QUALITY
Quality Measures for Pointsets in M dimensions

QUALITY is a library of MATLAB routines, using double precision arithmetic, which compute 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;

Related Data and Programs:

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

TABLE_QUALITY is an interactive program which can analyze a dataset that is stored in a file. This program is available in

• a C++ version,
• a FORTRAN90 version and
• a MATLAB version.

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

TRIANGULATION is a MATLAB library which includes some quality measures for triangulations.

Reference:

1. David Field,
   Qualitative Measures for Initial Meshes,
   International Journal of Numerical Methods in Engineering,
   Volume 47, 2000, pages 887-906.
2. Max Gunzburger, Uniformity Measures for Point Samples in Hypercubes,
   PDF version.
3. D P Hardin, Edward Saff,
   Discretizing Manifolds via Minimum Energy Points,
   Notices of the AMS,
   Volume 51, Number 10, November 2004, pages 1186-1194. PDF version.
4. Per-Olof Persson, Gilbert Strang, A Simple Mesh Generator in MATLAB, SIAM Review, Volume 46, Number 2, pages 329-345, June 2004.

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.
• dtable_data_read.m reads data from a double precision table file.
• dtable_header_read.m reads the header from a double precision table file.
• 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.
• 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.out, 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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