Capacity-Constrained Point Distributions

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In this presentation, we will speak about the main idea of the article entitled 'Capacity-Constrained Point Distributions: A Variant of Lloyd's Method' [Balzer, M. et al. 2009] and we will show some results obtained by applying of this method. In the aforementioned article the authors present a new general-purpose method for optimizing existing point sets. The resulting distributions possess high-quality blue noise characteristics and adapt precisely to given density functions. Among the results we can highlight the generation of distributions using samples guided by functions of type z=f(x, y) and samples from images (simulating stippling technique).

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Capacity-Constrained Point Distributions

  1. 1. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Capacity-Constrained Point Distributions A Variant of Lloyd’s Method Michel Alves dos Santos Pós-Graduação em Engenharia de Sistemas e Computação Universidade Federal do Rio de Janeiro - UFRJ - COPPE Cidade Universitária - Rio de Janeiro - CEP: 21941-972 Docentes Responsáveis: Prof. Dsc. Ricardo Marroquim & Prof. PhD. Cláudio Esperança {michel.mas, michel.santos.al}@gmail.com January, 2013 Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  2. 2. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Introduction Applications of Point Distributions... Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Presentation hosted on: http://www.lcg.ufrj.br/Members/malves/index Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  3. 3. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Introduction Applications of Point Distributions... Sampling Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  4. 4. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Introduction Applications of Point Distributions... Sampling Point-Based Rendering Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  5. 5. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Introduction Applications of Point Distributions... Sampling Point-Based Rendering Geometric Processing Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  6. 6. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Introduction Applications of Point Distributions... Sampling Point-Based Rendering Geometric Processing Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Halftoning etc... Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  7. 7. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Desidered Properties Desidered Properties for Point Distributions... Red Noise White Noise Blue Noise Blue noise features; Similar relative distance between points; No regular appearance (For most applications); Adaptation to the provided density functions. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  8. 8. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Desidered Properties Desidered Properties for Point Distributions... Red Noise White Noise Blue noise features; Similar relative distance between points; No regular appearance (For most applications); Adaptation to the provided density functions. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Blue Noise In this presentation we will discuss about a technique for optimal distribution of points! Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  9. 9. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Capacity-Constrained Point Distributions Capacity-Constrained Point Distributions: A Variant of Lloyd’s Method Michael Balzer Thomas Schl¨ mer o University of Konstanz, Germany Oliver Deussen Figure 1: (Left) 1024 points with constant density in a toroidal square and its spectral analysis to the right; (Center) 2048 points with the 2 2 density function ρ = e(−20x −20y ) + 0.2 sin2 (πx) sin2 (πy); (Right) 4096 points with a density function extracted from a grayscale image. Abstract New that point distributions adapt to density function in general-purpose method for optimizingpoints in an a givenpoint sets;density. existingis proportional to the the sense that the number of area We present a new general-purpose method for optimizing existing point sets. The resulting distributions possess high-quality blue noise characteristics and adapt precisely to given density functions. Our method is similar to the commonly used Lloyd’s method while avoiding its drawbacks. We achieve our results by utilizing the concept of capacity, which for each point is determined by the area of its Voronoi region weighted with an underlying density function. We demand that each point has the same capacity. In combination with a dedicated optimization algorithm, this capacity constraint enforces that each point obtains equal importance in the distribution. Our method can be used as a drop-in replacement for Lloyd’s method, and combines enhancement of blue Gráfica - LCG Michel Alves dos Santos: Laboratório de Computaçãonoise characteristics The iterative method by Lloyd [1982] is a powerful and Resulting distributions possess high-qualitycommonly noise featuresand flexible blue used to enhance the spectral properties technique that is adapt precisely to given density; of existing distributions of points or similar entities. However, the results from Lloyd’s method are satisfactory only to a limited ex- tent. First, if the method is not stopped at a Similar to the commonly used Lloyd’s Method while develop suitable iteration step, the resulting point distributions will avoiding its regularity artifacts, as shown in Figure 2. A reliable universal termination criterion to drawbacks; prevent this behavior is unknown. Second, the adaptation to given heterogenous density functions is suboptimal, requiring additional application-dependent optimizations to improve the results. We present a variant of Lloyd’s method which reliably converges toPós-Graduação em Engenharia de Sistemas e Computação - PESC
