Centroidal Voronoi Tessellations on Meshes Applications LLoyds - PowerPoint PPT Presentation

Centroidal Voronoi Tessellations on Meshes Definition and Centroidal Voronoi Tessellations on Meshes Applications LLoyd’s Algorithm Normal Leonardo K. Sacht & Thiago S. Pereira Clustering Mixing Both Approaches Visgraf - IMPA Flooding Approach November 18, 2008 Future Work References Centroidal Voronoi Tessellations on Meshes

Centroidal Voronoi Tessellations Centroidal Voronoi Tessellations on Meshes Definition and Applications Definition: Given Ω ⊆ R n , { z i } k i =1 ⊆ Ω, the Voronoi region of LLoyd’s each z i is defined by Algorithm Normal ˆ Clustering V i = { x ∈ Ω : � x − z i � < � x − z j � for j = 1 , . . . , k , j � = i } . Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Centroidal Voronoi Tessellations Centroidal Voronoi A Voronoi Diagram of [0 , 1] × [0 , 1] ⊆ R 2 Tessellations on Meshes Definition and Applications LLoyd’s Algorithm Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Centroidal Voronoi Tessellations Centroidal Definition: Let V ⊆ R n and ρ a density function on V . The Voronoi Tessellations mass centroid of V is z ∗ ∈ V such that on Meshes � � ρ ( y ) � z ∗ − y � 2 dy = inf ρ ( y ) � z − y � 2 dy , Definition and Applications z ∈ V ∗ V V LLoyd’s where V ∗ is the closure of V . Algorithm Normal Clustering Mixing Both The minimizer of this functional is Approaches Flooding � Approach y ρ ( y ) dy Future Work z ∗ = V . References � ρ ( y ) dy V Centroidal Voronoi Tessellations on Meshes

Centroidal Voronoi Tessellations Centroidal Voronoi Tessellations on Meshes Definition and Applications LLoyd’s Definition: { ˆ V i } n i =1 is a Centroidal Voronoi Tessellation of Algorithm Ω ⊆ R n if the generators of ˆ V i are their mass centroids. Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Centroidal Voronoi Tessellations Centroidal Voronoi A Centroidal Voronoi Tessellation of with ρ ≡ 1. Tessellations on Meshes Definition and Applications LLoyd’s Algorithm Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Centroidal Voronoi Tessellations Centroidal Voronoi ρ ( x , y ) = e − 40 x 2 − 40 y 2 Tessellations on Meshes Definition and Applications LLoyd’s Algorithm Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Applications Centroidal Voronoi Tessellations Data Compression in Image Processing; on Meshes Definition and Applications LLoyd’s Algorithm Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Applications Centroidal Voronoi Tessellations Data Compression in Image Processing; on Meshes Quadrature Rules; Definition and Applications LLoyd’s Algorithm Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Applications Centroidal Voronoi Tessellations Data Compression in Image Processing; on Meshes Quadrature Rules; Definition and Optimal Representation, Quantization, and Clustering; Applications LLoyd’s Algorithm Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Applications Centroidal Voronoi Tessellations Data Compression in Image Processing; on Meshes Quadrature Rules; Definition and Optimal Representation, Quantization, and Clustering; Applications Finite Difference Schemes Having Optimal Truncation LLoyd’s Algorithm Errors; Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Applications Centroidal Voronoi Tessellations Data Compression in Image Processing; on Meshes Quadrature Rules; Definition and Optimal Representation, Quantization, and Clustering; Applications Finite Difference Schemes Having Optimal Truncation LLoyd’s Algorithm Errors; Normal Clustering Optimal Placement of Resources; Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Applications Centroidal Voronoi Tessellations Data Compression in Image Processing; on Meshes Quadrature Rules; Definition and Optimal Representation, Quantization, and Clustering; Applications Finite Difference Schemes Having Optimal Truncation LLoyd’s Algorithm Errors; Normal Clustering Optimal Placement of Resources; Mixing Both Cell Division; Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Applications Centroidal Voronoi Tessellations Data Compression in Image Processing; on Meshes Quadrature Rules; Definition and Optimal Representation, Quantization, and Clustering; Applications Finite Difference Schemes Having Optimal Truncation LLoyd’s Algorithm Errors; Normal Clustering Optimal Placement of Resources; Mixing Both Cell Division; Approaches Flooding Territorial Behavior of Animals; Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Applications Centroidal Voronoi Tessellations Data Compression in Image Processing; on Meshes Quadrature Rules; Definition and Optimal Representation, Quantization, and Clustering; Applications Finite Difference Schemes Having Optimal Truncation LLoyd’s Algorithm Errors; Normal Clustering Optimal Placement of Resources; Mixing Both Cell Division; Approaches Flooding Territorial Behavior of Animals; Approach Future Work References Reference: [2]. Centroidal Voronoi Tessellations on Meshes

Applications Centroidal Voronoi Tessellations on Meshes Definition and Applications LLoyd’s Remeshing; Algorithm Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Applications Centroidal Voronoi Tessellations on Meshes Definition and Applications LLoyd’s Remeshing; Algorithm Normal Our applications... Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Framework of the Problem Centroidal Voronoi Tessellations on Meshes Given • a region Ω ⊆ R n , Definition and Applications • a positive integer k , LLoyd’s • a density function ρ , defined on Ω, Algorithm Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Framework of the Problem Centroidal Voronoi Tessellations on Meshes Given • a region Ω ⊆ R n , Definition and Applications • a positive integer k , LLoyd’s • a density function ρ , defined on Ω, Algorithm find Normal Clustering • k points z i ∈ Ω, Mixing Both • k regions V i that tesselate Ω, Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Framework of the Problem Centroidal Voronoi Tessellations on Meshes Given • a region Ω ⊆ R n , Definition and Applications • a positive integer k , LLoyd’s • a density function ρ , defined on Ω, Algorithm find Normal Clustering • k points z i ∈ Ω, Mixing Both • k regions V i that tesselate Ω, Approaches Flooding such that simultaneously for each i Approach • V i is the Voronoi region for z i , Future Work • z i is the mass centroid of V i . References Centroidal Voronoi Tessellations on Meshes

LLoyd’s Algorithm Centroidal Voronoi Tessellations on Meshes Definition and Applications LLoyd’s Algorithm Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Lloyd in Back to the Future III Centroidal Voronoi Tessellations on Meshes

Lloyd’s Algorithm Centroidal Voronoi Tessellations on Meshes It’s the most standard and intuitive algorithm to compute CVD’s. It’s described by the following steps: Definition and Applications (1) Compute randomly k points z i ∈ Ω; LLoyd’s Algorithm Normal Clustering Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes

Lloyd’s Algorithm Centroidal Voronoi Tessellations on Meshes It’s the most standard and intuitive algorithm to compute CVD’s. It’s described by the following steps: Definition and Applications (1) Compute randomly k points z i ∈ Ω; LLoyd’s Algorithm (2) Compute the Voronoi regions ˆ V i determined by the Normal Clustering points z i ; Mixing Both Approaches Flooding Approach Future Work References Centroidal Voronoi Tessellations on Meshes