CVT_1D_LLOYD
Centroidal Voronoi Tessellation in [0,1]

CVT_1D_LLOYD is a MATLAB program which allows the user to carry out Lloyd's algorithm for a centroidal Voronoi Tessellation in the interval [0,1].

The determination of the Voronoi regions is carried out using exact techniques.

For n generators, the solution is known in advance:

x(i) = ( 2 * i - 1 ) / ( 2 * n )

Usage:

cvt_1d_lloyd ( n, it_num, init )

• n is the number of generators;
• it_num is the number of iterative steps to take.
• init is 0 for random initialization, 1 for "squashed".

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

CCVT_BOX, a MATLAB program which constructs a modified CVT in which some points are forced to lie on the boundary.

CCVT_REFLECT, a MATLAB program which tries to construct a modified CVT in which some points are forced to lie on the boundary, using a reflection idea.

CVT, a MATLAB library which computes CVT's.

CVT, a dataset directory which contains a variety of examples of CVT datasets.

CVT_1D_NONUNIFORM, a MATLAB program which computes an N-point Centroidal Voronoi Tessellation in 1 dimension, under a nonuniform density, and plots the evolution of the locations of the generators during the iteration;

CVT_1D_SAMPLING, a MATLAB program which computes an N-point Centroidal Voronoi Tessellation (CVT) within the interval [0,1], under a uniform density, using sampling to estimate the Voronoi regions.

CVT_2D_SAMPLING, a MATLAB program which computes an N-point Centroidal Voronoi Tessellation (CVT) within the unit square [0,1]x[0,1], under a uniform density, using sampling to estimate the Voronoi regions.

CVT_3D_SAMPLING, a MATLAB program which computes an N-point Centroidal Voronoi Tessellation (CVT) within the unit cube [0,1]x[0,1]x[0,1], under a uniform density, using sampling to estimate the Voronoi regions.

CVT_DATASET, a MATLAB program which can create a CVT dataset.

CVT_DEMO, a MATLAB program which demonstrates a CVT calculation.

LCVT, a MATLAB library which computes a "Latinized" Centroidal Voronoi Tessellation.

Reference:

1. Franz Aurenhammer,
   Voronoi diagrams - a study of a fundamental geometric data structure,
   ACM Computing Surveys,
   Volume 23, Number 3, pages 345-405, September 1991.
2. Qiang Du, Vance Faber, Max Gunzburger,
   Centroidal Voronoi Tessellations: Applications and Algorithms,
   SIAM Review, Volume 41, 1999, pages 637-676.

Source Code:

• cvt_1d_lloyd.m

Examples and Tests:

TEST01 sets the random initial values, using 40 generators and 400 steps.

• test01_energy.png a PNG image of the "energy" of the partition, which is the averaged sum of squares of the distances between each sample point and its nearest generator.
• test01_evolution.png a PNG image of the evolution or trajectories of the generators as the iteration proceeds.
• test01_motion.png a PNG image of the "motion" of the generators, which is the averaged distance each generator moves during an iteration.

TEST02 sets the "squashed" initial values between 0.01 and 0.02, using 40 generators and 400 steps.

• test02_energy.png a PNG image of the "energy" of the partition, which is the averaged sum of squares of the distances between each sample point and its nearest generator.
• test02_evolution.png a PNG image of the evolution or trajectories of the generators as the iteration proceeds.
• test02_motion.png a PNG image of the "motion" of the generators, which is the averaged distance each generator moves during an iteration.

You can go up one level to the MATLAB source codes.