  10. 10. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Proposed Method initial point set Lloyd’s method α ≈ 0.75 α converged our method (converged) zone plate test function 1024 points and their Fourier amplitude sprectrum α ≈ 0.53 input sites initial state −→ capacity-constrained optimization −→ final state output sites Figure 3: Our method takes an existing site distribution and transfers it to a random discrete assignment in which each site has the same Figure 5:This initial set of is thenpoints is optimizedVoronoi regions are formed and sites are relocatedarethe centroids of their regions, while capacity. An assignment 1024 optimized so that by Lloyd’s method. After 40 iterations the points to well distributed with a normalized radius of α ≈ 0.75 Applications: characteristics. HDR Sampling an equilibriumspectral properties and introduces hexagonal and good blue noise for each site. The optimization stops deteriorates the state with the final site distribution. simultaneously maintaining the capacity Stippling, Further optimizationat Radiance/Luminance,2 etc. structures. In contrast, α ≈ 0.75 proves to be ill-suited for the sampling of the zone plate test function with 512 points as strong artifacts become apparent. Relying on the convergence of α is also not an option as only marginally fewer artifacts can be observed. In this sampling scenario, stopping Lloyd’s method after about 10 iterations with α ≈ 0.53 would provide the best sampling results. Our method converges 2. move each site siem Engenharia de Sistemas of Computação - PESC reliably to an equilibrium with better Voronoi Tessellation Michel AlvesAlgorithm 1: Capacity-Constrainedproperties in both - LCG dos Santos: Laboratório de Computação Gráfica scenarios. Pós-Graduação ∈ S to the center of mass e all points
  11. 11. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Proposed Method :: Steps and Details Density Function → Samples → Generation of Sites → Optimization → Optimized Sites Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  12. 12. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Lloyd’s Method Figure: (Left) Random dots (red) and polygons. (Right) Result after running approximate Lloyd relaxation twice - note the artifacts produced by technique. Used to enhance the spectral properties of existing point distributions. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  13. 13. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Lloyd’s Method Figure: (Left) Random dots (red) and polygons. (Right) Result after running approximate Lloyd relaxation twice - note the artifacts produced by technique. But this method presents regularity in distribution! Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  14. 14. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Lloyd’s Method Figure: (Left) Random dots (red) and polygons. (Right) Result after running approximate Lloyd relaxation twice - note the artifacts produced by technique. Difficulty in stopping criterion! Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  15. 15. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Lloyd’s Method Figure: (Left) Random dots (red) and polygons. (Right) Result after running approximate Lloyd relaxation twice - note the artifacts produced by technique. And poor adaptation to heterogeneous density functions! Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  16. 16. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Capacity Constrained Vs. Lloyd’s Method Capacity-Constrained Lloyd’s Method CCPD is a variation of the Lloyd’s Method that converges in a natural way and that in addition not presents the appearance of regularity still fits precisely to given density functions. CCPD Uses: Complexity Metrics or Distance Functions; Lloyd Centroidal Voronoi Tessellations; CCPD The Concept of Capacity; Minimization of Energy (through a Optimization Method). Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Memory Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  17. 17. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Metrics or Distance Functions Voronoi Tesselations Using Minkowski Metrics: L1 , L2 , L3 , L4 , L5 , L∞ . Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  18. 18. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Metrics or Distance Functions :: Manhattan or L1 d (x, y) = L1 (x, y) = Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG n i =1 |xi − yi | Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  19. 19. