CVT
Centroidal Voronoi Tessellations

CVT is a MATLAB library which creates Centroidal Voronoi Tessellation (CVT) datasets.

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 GNU LGPL license.

Languages:

CVT is available in a C++ version and a FORTRAN90 version and a MATLAB version.

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

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 program which constructs a CVT in one dimension, under a nonuniform density function.

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_CIRCLE_UNIFORM, a MATLAB program which calculates a Centroidal Voronoi Tessellation (CVT) over a circle with uniform density.

CVT_DATASET, a MATLAB program which creates a CVT dataset.

CVT_DEMO, a MATLAB program which demonstrates a CVT calculation.

CVT_SQUARE_NONUNIFORM, a MATLAB program which iteratively calculates a Centroidal Voronoi Tessellation (CVT) over a square, with a nonuniform density.

CVTM_1D, a MATLAB program which estimates a mirror-periodic centroidal Voronoi Tessellation (CVTM) in the periodic interval [0,1], using a version of Lloyd's iteration.

CVTP_1D, a MATLAB program 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 programs which explore the creation of a centroidal Voronoi Tessellation (CVT) of the state of Florida, based solely on geometric considerations.

GRID, a MATLAB library which computes elements of a grid dataset.

HALTON, a MATLAB library which computes elements of a Halton quasirandom sequence.

HAMMERSLEY, a MATLAB library which computes elements of a Hammersley quasirandom sequence.

HEX_GRID, a MATLAB library which computes elements of a hexagonal grid dataset.

IHS, a MATLAB library which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER, a MATLAB library which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE, a MATLAB library which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM, a MATLAB library which computes elements of a Latin Hypercube dataset, choosing points at random.

LCVT, a MATLAB library which computes a latinized Centroidal Voronoi Tessellation.

NEAREST_NEIGHBOR, a MATLAB library which works in a given M-dimensional space, seeking, for each point in a set S, the nearest point in a set R, by Richard Brown.

NIEDERREITER2, a MATLAB library which computes elements of a Niederreiter quasirandom sequence with base 2.

SOBOL, a MATLAB library which computes elements of a Sobol quasirandom sequence.

TEST_NEAREST, a MATLAB program which tests the time complexity of various procedures for solving the nearest neighbor problem.

UNIFORM, a MATLAB library which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a MATLAB library 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.
• get_seed.m returns a seed for the random number generator.
• halham_leap_check.m checks LEAP for a Halton or Hammersley sequence.
• halham_n_check.m checks N for a Halton or Hammersley sequence.
• halham_dim_num_check.m checks DIM_NUM for a Halton or Hammersley sequence.
• halham_seed_check.m checks SEED for a Halton or Hammersley sequence.
• halham_step_check.m checks STEP for a Halton or Hammersley sequence.
• halton_base_check.m checks BASE for a Halton sequence.
• i4_to_halton_sequence.m next N elements of an DIM_NUM-dimensional Halton sequence
• 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.
• random_initialize.m initializes the MATLAB random number seed.
• 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.
• timestamp.m prints the current YMDHMS date as a timestamp.
• 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_box_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.

Examples and Tests:

• cvt_test.m, runs all the tests.
• cvt_test.txt, the output file.
• cvt_test01.m works in 2D, computes 10 points, using UNIFORM initialization and UNIFORM sampling, IT_FIXED is 1, 10000 sample points are used, and the random number seed is 123456789.
• cvt_test02.m repeats test 1 with twice as many iterations.
• cvt_test03.m repeats test 1 with 100 times as many sample points.
• cvt_test04.m repeats test 1 with UNIFORM initialization and HALTON sampling.
• cvt_test05.m repeats test 1 with UNIFORM initialization and GRID sampling.
• cvt_test06.m repeats test 1 with UNIFORM initialization and RANDOM sampling.
• cvt_test07.m repeats test 1 with a different seed.
• cvt_test08.m repeats test 1 with a different batch size.
• cvt_test09.m repeats test 1 with IT_FIXED = IT_MAX.
• cvt_test10.m generates 100 points in 3D.
• cvt_test11.m experiments with a Cartesian grid as starting point.
• cvt_test12.m tests the 'RANDOM' initialization option.
• cvt_test13.m works in 2D, computes 100 points, using USER initialization and USER sampling, IT_FIXED is 1, 10000 sample points are used, and the random number seed is 123456789. The user geometry is the unit circle, with uniform density.
• cvt_test14.m works in 1D (where we know the exact answer) and shows that it can take a long time for the computation to get close to the exact answer.
• cvt_circle.txt, a set of 100 CVT points in a circle, generated using the built in USER routine.
• cvt_circle.png, a PNG image of the 100 CVT points in a circle.

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