Five Minute Speech: Activities Developed in Computational Geometry Discipline

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Five Minute Speech: An Overview of Activities Developed in Computational Geometry Discipline. In this presentation, I spoke about the main idea of the article entitled 'Capacity-Constrained Point Distributions: A Variant of Lloyd's Method' [Balzer, M. et al. 2009]. In this 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.This method is similar to the commonly used Lloyd's method while avoiding its drawbacks.

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Five Minute Speech: Activities Developed in Computational Geometry Discipline

  1. 1. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Five Minute Speech :: An Overview of Activities Developed in Computational Geometry Discipline :: Laboratory Seminars and Meetings :: November, 2013 Five Minute Speech An Overview of Activities Developed in Computational Geometry Discipline 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 November, 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 Five Minute Speech :: An Overview of Activities Developed in Computational Geometry Discipline :: Laboratory Seminars and Meetings :: November, 2013 Introduction 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 noise characteristics Michel Alves dos Santos: Laboratório de Computação Gráfica - LCG The iterative method by Lloyd [1982] is a powerful and flexible Resulting distributions possess high-qualitycommonly noise characteristics blue used to enhance the spectral properties technique that is of existing distributions of points or similar entities. However, the and adapt precisely to given density; 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
  3. 3. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Five Minute Speech :: An Overview of Activities Developed in Computational Geometry Discipline :: Laboratory Seminars and Meetings :: November, 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 withde ComputaçãoTessellationLCG Michel AlvesAlgorithm 1: Capacity-Constrainedproperties Gráfica scenarios. dos Santos: Laboratório better Voronoi in both Pós-Graduação ∈ S to the center of mass e all points
  4. 4. Universidade Federal do Rio de Janeiro - UFRJ - Campus Cidade Universitária - Rio de Janeiro - Ilha do Fundão, CEP: 21941-972 - COPPE/PESC/LCG Five Minute Speech :: An Overview of Activities Developed in Computational Geometry Discipline :: Laboratory Seminars and Meetings :: November, 2013 Expected Results Figure: Stippling Example. From left to right: original grayscale image, [Secord 2002], proposed method. Each stipple drawing uses 20’000 points with the same draw radius. A grayscale image is used as the density function to generate stipple drawings. The result of proposed unmodified method exhibits no regularities and higher local contrast than the result by [Secord 2002]. 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 Five Minute Speech :: An Overview of Activities Developed in Computational Geometry Discipline :: Laboratory Seminars and Meetings :: November, 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

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