cvt

cvt, an Octave code which creates Centroidal Voronoi Tessellations (CVT).

The generation of a CVT dataset is of necessity more complicated than for a quasirandom sequence. An iteration is involved, so there must be an initial assignment for the generators, and then a number of iterations. Moreover, in each iteration, estimates must be made of the volume and location of the Voronoi cells. This is typically done by Monte Carlo sampling. The accuracy of the resulting CVT depends in part on the number of sampling points and the number of iterations taken.

The library is mostly used to generate a dataset of points uniformly distributed in the unit hypersquare. However, a user may be interested in computations with other geometries or point densities. To do this, the user needs to replace the USER() routine in the CVT library, and then specify the appropriate values init=3 and sample=3.

The USER routine returns a set of sample points from the region of interest. The default USER routine samples points uniformly from the unit circle. But other geometries are easy to set up. Changing the point density simply requires weighting the sampling in the region.

Licensing:

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

Languages:

cvt is available in a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

Related Data and Programs:

cvt_test

cvt_box, an Octave code which constructs a modified cvt in which some points are forced to lie on the boundary.

ccvt_reflect, an Octave code which tries to construct a modified cvt in which some points are forced to lie on the boundary, using a reflection idea.

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

cvt_1d_lloyd, an Octave code which computes an n-point centroidal voronoi tessellation (cvt) within the interval [0,1], under a uniform density.

cvt_1d_nonuniform, an Octave code which constructs a cvt in one dimension, under a nonuniform density function.

cvt_1d_sampling, an Octave code 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, an Octave code 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_circle_uniform, an Octave code which calculates a centroidal voronoi tessellation (cvt) over a circle with uniform density.

cvt_square_nonuniform, an Octave code which iteratively calculates a centroidal voronoi tessellation (cvt) over a square, with a nonuniform density.

cvtm_1d, an Octave code which estimates a mirror-periodic centroidal voronoi tessellation (cvtm) in the periodic interval [0,1], using a version of lloyd's iteration.

cvtp_1d, an Octave code which estimates a periodic centroidal voronoi tessellation (cvtp) in the periodic interval [0,1], using a version of lloyd's iteration.

florida_cvt_geo, MATLAB codes which explore the creation of a centroidal voronoi tessellation (cvt) of the state of florida, based solely on geometric considerations.

halton, an Octave code which computes elements of a halton quasirandom sequence.

hammersley, an Octave code which computes elements of a hammersley quasirandom sequence.

latin_center, an Octave code which computes elements of a latin hypercube dataset, choosing center points.

latin_edge, an Octave code which computes elements of a latin hypercube dataset, choosing edge points.

latin_random, an Octave code which computes elements of a latin hypercube dataset, choosing points at random.

niederreiter2, an Octave code which computes elements of a niederreiter quasirandom sequence with base 2.

sobol, an Octave code which computes elements of a sobol quasirandom sequence.

uniform, an Octave code which computes elements of a uniform pseudorandom sequence.

van_der_corput, an Octave code which computes elements of a van der corput quasirandom sequence.

Reference:

1. Franz Aurenhammer,
   Voronoi diagrams - a study of a fundamental geometric data structure,
   ACM Computing Surveys,
   Volume 23, Number 3, September 1991, pages 345-405.
2. Paul Bratley, Bennett Fox, Linus Schrage,
   A Guide to Simulation,
   Second Edition,
   Springer, 1987,
   ISBN: 0387964673.
3. John Burkardt, Max Gunzburger, Janet Peterson, Rebecca Brannon,
   User Manual and Supporting Information for Library of Codes for Centroidal Voronoi Placement and Associated Zeroth, First, and Second Moment Determination,
   Sandia National Laboratories Technical Report SAND2002-0099,
   February 2002.
4. Qiang Du, Vance Faber, Max Gunzburger,
   Centroidal Voronoi Tessellations: Applications and Algorithms,
   SIAM Review,
   Volume 41, Number 4, December 1999, pages 637-676.
5. Bennett Fox,
   Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators,
   ACM Transactions on Mathematical Software,
   Volume 12, Number 4, December 1986, pages 362-376.
6. John Halton,
   On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals,
   Numerische Mathematik,
   Volume 2, Number 1, December 1960, pages 84-90.
7. Lili Ju, Qiang Du, Max Gunzburger,
   Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations,
   Parallel Computing,
   Volume 28, 2002, pages 1477-1500.

Source Code:

• ch_cap.m capitalizes a single character.
• cvt.m computes a Centroidal Voronoi Tessellation.
• cvt_energy.m computes the CVT "energy" of a given dataset.
• cvt_iterate.m takes one step of the CVT iteration.
• cvt_sample.m returns sample points.
• data_read.m reads data from a file.
• find_closest.m finds the Voronoi cell generator closest to a point X.
• prime.m returns the N-th prime number.
• r8mat_transpose_print.m prints the transpose of an R8MAT, with an optional title.
• r8mat_transpose_print_some.m prints the transpose of some of an R8MAT, with an optional title.
• r8mat_uniform_01.m returns a unit pseudorandom R8MAT.
• r8mat_write.m writes an R8MAT to a file.
• s_eqi.m is TRUE if two strings are equal, ignoring case.
• s_len_trim.m returns the length of a string to the last nonblank.
• tuple_next_fast.m computes the next element of a tuple space, "fast".
• user.m a sample version of the USER routine, which can be used to specify a region which is not the unit hypercube.
• user_circle_2d.m a sample version of the USER routine, which in this case, returns points in the unit circle, under a uniform density.
• user_rectangle_3d.m a sample version of the USER routine, which in this case, returns points in a 3D box of dimension [0,5]x[-1,+1]x[2,4], under a uniform density.