional Poisson-Disk Sampling (TOG) • Efficient Maximal Poisson-Disk Sampling • Blue-Noise Point Sampling using Kernel Density Model • Differential Domain Analysis for Non-uniform Sampling Noise & filtering • Filtering Solid Gabor Noise • Accelerating Spatially Varying Gaussian

3. Spectral Sampling of Manifolds A. Cengiz Öztireli Marc Alexa Markus Gross ETH Zürich TU Berlin ETH Zürich

4. Authors CENGİZ ÖZTİRELİ PhD CandidateComputer Graphics LaboratoryETH Zürich Research interests: Reconstruction, sampling and processing of surfaces, and sketch based modeling. Marc Alexa Professor Electrical Engineering & Computer Science TU Berlin Markus Gross ProfessorDepartment of Computer Science ETH Zürich Research interests: Computer graphics, image generation, geometric modeling, computer animation, and scientific visualization

5. Motivation • Goal: finding optimal sampling conditions for a given surface representation • Work: propose a new method to solve this problem based on spectral analysis of manifolds, kernel methods and matrix perturbation theory

6. Contributions • Efficient, simple to implement, easy to control through intuitive parameters, feature sensitive • Result in accurate reconstructions with kernel based approximation methods and high quality isotropic samplings • A discrete spectral analysis of manifolds using results from kernel methods and matrix perturbation theory

7. Main Algorithm • Input: a set of points lying near a manifold with normals + a kernel function definition = a continuous surface

8. Algorithms for Sampling • Subsampling: measuring the effect of a point on the manifold using the Laplace-Beltrami spectrum measures the contribution of a point to the manifold definition

9. Algorithms for Sampling • Resampling: maximizing and equalizing s (x) for all points • use local operations and move points in a simple gradient ascent

10. Results

11. Results

12. Results

13. Conclusions • New algorithms for the simplification and resampling of manifolds depending on a measure that restricts changes to the Laplace-Beltrami spectrum • Limitations: the algorithms are greedy and thus not theoretically guaranteed to give the optimal sampling • Future Directions: • texture on a surface • isotropic adaptive remeshing

14. Accurate Multi-Dimensional Poisson-Disk Sampling Manuel N. Gamito   Steve Maddock Lightwork Design Ltd The University of Sheffield

15. Authors Manuel Noronha Gamito software engineerLightwork Design Ltd Steve Maddock Senior Lecturer The University of Sheffield Research interests: character animation, specifically modelling and animating faces

16. Poisson-Disk Sampling • Definition: • Each sample is placed with uniform probability density • No two samples are closer than , where is some chosen distribution 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]

17. 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]

18. Main Algorithm

19. Radius vs. Number of Samples • A distribution can be specified by supplying either • The distribution radius r • The desired number of samples N • When the number of samples is specified • The algorithm uses a radius r based on N and on the measured packing density of sample disks • 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%)

20. Results • Number of samples • Sampling time • Samples per second

21. Results

22. Conclusions • 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

23. Future Work • Make it multi-threaded • Distant parts of the domain can be sampled in parallel with different threads • Some synchronisation between threads is still required • Generate non-uniform distributions • Have the distribution radius be a function of the position in the domain • Work over irregular domains • Discard subdivided tree nodes that fall outside the domain

24. Efficient Maximal Poisson-Disk Sampling

25. Authors Mohamed S. Ebeida post-doctor Carnegie Mellon university Andrew Davidson PhD Carnegie University of California, Davis Patrick M. Knupp Distinguished Member Technical Staff Sandia National Laboratories Anjul Patney PhD Carnegie University of California, Davis Scott A. Mitchell Principal Member of Technical Staff Sandia National Laboratories John D. Owens Associate Professor Carnegie University of California, Davis

26. Work • generating a uniform Poisson-disk sampling that is both maximal and unbiased over bounded non-convex domains

27. Motivation • Maximal Poisson-disk sampling distributions: • Avoid aliasing • Have blue noise property • Bias-free: • Crucial in fracture propagation simulations

28. Conditions • Maximal : the sample disks overlap cover the whole domain leaving no room to insert an additional point • Bias-free the likelihood of a sample being inside any subdomain is proportional to the area of the subdomain, provided the subdomain is completely outside all prior samples’ disks

29. Previous methods • relax the unbiased or maximal conditions, or require potentially unbounded time or space • Dart-throwing • unbiased but also not maximal • Tile-based • biased and require relatively large storage.

30. Main Algorithm • First phase: • an unbiased, near-maximal covering • voids: the part of a grid cell outside all circles • Second phase: • completes the maximal covering • darts are thrown directly into the voids, maintaining the bias-free condition • A maximal distribution is achieved when the domain is completely covered, leaving no room for new points to be selected

31. Sequential Sampling • Generate a background grid; mark interior and boundary cells • Phase I. Throw darts into square cells; remove hit cells • Generate polygonal approximations to the remaining voids • Phase II. Throw darts into voids; update remaining areas

32. Algorithm through Phase I

33. Voids • Polygonal approximations to arc-voids

34. Results

35. Implementation Performance

36. Conclusions • An efficient algorithm for maximal Poisson-disk sampling in two-dimensions • the final result is provably maximal • the sampling is unbiased • it is O(n log n) in expected time • it is O(n) in deterministic memory required • not limited to convex domains • efficiently implemented in both sequential and parallel forms • Future work: 3D maximal Poisson-disk sampling algorithm

37. Blue-Noise Point Sampling using Kernel Density Model Raanan Fattal Hebrew University of Jerusalem, Israel

38. Author Raanan Fattal Alon faculty member School of Computer Science and Engineering The Hebrew University of Jerusalem

39. Work • A new approach for generating point sets with high-quality blue noise properties that formulates the problem using a statistical mechanics interacting particle model

40. Contributions • present a new approach that formulates the problem using a statistical mechanics interacting particle model • derive a highly efficient multi-scale sampling scheme for drawing random point distributions • avoids the critical slowing down phenomena that plagues this type of models

41. Previous work • Dart throwing • constrain a minimal distance between every pair of points • Relaxation • follow a greedy strategy that maximizes this distance • two main shortcomings: • teriminatin • impreciseness : apparent blur

42. New Approach • model the target density as a sum of nonnegative radially-symmetric kernels The j-th kernel centered around the point xj:

43. New Approach • The error of this approximation • Minimizing E, with respect to the kernel centers • equivalent to the one obtained by converged Lloyd’s iterations • has the ability to achieve spectral enhancement • We unify error minimization and randomness by defining a statistical mechanics particle model using E

44. New Approach • Assigning each configuration a probability density according to the following Boltzmann-Gibbs distribution:

45. Drawing samples • Markov-chain Monte Carlo(MCMC) • Gibbs sampler • Langevin method • Metropolis-Hastings(MH) test

46. Critical slowing down

47. Multi-scale sampling

48. Results

49. Results

50. Differential Domain Analysis for Non-uniform Sampling Li-Yi Wei Rui Wang Microsoft Research University of Massachusetts Amherst