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.

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. 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. 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. 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. 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