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Metrics or Distance Functions :: Euclidean or L2 d (x, y) = L2 (x, y) = ( Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG n i =1 |xi − yi |2 )1/2 Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  20. 20. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Metrics or Distance Functions :: L3 d (x, y) = L3 (x, y) = ( Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG n i =1 |xi − yi |3 )1/3 Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  21. 21. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Metrics or Distance Functions :: L4 d (x, y) = L4 (x, y) = ( Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG n i =1 |xi − yi |4 )1/4 Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  22. 22. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Metrics or Distance Functions :: L5 d (x, y) = L5 (x, y) = ( Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG n i =1 |xi − yi |5 )1/5 Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  23. 23. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Metrics or Distance Functions :: Chebyshev or L∞ d (x, y) = L∞ (x, y) = limp→∞ ( Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG n i =1 |xi − yi |p )1/p = maxin (|xi − yi |) =1 Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  24. 24. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Toroidal Square Distance 1 1 2 3 4 6 1 2 4 5 7 6 1 For the current work we used a metric based on a toroidal square. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  25. 25. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Toroidal Square Distance :: Algorithm § 1 2 3 4 5 6 7 8 9 10 11 12 13 /∗ ∗ ∗ Method r e s p o n s i b l e by d i s t a n c e c a l c u l a t i o n s [ t o r o i d a l s q u a r e ] . ∗ @ v a r i a b l e p1 and p2 : p o i n t s on t o r o i d a l s q u a r e . ∗ @ v a r i a b l e s i z e : keeps the dimensions of the input square . ∗/ d o u b l e TSD( c o n s t P o i n t 2& p1 , c o n s t P o i n t 2& p2 , c o n s t P o i n t 2& s i z e ) { d o u b l e dx = p1 . x − p2 . x ; i f ( f a b s ( dx ) > s i z e . x / 2 ) { i f ( p1 . x < s i z e . x / 2 ) dx = p1 . x − ( p2 . x − s i z e . x ) ; e l s e dx = p1 . x − ( p2 . x + s i z e . x ) ; } ¤ 14 d o u b l e dy = p1 . y − p2 . y ; i f ( f a b s ( dy ) > s i z e . y / 2 ) { i f ( p1 . y < s i z e . y / 2 ) dy = p1 . y − ( p2 . y − s i z e . y ) ; e l s e dy = p1 . y − ( p2 . y + s i z e . y ) ; } 15 16 17 18 19 20 21 22 23 } return s q r t ( dx ∗ dx + dy ∗ dy ) ; ¦ Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG ¥ Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  26. 26. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Toroidal Square Distance :: Example Optimized Sites Voronoi Tessellation Note the regions that lie within the limits. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  27. 27. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Toroidal Square Distance :: Example Optimized Sites Voronoi Tessellation Note the regions that lie within the limits. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  28. 28. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Toroidal Square Distance :: Example Optimized Sites Voronoi Tessellation Note the regions that lie within the limits. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  29. 29. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Toroidal Square Distance :: Example Optimized Sites Voronoi Tessellation Note the regions that lie within the limits. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  30. 30. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Toroidal Square Distance :: Example Optimized Sites Voronoi Tessellation Note the regions that lie within the limits. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  31. 31. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Toroidal Square Distance :: Example Optimized Sites Voronoi Tessellation Note the regions that lie within the limits. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  32. 32. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Centroidal Voronoi Tessellation Non-Centroidal Voronoi Centroidal Voronoi 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 CVT is a Voronoi Tesselation with the property that each site itself coincides with the centroid of their respective Voronoi region. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  33. 33. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Centroidal Voronoi Tessellation :: Applications Optimal quadrature rules; Covolume and finite difference methods for PDE’s; Optimal representation, quantization, and clustering; Optimal placement of sensors and actuators; Optimal distribution of resources; Cell division; Finite volume methods for PDE’s; Territorial behavior of animals; Data compression; Image segmentation; Meshfree methods; Grid generation; Point distributions and grid generation on surfaces; Hypercube point sampling; Reduced-order modeling; Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  34. 34. