1. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Capacity-Constrained Point Distributions A Variant of Lloyds MethodMichel Alves dos Santos Ps-Graduao em Engenharia de Sistemas e Computao Universidade Federal do Rio de Janeiro - UFRJ - COPPE Cidade Universitria - Rio de Janeiro - CEP: 21941-972 Docentes Responsveis: Prof. Dsc. Ricardo Marroquim & Prof. PhD. Cludio Esperana{michel.mas, michel.santos.al}@gmail.comJanuary, 2013 Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC

2. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Introduction Applications of Point Distributions...Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPresentation hosted on: http://www.lcg.ufrj.br/Members/malves/indexPs-Graduao em Engenharia de Sistemas e Computao - PESC 3. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Introduction Applications of Point Distributions...Sampling Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 4. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Introduction Applications of Point Distributions...SamplingPoint-Based RenderingMichel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 5. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Introduction Applications of Point Distributions...SamplingPoint-Based RenderingGeometric ProcessingMichel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 6. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Introduction Applications of Point Distributions...SamplingPoint-Based RenderingGeometric ProcessingMichel Alves dos Santos: Laboratrio de Computao Grca - LCGHalftoningetc...Ps-Graduao em Engenharia de Sistemas e Computao - PESC 7. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Desidered Properties Desidered Properties for Point Distributions...Red NoiseWhite NoiseBlue NoiseBlue noise features; Similar relative distance between points; No regular appearance (For most applications); Adaptation to the provided density functions. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 8. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Desidered Properties Desidered Properties for Point Distributions...Red NoiseWhite NoiseBlue noise features; Similar relative distance between points; No regular appearance (For most applications); Adaptation to the provided density functions. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGBlue NoiseIn this presentation we will discuss about a technique for optimal distribution of points!Ps-Graduao em Engenharia de Sistemas e Computao - PESC 9. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Capacity-Constrained Point Distributions Capacity-Constrained Point Distributions: A Variant of Lloyds Method Michael BalzerThomas Schl mer o University of Konstanz, GermanyOliver DeussenFigure 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 = e(20x 20y ) + 0.2 sin2 (x) sin2 (y); (Right) 4096 points with a density function extracted from a grayscale image.Abstract Newthat 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 areaWe 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 Lloyds 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 Lloyds method, and combines enhancement of blue Grca - LCG Michel Alves dos Santos: Laboratrio de Computaonoise characteristicsThe iterative method by Lloyd [1982] is a powerful and Resulting distributions possess high-qualitycommonly noise featuresand exible blue used to enhance the spectral properties technique that is adapt precisely to given density; of existing distributions of points or similar entities. However, theresults from Lloyds method are satisfactory only to a limited ex-tent. First, if the method is not stopped at a Similar to the commonly used Lloyds 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 Lloyds method which reliably converges toPs-Graduao em Engenharia de Sistemas e Computao - PESC 10. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Proposed Method initial point setLloyds method 0.75 convergedour method (converged)zone plate test function1024 points and their Fourier amplitude sprectrum 0.53input sitesinitial statecapacity-constrained optimizationnal stateoutput sitesFigure 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 Lloyds method. After 40 iterations the points to well distributed with a normalized radius of 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 nal site distribution. simultaneously maintaining the capacity Stippling, Further optimizationat Radiance/Luminance,2 etc. structures. In contrast, 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 is also not an option as only marginally fewer artifacts can be observed. In this sampling scenario, stopping Lloyds method after about 10 iterations with 0.53 would provide the best sampling results. Our method converges 2. move each site siem Engenharia de Sistemas of Computao - PESC reliably to an equilibrium with better Voronoi Tessellation Michel AlvesAlgorithm 1: Capacity-Constrainedproperties in both - LCG dos Santos: Laboratrio de Computao Grca scenarios. Ps-Graduao S to the center of mass e all points 11. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Proposed Method :: Steps and DetailsDensity Function Samples Generation of Sites Optimization Optimized Sites Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 12. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Lloyds MethodFigure: (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: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 13. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Lloyds MethodFigure: (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: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 14. