CVT_METRIC
CVT Calculation With Varying Metric

CVT_METRIC is a MATLAB program which computes a CVT (Centroidal Voronoi Tessellation) calculation under a spatially varying metric.

What we are saying is that the distance between two vectors a and b is no longer simply the Euclidean distance, but rather a quadratic form involving a spatially varying, positive definite symmetric matrix that represents the metric:


        d(a,b) = ( a - b )' * A * ( a - b )

We assume that A varies spatially, but we would prefer to simplify this variation in a manner that saves us computational effort, while allowing us to recover the variational behavior if we are willing to use a finer spatial sampling.

The metric function is distinct from the density function, which can also be used in these kinds of problems. The density function is weaker, and corresponds to a metric whose matrix at any point is a multiple of the identity matrix. A density function approach cannot result in the more interesting anisotropic effects that an arbitrary metric produces.

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:

CVT_1D_LLOYD, a MATLAB program which computes an N-point Centroidal Voronoi Tessellation (CVT) within the interval [0,1], under a uniform density.

CVT_1D_NONUNIFORM, a MATLAB library which allows the user to watch the evolution of a CVT computed over a 1D interval with a nonuniform density.

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_DEMO, a MATLAB library which allows the user to generate a CVT over several geometric regions using uniform or nonuniform density.

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

TEST_TRIANGULATION, a MATLAB library which defines the geometry of a number of sample regions.

Reference:

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

Source Code:

cvt_square_metric.m, the text of the MATLAB function.
square_uniform.m, the routine which returns sample points from the square.
timestamp.m, prints the YMDHMS date as a timestamp.

Examples and Tests:

The following files contain functions to evaluate various simple metrics:

metric_01.m, the identity matrix.
metric_02.m, the matrix [9,0;0,1].
metric_03.m, the matrix [1,0;0,100].
metric_04.m, the matrix [1,0;0,1] * (0.2+sin(2*pi*x)**2 * sin(2*pi*y)**2).
metric_05.m, the matrix [2,3;3,5].

Here are images of CVT point sets. These are crude calculations, with a relatively low number of sample points and iterations. This is partly because I haven't figured out how to optimize these calculations in MATLAB.

metric_03.png, 200 generators, using 1600 sample points, and 20 iterations, with the constant metric [1,0;0,100].
metric_04.png, 200 generators, using 1600 sample points, and 20 iterations, with the metric matrix [1,0;0,1] * (0.2+sin(2*pi*x)**2 * sin(2*pi*y)**2).

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