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Centroidal Voronoi Tessellation :: Centroids The centroid of a Voronoi region is nothing but the center of mass of a region weighted by the density function defined in V area. z= Vi Vi xρ(x)dx ρ(x)dx zi∗ = M i =1 xi ρ(xi ) M ρ(x ) i i =1 For discrete sets of points we have V = {xi }M in Rn and a density i =1 function ρ(xi ), i = 1, · · · , M. The center of mass is given by zi∗ . The importance of centroidal Voronoi tessellation is established by its relationship with the energy function: M F (S, V ) = i =1 Vi ρ(x)|x − si |2 dx S → sites; V → voronoi regions; ρ(x) → density function; F (S, V ) → energy function Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  35. 35. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Centroidal Voronoi Tessellation :: Using Lloyd Steps for generating a Centroidal Voronoi Tessellation: 1 2 3 Generate a Voronoi tessellation V (S) in a region Ω; Move each site si ∈ S to the centroid pi of corresponding Voronoi region vi ∈ V ; Repeat the previous steps until the sites reach a convergence criterion. Energy Equation: m |xi − A(xi )|2 F (X, A) = i =1 1. The relocation of the sites in the centroid position reduces energy F . 2. The algorithm converges to a local minimum F, where each site coincides with the centroid of the region. 3. In the discrete case, the limited space Ω with density function ρ is represented by a set X with m samples. A : X → S takes each point in X to the nearest site in S. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  36. 36. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 The Concept of Capacity :: Capacity-Constrained Let S be a set with n sites which determine a Voronoi tessellation V (S) in limited space Ω with function density ρ(x ). Definition: The capacity c(si ) of a site si ∈ S with respect to its respective Voronoi region Vi ∈ V is defined as: c(si ) = ρ(x)dx Vi We say that a distribution of sites in S adapts optimally to a density function if the capacity of each site follows: ρ(x)dx c(si ) = c ∗ , where c ∗ is defined as c ∗ = Ω n In other words, the capacity of a site is equivalent to the area of Voronoi region weighted by the density function. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  37. 37. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Minimization of Energy :: Voronoi Tessellation Steps for Minimization: 1 2 Generate the Voronoi tessellation doing the assignment A : X → S of m points in X for n sites in S with capacity c ∗ . Minimize the function F (X , A) by swap between two points in X that belong to different sites in S, with the condition that the energy is reduced; Restrictions to make the swap ensure that capacity is maintained even after the minimization. 3 Repeat the exchange until a stage of stability is achieved. input sites initial state −→ capacity-constrained optimization −→ final state output sites Figure 3: Our method takes an existing site distribution and transfers it to a random discrete assignment in which each site has the same capacity. This assignment is then optimized so that Voronoi regions are formed and sites are relocated to the centroids of their regions, while simultaneously maintaining the capacity for each site. The optimization stops at an equilibrium state with the final site distribution. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  38. 38. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Overview of Capacity-Constrained Method Capacity-Constrained: The Capacity-Constrained method differs from the usual Voronoi tessellation because it is generated taking into account the capacity of each site and only optimizing their locations. Steps to generate the capacity-constrained distribution: 1 Create a set X with m points weighted by the density function ρ(x ); 2 Generate the Voronoi tesselation V (S) with conditioned capacity for the set of n sites S, where each site si has capacity c(si ) = m/n; 3 Move each site si ∈ S to the center of mass of all points xi ∈ X ; 4 Repeat steps 2 and 3 until the new sites achieve the convergence criterion. In possession of present provided theory, we will see some evaluations of results obtained according to Balzer et al. (2009). Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  39. 39. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Evaluation of Results The method presented here achieves better results when compared to the Lloyd’s Method, in some respects: Blue-noise features; Number of neighbors; Stopping criteria; Measuring the quality of the adaptation. 100 % percentage 60 % 40 % 20 % 0% 4 5 6 7 8 0.95 1.1 0.85 our method 1.0 0.9 Lloyd’s method 0.7 our method 4 number of neighbors (a) number of neighbors percentage 5 6 7 number of neighbors (b) normalized Voronoi region area Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG 8 normalized radius α our method normalized Voronoi region area 1.2 Lloyd’s method 80 % 0.75 0.65 0.55 16 64 256 1024 4096 16384 number of sites (c) normalized radius α Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  40. 40. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Evaluation of Results :: Blue-noise and Neighbors 1024 optimized points number of neighbors 4 5 6 7 spectral analysis 8 2 power 1.5 1 Lloyd’s method 0.5 0 0 frequency fc +10 anisotropy +5 0 -5 -10 0 frequency fc 2 power 1.5 1 0.5 our method 0 0 frequency fc +10 anisotropy +5 0 -5 -10 0 frequency fc Lloyd’s method generates point distributions with regular structures. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  41. 41. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Evaluation of Results :: Stopping and Quality Stopping Criteria: In the Lloyd’s Method is necessary a manual intervention or a specific criterion determined by the application. Quality of the adaptation: The capacity offers the opportunity to measure the quality of adaptation by a distribution of sites through the errors of the capacity given by: δc = 1 n i =1  c(si )  n c∗ − 2 1 In respect of capacity: Constant Density: Lloyd generates a uniform distribution with small errors. Non-Constant Density: Lloyd generates distribution of sites with large errors. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  42. 42. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results Using: Regular Density Functions Custom Density Functions Images as Density Functions Now we will see some results obtained with the technique... Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  43. 43. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Regular Density Functions 1.01 1.005 1.01 1 1.005 0.995 1 0.99 0.995 0.99 f(x,y) = c f(x,y) = c {(x,y) | x ∈ R, y ∈ R} (x,y)-> random choice All tabled numerical results shown in this presentation are an average of 15 executions for each set of points. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  44. 44. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Constant Regular Grid Optimized Sites f (x, y) = c; Voronoi Tessellation {(x, y)|x ∈ R, y ∈ R} Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  45. 45. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Constant Regular Grid 1.01 1.005 1 1.01 0.995 1.005 1 1 0.995 0.99 0.5 0.99 0 -1 -0.5 -0.5 0 0.5 1 -1 Regular Density f(x,y) --> samples 512 Points 1024 Points 2048 Points 16384 Samples 32768 Samples 65536 Samples Figure: The figure above shows the number of points and samples for each set. In the table below we can view times of generation, times of optimization and steps until convergence. Amount of Points 512 1024 2048 Generation Time (<) 00.01 seconds (<) 00.01 seconds (<) 00.01 seconds Optimization Time 00.03 seconds 00.08 seconds 00.25 seconds Optimization Steps* 12 13 16 Optimization Steps*: number of iterations until convergence. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  46. 46. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Constant Regular Random Optimized Sites f (x, y) = c; Voronoi Tessellation (x, y) −→ random choice Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  47. 47. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Constant Regular Random Lattice test for random numbers 1.01 1.00 0.80 0.60 0.40 0.20 0.00 1.005 1 0.995 0.99 0.00 1.00 0.20 0.80 0.40 0.60 0.60 0.40 0.80 0.20 1.00 0.00 Random Density f(x,y) --> samples 512 Points 1024 Points 2048 Points 16384 Samples 32768 Samples 65536 Samples Figure: The figure above shows the number of points and samples for each set. In the table below we can view times of generation, times of optimization and steps until convergence. Amount of Points 512 1024 2048 Generation Time (<) 00.01 seconds (<) 00.01 seconds (<) 00.01 seconds Optimization Time 00.04 seconds 00.09 seconds 00.31 seconds Optimization Steps* 15 13 19 Optimization Steps*: number of iterations until convergence. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  48. 48. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Regular Grid Vs. Regular Random Grid Random 128 Optimized Sites 128 Optimized Sites Comparison between distributions obtained. Each experiment used 65536 samples. Iterations: 66 for grid density and 99 for random density. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  49. 49. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Custom Density Functions Five Mountains SinSquare SinXCosY Shadow Torus The Waves All tabled numerical results shown in this presentation are an average of 15 executions for each set of points. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  50. 50. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Five Mountains Optimized Sites f (x, y) = e(−20x 2 Voronoi Tessellation −20y 2 ) Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG + 0.2 sin2 (πx) sin2 (πy) Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  51. 51. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Five Mountains Five Mountains f(x,y) --> samples 1024 Points 2048 Points 4096 Points 32768 Samples 65536 Samples 131072 Samples Figure: The figure above shows the number of points and samples for each set. In the table below we can view times of generation, times of optimization and steps until convergence. Amount of Points 1024 2048 4096 Generation Time 15.30 seconds 75.90 seconds 355.60 seconds Optimization Time 00.12 seconds 00.39 seconds 01.29 seconds Optimization Steps* 20 13 19 Optimization Steps*: number of iterations until convergence. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  52. 52. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: SinSquare Optimized Sites Voronoi Tessellation f (x, y) = sin (x 2 y 2 ) Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  53. 53. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: SinSquare SinSquare f(x,y) --> samples 1024 Points 2048 Points 4096 Points 32768 Samples 65536 Samples 131072 Samples Figure: The figure above shows the number of points and samples for each set. In the table below we can view times of generation, times of optimization and steps until convergence. Amount of Points 1024 2048 4096 Generation Time 12.30 seconds 58.90 seconds 289.00 seconds Optimization Time 00.18 seconds 00.65 seconds 02.31 seconds Optimization Steps* 14 17 17 Optimization Steps*: number of iterations until convergence. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  54. 54. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Shadow Torus Optimized Sites Voronoi Tessellation f (x, y) = (0.16 − (0.6 − x 2 + y 2 )2 )1/2 Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  55. 55. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Shadow Torus Shadow Torus f(x,y) --> samples 1024 Points 2048 Points 4096 Points 32768 Samples 65536 Samples 131072 Samples Figure: The figure above shows the number of points and samples for each set. In the table below we can view times of generation, times of optimization and steps until convergence. Amount of Points 1024 2048 4096 Generation Time 05.60 seconds 27.10 seconds 135.20 seconds Optimization Time 00.11 seconds 00.29 seconds 01.20 seconds Optimization Steps* 20 13 23 Optimization Steps*: number of iterations until convergence. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  56. 56. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: SinXCosY Optimized Sites Voronoi Tessellation f (x, y) = 0.2 sin (5x) cos (5y) Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  57. 57. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: SinXCosY SinXCosY f(x,y) --> samples 1024 Points 2048 Points 4096 Points 32768 Samples 65536 Samples 131072 Samples Figure: The figure above shows the number of points and samples for each set. In the table below we can view times of generation, times of optimization and steps until convergence. Amount of Points 1024 2048 4096 Generation Time 34.00 seconds 151.40 seconds 813.20 seconds Optimization Time 00.12 seconds 00.36 seconds 01.33 seconds Optimization Steps* 14 14 20 Optimization Steps*: number of iterations until convergence. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  58. 58. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: The Waves Optimized Sites Voronoi Tessellation f (x, y) = x 3 − 3xy 2 Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  59. 59. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: The Waves The Waves f(x,y) --> samples 1024 Points 2048 Points 4096 Points 32768 Samples 65536 Samples 131072 Samples Figure: The figure above shows the number of points and samples for each set. In the table below we can view times of generation, times of optimization and steps until convergence. Amount of Points 1024 2048 4096 Generation Time 07.30 seconds 31.90 seconds 168.60 seconds Optimization Time 00.15 seconds 00.55 seconds 01.72 seconds Optimization Steps* 16 28 16 Optimization Steps*: number of iterations until convergence. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  60. 60. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Points and Optimization Time Number of Points and Samples Results :: Stippling :: Image as Density Function 4096 Points 16384 Points 20000 Points 393216 Samples 786432 Samples 1280000 Samples 4096 Points 8192 Points 12288 Points 02.97 seconds 09.62 seconds 17.57 seconds Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  61. 61. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Stippling - Corn Plant/Dracaena 4096 Points 16384 Points 20000 Points 393216 Samples 786432 Samples 1280000 Samples Figure: Input image with 1000x1000 pixels. The figure above shows the number of points and the number of samples for each set. In the table below we can visualize times of generation, times of optimization and steps until convergence. Amount of Points 4096 16384 20000 Generation Time 25.50 minutes 101.70 minutes 271.20 minutes Optimization Time 4.41 seconds 20.45 seconds 42.43 seconds Optimization Steps* 57 24 35 Optimization Steps*: number of iterations until convergence. Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  62. 62. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Results :: Stippling - Madonna’s Face 4096 Points 8192 Points 12288 Points 02.97 seconds 09.62 seconds 17.57 seconds Figure: Madonna’s Face. Input image with 1000x1000 pixels. The figure shows the number of points and the optimization time for each set. Amount of Points 4096 8192 12288 Generation Time 13.40 minutes 55.37 minutes 124.99 minutes Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Optimization Time 02.97 seconds 09.62 seconds 17.57 seconds Optimization Steps 38 36 38 Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  63. 63. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Conclusions Some conclusions about the method: 1.01 1.00 0.80 0.60 0.40 0.20 0.00 1.005 1 0.995 Performs distribution points optimally. It is more stable than Lloyd’s Method therefore uses the concept of capacity as a form of optimization. Improves the characteristics of the blue-noise and has no apparent regularities in the arrangement of sites. Displays ‘precise’ adaptation to arbitrary distribution functions. No manual intervention is required and neither depends on the initial distribution to generate good quality results. 