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Lloyds MethodFigure: (Left) Random dots (red) and polygons. (Right) Result after running approximate Lloyd relaxation twice - note the artifacts produced by technique. Diculty in stopping criterion! Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 15. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Lloyds MethodFigure: (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: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 16. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Capacity Constrained Vs. Lloyds Method Capacity-ConstrainedLloyds MethodCCPD is a variation of the Lloyds Method that converges in a natural way and that in addition not presents the appearance of regularity still ts precisely to given density functions. CCPD Uses:ComplexityMetrics or Distance Functions; Lloyd Centroidal Voronoi Tessellations; CCPD The Concept of Capacity; Minimization of Energy (through a Optimization Method). Michel Alves dos Santos: Laboratrio de Computao Grca - LCGMemoryPs-Graduao em Engenharia de Sistemas e Computao - PESC 17. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Metrics or Distance FunctionsVoronoi Tesselations Using Minkowski Metrics: L1 , L2 , L3 , L4 , L5 , L . Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 18. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Metrics or Distance Functions :: Manhattan or L1d (x, y) = L1 (x, y) = Michel Alves dos Santos: Laboratrio de Computao Grca - LCGn i =1|xi yi |Ps-Graduao em Engenharia de Sistemas e Computao - PESC 19. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Metrics or Distance Functions :: Euclidean or L2d (x, y) = L2 (x, y) = ( Michel Alves dos Santos: Laboratrio de Computao Grca - LCGn i =1|xi yi |2 )1/2Ps-Graduao em Engenharia de Sistemas e Computao - PESC 20. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Metrics or Distance Functions :: L3d (x, y) = L3 (x, y) = ( Michel Alves dos Santos: Laboratrio de Computao Grca - LCGn i =1|xi yi |3 )1/3Ps-Graduao em Engenharia de Sistemas e Computao - PESC 21. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Metrics or Distance Functions :: L4d (x, y) = L4 (x, y) = ( Michel Alves dos Santos: Laboratrio de Computao Grca - LCGn i =1|xi yi |4 )1/4Ps-Graduao em Engenharia de Sistemas e Computao - PESC 22. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Metrics or Distance Functions :: L5d (x, y) = L5 (x, y) = ( Michel Alves dos Santos: Laboratrio de Computao Grca - LCGn i =1|xi yi |5 )1/5Ps-Graduao em Engenharia de Sistemas e Computao - PESC 23. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Metrics or Distance Functions :: Chebyshev or Ld (x, y) = L (x, y) = limp ( Michel Alves dos Santos: Laboratrio de Computao Grca - LCGn i =1|xi yi |p )1/p = maxin (|xi yi |) =1 Ps-Graduao em Engenharia de Sistemas e Computao - PESC 24. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Toroidal Square Distance 11234 6 1245 76 1For the current work we used a metric based on a toroidal square. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 25. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Toroidal 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 ) ; }14d 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}returns q r t ( dx dx + dy dy ) ;Michel Alves dos Santos: Laboratrio de Computao Grca - LCG Ps-Graduao em Engenharia de Sistemas e Computao - PESC 26. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Toroidal Square Distance :: ExampleOptimized SitesVoronoi TessellationNote the regions that lie within the limits. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 27. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Toroidal Square Distance :: ExampleOptimized SitesVoronoi TessellationNote the regions that lie within the limits. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 28. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Toroidal Square Distance :: ExampleOptimized SitesVoronoi TessellationNote the regions that lie within the limits. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 29. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Toroidal Square Distance :: ExampleOptimized SitesVoronoi TessellationNote the regions that lie within the limits. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 30. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Toroidal Square Distance :: ExampleOptimized SitesVoronoi TessellationNote the regions that lie within the limits. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 31. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Toroidal Square Distance :: ExampleOptimized SitesVoronoi TessellationNote the regions that lie within the limits. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 32. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Centroidal Voronoi Tessellation Non-Centroidal VoronoiCentroidal Voronoi110.80.80.60.60.40.40.20.2000.20.20.40.40.60.60.80.8110.80.60.40.200.20.40.60.81110.80.60.40.200.20.40.60.81CVT is a Voronoi Tesselation with the property that each site itself coincides with the centroid of their respective Voronoi region. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 33. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Centroidal Voronoi Tessellation :: Applications Optimal quadrature rules; Covolume and nite dierence methods for PDEs; Optimal representation, quantization, and clustering; Optimal placement of sensors and actuators; Optimal distribution of resources; Cell division; Finite volume methods for PDEs; 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: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 34. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Centroidal Voronoi Tessellation :: Centroids The centroid of a Voronoi region is nothing but the center of mass of a region weighted by the density function dened in V area.z=Vi Vix(x)dx (x)dxzi=M i =1 xi (xi ) M (x ) i i =1For discrete sets of points we have V = {xi }M in Rn and a density i =1 function (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: MF (S, V ) = i =1 Vi(x)|x si |2 dxS sites; V voronoi regions; (x) density function; F (S, V ) energy function Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 35. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Centroidal Voronoi Tessellation :: Using Lloyd Steps for generating a Centroidal Voronoi Tessellation: 1 2 3Generate a Voronoi tessellation V (S) in a region ; 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 )|2F (X, A) = i =11. 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 with density function 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: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 36. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013The Concept of Capacity :: Capacity-Constrained Let S be a set with n sites which determine a Voronoi tessellation V (S) in limited space with function density (x ). Denition: The capacity c(si ) of a site si S with respect to its respective Voronoi region Vi V is dened as: c(si ) =(x)dx ViWe say that a distribution of sites in S adapts optimally to a density function if the capacity of each site follows: (x)dx c(si ) = c , where c is dened as c =nIn other words, the capacity of a site is equivalent to the area of Voronoi region weighted by the density function. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 37. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Minimization of Energy :: Voronoi Tessellation Steps for Minimization: 1 2Generate 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 dierent 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.3Repeat the exchange until a stage of stability is achieved.input sitesinitial statecapacity-constrained optimizationnal stateoutput sitesFigure 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 nal site distribution. Michel Alves dos Santos: Laboratrio de Computao Grca - LCG Ps-Graduao em Engenharia de Sistemas e Computao - PESC 38. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Overview of Capacity-Constrained Method Capacity-Constrained: The Capacity-Constrained method diers 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: 1Create a set X with m points weighted by the density function (x );2Generate 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;3Move each site si S to the center of mass of all points xi X ;4Repeat 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: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 39. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Evaluation of Results The method presented here achieves better results when compared to the Lloyds Method, in some respects:Blue-noise features; Number of neighbors; Stopping criteria; Measuring the quality of the adaptation. 100 %percentage60 %40 %20 %0%456780.951.10.85our method1.00.9Lloyds method 0.7our method 4number of neighbors(a) number of neighbors percentage567number of neighbors(b) normalized Voronoi region areaMichel Alves dos Santos: Laboratrio de Computao Grca - LCG8normalized radius our methodnormalized Voronoi region area1.2Lloyds method 80 %0.750.650.5516642561024409616384number of sites(c) normalized radius Ps-Graduao em Engenharia de Sistemas e Computao - PESC 40. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Evaluation of Results :: Blue-noise and Neighbors 1024 optimized pointsnumber of neighbors 4567spectral analysis 8 2power1.5 1Lloyds method0.5 0 0frequencyfc+10 anisotropy+5 0-5-10 0frequencyfc2power1.5 10.5our method0 0frequencyfc+10 anisotropy+5 0-5-10 0frequencyfcLloyds method generates point distributions with regular structures. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 41. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Evaluation of Results :: Stopping and Quality Stopping Criteria: In the Lloyds Method is necessary a manual intervention or a specic criterion determined by the application. Quality of the adaptation: The capacity oers the opportunity to measure the quality of adaptation by a distribution of sites through the errors of the capacity given by:c =1 ni =1c(si )nc2 1In 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: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 42. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013ResultsUsing:Regular Density Functions Custom Density Functions Images as Density Functions Now we will see some results obtained with the technique... Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 43. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Results :: Regular Density Functions 1.01 1.005 1.0111.0050.99510.990.995 0.99f(x,y) = cf(x,y) = c{(x,y) | x R, y R}(x,y)-> random choiceAll tabled numerical results shown in this presentation are an average of 15 executions for each set of points. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 44. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Results :: Constant Regular GridOptimized Sitesf (x, y) = c;Voronoi Tessellation{(x, y)|x R, y R}Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 45. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Results :: Constant Regular Grid1.01 1.005 1 1.01 0.9951.005 110.9950.990.50.99 0 -1 -0.5-0.50 0.5 1-1Regular Density f(x,y) --> samples512 Points1024 Points2048 Points16384 Samples32768 Samples65536 SamplesFigure: The gure 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 2048Generation Time ( samples1024 Points2048 Points4096 Points32768 Samples65536 Samples131072 SamplesFigure: The gure 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 4096Generation Time 07.30 seconds 31.90 seconds 168.60 secondsOptimization Time 00.15 seconds 00.55 seconds 01.72 secondsOptimization Steps* 16 28 16Optimization Steps*: number of iterations until convergence. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 60. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Points and Optimization TimeNumber of Points and SamplesResults :: Stippling :: Image as Density Function4096 Points16384 Points20000 Points393216 Samples786432 Samples1280000 Samples4096 Points8192 Points12288 Points02.97 seconds09.62 seconds17.57 secondsMichel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 61. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Results :: Stippling - Corn Plant/Dracaena4096 Points16384 Points20000 Points393216 Samples786432 Samples1280000 SamplesFigure: Input image with 1000x1000 pixels. The gure 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 20000Generation Time 25.50 minutes 101.70 minutes 271.20 minutesOptimization Time 4.41 seconds 20.45 seconds 42.43 secondsOptimization Steps* 57 24 35Optimization Steps*: number of iterations until convergence. Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 62. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Results :: Stippling - Madonnas Face4096 Points8192 Points12288 Points02.97 seconds09.62 seconds17.57 secondsFigure: Madonnas Face. Input image with 1000x1000 pixels. The gure shows the number of points and the optimization time for each set. Amount of Points 4096 8192 12288Generation Time 13.40 minutes 55.37 minutes 124.99 minutesMichel Alves dos Santos: Laboratrio de Computao Grca - LCGOptimization Time 02.97 seconds 09.62 seconds 17.57 secondsOptimization Steps 38 36 38Ps-Graduao em Engenharia de Sistemas e Computao - PESC 63. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Conclusions Some conclusions about the method:1.01 1.00 0.80 0.60 0.40 0.20 0.001.005 1 0.995Performs distribution points optimally. It is more stable than Lloyds 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.0.990.001.000.200.800.400.600.600.400.80Michel Alves dos Santos: Laboratrio de Computao Grca - LCG0.20 1.00 0.00Ps-Graduao em Engenharia de Sistemas e Computao - PESC 64. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Conclusions Some conclusions about the method:1.01 1.00 0.80 0.60 0.40 0.20 0.001.005 1 0.995Performs distribution points optimally. It is more stable than Lloyds 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.0.990.001.000.200.800.400.600.600.400.80Michel Alves dos Santos: Laboratrio de Computao Grca - LCG0.20 1.00 0.00Ps-Graduao em Engenharia de Sistemas e Computao - PESC 65. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Conclusions Some conclusions about the method:1.01 1.00 0.80 0.60 0.40 0.20 0.001.005 1 0.995Performs distribution points optimally. It is more stable than Lloyds 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.0.990.001.000.200.800.400.600.600.400.80Michel Alves dos Santos: Laboratrio de Computao Grca - LCG0.20 1.00 0.00Ps-Graduao em Engenharia de Sistemas e Computao - PESC 66. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Conclusions Some conclusions about the method:1.01 1.00 0.80 0.60 0.40 0.20 0.001.005 1 0.995Performs distribution points optimally. It is more stable than Lloyds 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.0.990.001.000.200.800.400.600.600.400.80Michel Alves dos Santos: Laboratrio de Computao Grca - LCG0.20 1.00 0.00Ps-Graduao em Engenharia de Sistemas e Computao - PESC 67. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Conclusions Some conclusions about the method:1.01 1.00 0.80 0.60 0.40 0.20 0.001.005 1 0.995Performs distribution points optimally. It is more stable than Lloyds 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.0.990.001.000.200.800.400.600.600.400.80Michel Alves dos Santos: Laboratrio de Computao Grca - LCG0.20 1.00 0.00Ps-Graduao em Engenharia de Sistemas e Computao - PESC 68. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013ThanksThanks 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/prole/view?id=26542507 Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 69. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Samples Voronoi Sites Thank you for your attention! Michel Alves dos Santos: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC 70. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitria - Rio de Janeiro - Ilha do Fundo, CEP: 21941-972 - COPPE/PESC/LCG Capacity-Constrained Point Distributions :: A Variant of Lloyds Method :: Computational Geometry Discipline :: Laboratory Seminar :: January, 2013Bibliography M. Balzer, T. Schlmer, and O. Deussen. Capacity-constrained point distributions: A variant of Lloyds method. ACM Transactions on Graphics (Proceedings of SIGGRAPH 2009), 28(3):86:18, 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):110, 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 13:113:1, 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 3743. 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: Laboratrio de Computao Grca - LCGPs-Graduao em Engenharia de Sistemas e Computao - PESC