QUALITY
Quality Measures for Pointsets in M dimensions

QUALITY is a MATLAB library which computes some measures of the quality of dispersion of a set of N points in an M dimensional region.

The region to be analyzed is defined by a single routine which the user may specify. We will refer to that routine as sample_routine. This routine is simply required to return sample points from the region.

The library includes routines for two simple regions, namely the unit hypercube and the unit hypersphere. If either of these regions is of interest, then the user simply has to pass the appropriate name, either sample_hypercube_uniform or sample_sphere_uniform. For other regions, the user should write the appropriate version of a sampling routine.

The unit hypercube in M dimensional space is simply the set of all vectors whose entries are between 0 and 1. It is a natural abstraction of the unit line segment, the unit square, and the unit cube, and represents a very simple chunk of M dimensional space.

The unit hypersphere in M dimensional space is simply the set of all points whose distance from the origin is 1 or less.

The quality measures computed include:

• Alpha, the minimum angle divided by the maximum possible minimum angle (for triangulated 2D datasets only);
• Area, the ratio of the minimum and maximum triangle areas (for triangulated 2D datasets only);
• Beta, the standard deviation of the minimum spacing between nodes, normalized by the average spacing between nodes;
• Chi, the regularity measure;
• D, the second moment determinant measure;
• E, the Voronoi energy measure;
• Gamma, the mesh ratio;
• H, the point distribution norm;
• Lambda, the COV measure;
• Mu, the point distribution ratio;
• Nu, the cell volume deviation;
• Q, two times the radius of the inscribed circle divided by the radius of the circumscribed circle (for triangulated 2D datasets only);
• R0, the Riesz S=0 energy;
• S, the sphere volume measure.
• Tau, the second moment trace measure;

Licensing:

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

Languages:

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

Related Data and Programs:

DIAPHONY, a MATLAB program which reads a file of N points in M dimensions and computes its diaphony, a measure of point dispersion.

STAR_DISCREPANCY, a C++ program which reads a TABLE file of M points in N dimensions (presumed to lie in the unit hypercube) and computes bounds on the star discrepancy, a measure of dispersion, by Eric Thiemard.

TABLE_QUALITY, a MATLAB program which reads a data file and computes the quality of the points.

TET_MESH, a MATLAB library which includes some quality measures for tetrahedral meshes.

Reference:

1. David Field,
   Qualitative Measures for Initial Meshes,
   International Journal of Numerical Methods in Engineering,
   Volume 47, 2000, pages 887-906.
2. D P Hardin, Edward Saff,
   Discretizing Manifolds via Minimum Energy Points,
   Notices of the AMS,
   Volume 51, Number 10, November 2004, pages 1186-1194.
3. Per-Olof Persson, Gilbert Strang, A Simple Mesh Generator in MATLAB, SIAM Review, Volume 46, Number 2, pages 329-345, June 2004.
4. Yuki Saka, Max Gunzburger, John Burkardt,
   Latinized, Improved LHS, and CVT Point Sets in Hypercubes,
   International Journal of Numerical Analysis and Modeling,
   Volume 4, Number 3-4, 2007, pages 729-743,

Source Code:

• alpha_measure.m determines the minimum angle measure ALPHA.
• area_measure.m determines the area ratio measure AREA.
• arc_cosine.m computes the arc cosine of an angle, but forces the argument to be between -1 and 1.
• bandwidth_mesh.m determines the bandwidth of the coefficient matrix.
• beta_measure.m determines the relative spacing deviation BETA.
• chi_measure.m determines the pointset quality measure CHI.
• d_measure.m determines the pointset quality measure D.
• dge_det.m computes the determinant of a square matrix in DGE storage.
• e_measure.m determines the pointset quality measure E.
• file_column_count.m counts the number of columns in the first line of a file.
• file_row_count.m counts the number of row records in a file.
• find_closest.m finds the Voronoi cell generator closest to a point X.
• gamma_measure.m determines the pointset quality measure GAMMA.
• h_measure.m determines the pointset quality measure H.
• lambda_measure.m determines the pointset quality measure LAMBDA.
• mu_measure.m determines the pointset quality measure MU.
• nu_measure.m determines the pointset quality measure NU.
• pointset_spacing.m determines the minimum spacing between points in the set.
• q_measure.m determines the pointset quality measure Q.
• r0_measure.m determines the pointset quality measure R0.
• r8_huge.m returns a "huge" R8.
• r8_uniform_01.m returns a unit pseudorandom R8.
• r8mat_in_01.m is TRUE if the entries of an R8MAT are all in the unit interval.
• r8mat_transpose_print.m prints an R8MAT in transposed form.
• r8mat_transpose_print_some.m prints some of an R8MAT in transposed form.
• r8mat_uniform_01.m returns a unit pseudorandom R8MAT.
• r8vec_normal_01.m fills a unit pseudonormal R8VEC.
• r8vec_uniform_01.m fills a unit pseudorandom R8VEC.
• radius_maximus.m finds the biggest possible nonintersecting sphere.
• r8mat_data_read.m reads data from an R8MAT file.
• r8mat_header_read.m reads the header from an R8MAT file.
• s_len_trim.m returns the length of a string to the last nonblank.
• s_word_count.m counts the number of "words" in a string.
• sample_hypercube_uniform.m returns uniformly distributed sample points from the unit hypercube.
• sample_sphere_uniform.m returns uniformly distributed sample points from the unit sphere.
• sphere_measure.m determines the pointset quality measure S.
• sphere_volume_nd.m computes the volume of a sphere in ND.
• tau_measure.m determines the pointset quality measure TAU.
• timestamp.m prints the current YMDHMS date as a time stamp.

Examples and Tests:

• quality_test.m, calls all the tests for all the datasets.
• quality_test_cvt.m, calls all the tests for the CVT dataset.
• quality_test_halton.m, calls all the tests for the HALTON dataset.
• quality_test_sphere.m, calls all the tests for the SPHERE dataset.
• quality_test005.m, tests ALPHA_MEASURE.
• quality_test006.m, tests AREA_MEASURE.
• quality_test007.m, tests BETA_MEASURE.
• quality_test01.m, tests CHI_MEASURE.
• quality_test02.m, tests D_MEASURE.
• quality_test03.m, tests E_MEASURE.
• quality_test04.m, tests GAMMA_MEASURE.
• quality_test05.m, tests H_MEASURE.
• quality_test06.m, tests LAMBDA_MEASURE.
• quality_test07.m, tests MU_MEASURE.
• quality_test08.m, tests NU_MEASURE.
• quality_test085.m, tests R0_MEASURE.
• quality_test09.m, tests SPHERE_MEASURE.
• quality_test10.m, tests TAU_MEASURE.
• quality_test11.m, tests POINTSET_SPACING.
• quality_test_output.txt, printed output from a run of the test code.
• cvt_02_00100.txt, a set of 100 CVT points in 2D, to be analyzed using the test code.
• halton_02_00100.txt, a set of 100 Halton points in 2D, to be analyzed using the test code.
• sphere_02_00100.txt, a set of 100 points in 2D, in the unit sphere.

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