QUALITY
Quality Measures for Pointsets in M dimensions

QUALITY is a C++ library which computes some measures of the quality of dispersion of a set of N points in the unit hypercube in M dimensional space.

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 (unit hypercube datasets only).
• 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 Programs:

DIAPHONY, a FORTRAN90 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 N points in M 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 C++ program which analyzes the point spacing quality of data stored in a file.

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

TET_MESH_QUALITY, a C++ program which computes quality measures of a tetrahedral mesh.

TRIANGULATION_QUALITY, a C++ program which computes quality measures of a triangulation.

Reference:

1. David Field,
   Qualitative Measures for Initial Meshes,
   International Journal of Numerical Methods in Engineering,
   Volume 47, 2000, pages 887-906.
2. DP 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:

• quality.cpp, the source code.
• quality.hpp, the include file.

Examples and Tests:

• quality_test.cpp, the test code.
• cvt_02_00100.txt, a set of 100 CVT points in the 2D unit hypercube.
• quality_test.txt, the output file.
• halton_02_00100.txt, a set of 100 Halton points in the 2D unit hypercube.
• sphere_02_00100.txt, a set of 100 points in the 2D unit sphere.

List of Routines:

• ALPHA_MEASURE determines the triangulated pointset quality measure ALPHA.
• ARC_COSINE computes the arc cosine function, with argument truncation.
• AREA_MEASURE determines the area ratio quality measure.
• BETA_MEASURE determines the pointset quality measure BETA.
• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• CHI_MEASURE determines the pointset quality measure CHI.
• D_MEASURE determines the pointset quality measure D.
• DGE_DET computes the determinant of a square matrix in DGE storage.
• DIAEDG chooses a diagonal edge.
• DTABLE_DATA_READ reads the data from a real TABLE file.
• DTABLE_HEADER_READ reads the header from a real TABLE file.
• DTRIS2 constructs a Delaunay triangulation of 2D vertices.
• E_MEASURE determines the pointset quality measure E.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• FIND_CLOSEST finds the nearest R point to each S point.
• GAMMA_MEASURE determines the pointset quality measure GAMMA.
• H_MEASURE determines the pointset quality measure H.
• I4_MAX returns the maximum of two integers.
• I4_MIN returns the smaller of two integers.
• I4_MODP returns the nonnegative remainder of integer division.
• I4_SIGN returns the sign of an integer.
• I4_WRAP forces an integer to lie between given limits by wrapping.
• I4VEC_INDICATOR sets an integer vector to the indicator vector.
• I4VEC_PRINT prints an integer vector.
• LAMBDA_MEASURE determines the pointset quality measure LAMBDA.
• LRLINE determines where a point lies in relation to a directed line.
• MU_MEASURE determines the pointset quality measure MU.
• NU_MEASURE determines the pointset quality measure NU.
• PERM_INV inverts a permutation "in place".
• POINTSET_SPACING determines the minimum spacing between points in the set.
• Q_MEASURE determines the triangulated pointset quality measure Q.
• R0_MEASURE determines the pointset quality measure R0.
• R8_EPSILON returns the round off unit for double precision arithmetic.
• R8_HUGE returns a "huge" double precision value.
• R8_MAX returns the maximum of two double precision values.
• R8_MIN returns the minimum of two double precision values.
• R8_UNIFORM_01 is a portable pseudorandom number generator.
• R82VEC_PERMUTE permutes an R2 vector in place.
• R82VEC_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R2 vector.
• R8MAT_IN_01 is TRUE if the entries of an array are in the range [0,1].
• R8MAT_TRANSPOSE_PRINT prints a real matrix, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of a real matrix, transposed.
• R8MAT_UNIFORM_01 fills a double precision array with pseudorandom values.
• R8VEC_MAX returns the value of the maximum element in a double precision array.
• R8VEC_MIN returns the value of the minimum element in a double precision array.
• R8VEC_NORMAL_01 samples the standard normal probability distribution.
• R8VEC_UNIFORM_01 fills a double precision vector with pseudorandom values.
• RADIUS_MAXIMUS finds the biggest possible nonintersecting sphere.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• SAMPLE_HYPERCUBE_UNIFORM returns sample points in the unit hypercube.
• SAMPLE_SPHERE_UNIFORM samples points inside the unit sphere.
• SPHERE_MEASURE determines the pointset quality measure S.
• SPHERE_VOLUME_ND computes the volume of a sphere in ND.
• SWAPEC swaps diagonal edges until all triangles are Delaunay.
• TAU_MEASURE determines the pointset quality measure TAU.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.
• VBEDG determines which boundary edges are visible to a point.

You can go up one level to the C++ source codes.