0.99 0.00 1.00 0.20 0.80 0.40 0.60 0.60 0.40 0.80 Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG 0.20 1.00 0.00 Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  64. 64. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Conclusions Some conclusions about the method: 1.01 1.00 0.80 0.60 0.40 0.20 0.00 1.005 1 0.995 Performs distribution points optimally. It is more stable than Lloyd’s Method therefore uses the concept of capacity as a form of optimization. Improves the characteristics of the blue-noise and has no apparent regularities in the arrangement of sites. Displays ‘precise’ adaptation to arbitrary distribution functions. No manual intervention is required and neither depends on the initial distribution to generate good quality results. 0.99 0.00 1.00 0.20 0.80 0.40 0.60 0.60 0.40 0.80 Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG 0.20 1.00 0.00 Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  65. 65. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Conclusions Some conclusions about the method: 1.01 1.00 0.80 0.60 0.40 0.20 0.00 1.005 1 0.995 Performs distribution points optimally. It is more stable than Lloyd’s Method therefore uses the concept of capacity as a form of optimization. Improves the characteristics of the blue-noise and has no apparent regularities in the arrangement of sites. Displays ‘precise’ adaptation to arbitrary distribution functions. No manual intervention is required and neither depends on the initial distribution to generate good quality results. 0.99 0.00 1.00 0.20 0.80 0.40 0.60 0.60 0.40 0.80 Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG 0.20 1.00 0.00 Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  66. 66. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Conclusions Some conclusions about the method: 1.01 1.00 0.80 0.60 0.40 0.20 0.00 1.005 1 0.995 Performs distribution points optimally. It is more stable than Lloyd’s Method therefore uses the concept of capacity as a form of optimization. Improves the characteristics of the blue-noise and has no apparent regularities in the arrangement of sites. Displays ‘precise’ adaptation to arbitrary distribution functions. No manual intervention is required and neither depends on the initial distribution to generate good quality results. 0.99 0.00 1.00 0.20 0.80 0.40 0.60 0.60 0.40 0.80 Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG 0.20 1.00 0.00 Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  67. 67. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Conclusions Some conclusions about the method: 1.01 1.00 0.80 0.60 0.40 0.20 0.00 1.005 1 0.995 Performs distribution points optimally. It is more stable than Lloyd’s Method therefore uses the concept of capacity as a form of optimization. Improves the characteristics of the blue-noise and has no apparent regularities in the arrangement of sites. Displays ‘precise’ adaptation to arbitrary distribution functions. No manual intervention is required and neither depends on the initial distribution to generate good quality results. 0.99 0.00 1.00 0.20 0.80 0.40 0.60 0.60 0.40 0.80 Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG 0.20 1.00 0.00 Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  68. 68. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Thanks Thanks for your attention! Michel Alves dos Santos - michel.mas@gmail.com Michel Alves dos Santos - (Alves, M.) MSc Candidate at Federal University of Rio de Janeiro. E-mail: michel.mas@gmail.com, malves@cos.ufrj.br Lattes: http://lattes.cnpq.br/7295977425362370 Home: http://www.michelalves.com Phone: +55 21 2562 8572 (Institutional Phone Number) http://www.facebook.com/michel.alves.santos http://www.linkedin.com/profile/view?id=26542507 Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  69. 69. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Samples Voronoi Sites Thank you for your attention! Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC
  70. 70. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyd’s Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013 Bibliography M. Balzer, T. Schlömer, and O. Deussen. Capacity-constrained point distributions: A variant of Lloyd’s method. ACM Transactions on Graphics (Proceedings of SIGGRAPH 2009), 28(3):86:1–8, 2009. F. de Goes, K. Breeden, V. Ostromoukhov, and M. Desbrun. Blue noise through optimal transport. ACM Trans. Graph. (SIGGRAPH Asia), 31, 2012. R. Fattal. Blue-noise point sampling using kernel density model. ACM SIGGRAPH 2011 papers, 28(3):1–10, 2011. H. Li, D. Nehab, L.-Y. Wei, P. V. Sander, and C.-W. Fu. Fast capacity constrained voronoi tessellation. In Proceedings of the 2010 ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games, I3D ’10, pages 13:1–13:1, New York, NY, USA, 2010. ACM. A. Secord. Weighted Voronoi stippling. In Proceedings of the second international symposium on Non-photorealistic animation and rendering, pages 37–43. ACM Press, 2002. R. Ulichney. Digital Halftoning. MIT Press, 1987. ISBN 9780262210096. I dedicate this presentation to Renata Thomaz Lins do Nascimento, my love, my life! Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG Pós-Graduação em Engenharia de Sistemas e Computação - PESC

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