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<html>
<head>
<title>
QUALITY - Quality Measures for Pointsets in M dimensions
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
QUALITY <br> Quality Measures for Pointsets in M dimensions
</h1>
<hr>
<p>
<b>QUALITY</b>
is a FORTRAN90 library which
computes measures of the quality
of dispersion of a set of N points in an M dimensional region.
</p>
<p>
The region to be analyzed is defined by a single routine which
the user may specify. We will refer to that routine as
<b>sample_routine</b>. This routine is simply required to return
sample points from the region.
</p>
<p>
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 <b>sample_hypercube_uniform</b> or
<b>sample_sphere_uniform</b>. For other regions, the user should
write the appropriate version of a sampling routine.
</p>
<p>
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 <b>M</b>
dimensional space.
</p>
<p>
The unit hypersphere in M dimensional space is simply the set of
all points whose distance from the origin is 1 or less.
</p>
<p>
The quality measures computed include:
<ul>
<li>
<b>Alpha</b>, the minimum angle divided by the maximum possible
minimum angle (for triangulated 2D datasets only);
</li>
<li>
<b>Area</b>, the ratio of the minimum and maximum triangle
areas (for triangulated 2D datasets only);
</li>
<li>
<b>Beta</b>, the standard deviation of the minimum spacing
between nodes, normalized by the average spacing between nodes;
</li>
<li>
<b>Chi</b>, the regularity measure;
</li>
<li>
<b>D</b>, the second moment determinant measure;
</li>
<li>
<b>E</b>, the Voronoi energy measure;
</li>
<li>
<b>Gamma</b>, the mesh ratio;
</li>
<li>
<b>H</b>, the point distribution norm;
</li>
<li>
<b>Lambda</b>, the COV measure;
</li>
<li>
<b>Mu</b>, the point distribution ratio;
</li>
<li>
<b>Nu</b>, the cell volume deviation;
</li>
<li>
<b>Q</b>, two times the radius of the inscribed circle
divided by the radius of the circumscribed circle
(for triangulated 2D datasets only);
</li>
<li>
<b>R0</b>, the Riesz S=0 energy;
</li>
<li>
<b>S</b>, the sphere volume measure
(unit hypercube datasets only).
</li>
<li>
<b>Tau</b>, the second moment trace measure;
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>QUALITY</b> is available in
<a href = "../../cpp_src/quality/quality.html">a C++ version</a> and
<a href = "../../f_src/quality/quality.html">a FORTRAN90 version</a> and
<a href = "../../m_src/quality/quality.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../f_src/diaphony/diaphony.html">
DIAPHONY</a>,
a FORTRAN90 program which
reads a file of N points in M dimensions and computes its diaphony, a measure
of point dispersion.
</p>
<p>
<a href = "../../f_src/point_merge/point_merge.html">
POINT_MERGE</a>,
a FORTRAN90 library which
considers N points in M dimensional space, and counts or indexes
the unique or "tolerably unique" items.
</p>
<p>
<a href = "../../cpp_src/star_discrepancy/star_discrepancy.html">
STAR_DISCREPANCY</a>,
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.
</p>
<p>
<a href = "../../f_src/table_quality/table_quality.html">
TABLE_QUALITY</a>,
a FORTRAN90 program which
computes the quality measures of a dataset read from a file.
which can analyze a dataset that is stored in a file.
</p>
<p>
<a href = "../../f_src/tet_mesh/tet_mesh.html">
TET_MESH</a>,
a FORTRAN90 library which
carries out various operations on tetrahedral meshes.
</p>
<p>
<a href = "../../f_src/tet_mesh_quality/tet_mesh_quality.html">
TET_MESH_QUALITY</a>,
a FORTRAN90 program which
computes quality measures of a tetrahedral mesh.
</p>
<p>
<a href = "../../f_src/triangulation_quality/triangulation_quality.html">
TRIANGULATION_QUALITY</a>,
a FORTRAN90 program which
computes quality measures of a triangulation.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
David Field,<br>
Qualitative Measures for Initial Meshes,<br>
International Journal of Numerical Methods in Engineering,<br>
Volume 47, 2000, pages 887-906.
</li>
<li>
Douglas Hardin, Edward Saff,<br>
Discretizing Manifolds via Minimum Energy Points,<br>
Notices of the AMS,<br>
Volume 51, Number 10, November 2004, pages 1186-1194.
</li>
<li>
Per-Olof Persson, Gilbert Strang,<br>
A Simple Mesh Generator in MATLAB,<br>
SIAM Review,<br>
Volume 46, Number 2, pages 329-345, June 2004.
</li>
<li>
Yuki Saka, Max Gunzburger, John Burkardt,<br>
Latinized, Improved LHS, and CVT Point Sets in Hypercubes, <br>
International Journal of Numerical Analysis and Modeling,<br>
Volume 4, Number 3-4, 2007, pages 729-743,
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "quality.f90">quality.f90</a>, the source code.
</li>
<li>
<a href = "quality.sh">quality.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "quality_prb.f90">quality_prb.f90</a>, the test code.
</li>
<li>
<a href = "quality_prb.sh">quality_prb.sh</a>,
commands to compile, link, load and run the test code.
</li>
<li>
<a href = "quality_prb_output.txt">quality_prb_output.txt</a>,
the output file.
</li>
<li>
<a href = "cvt_02_00100.txt">cvt_02_00100.txt</a>,
a set of 100 CVT points in the 2D unit hypercube.
</li>
<li>
<a href = "halton_02_00100.txt">halton_02_00100.txt</a>,
a set of 100 Halton points in the 2D unit hypercube.
</li>
<li>
<a href = "sphere_02_00100.txt">sphere_02_00100.txt</a>,
a set of 100 random points in the 2D unit sphere.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ALPHA_MEASURE</b> determines the triangulated pointset quality measure ALPHA.
</li>
<li>
<b>ARC_COSINE</b> computes the arc cosine function, with argument truncation.
</li>
<li>
<b>AREA_MEASURE</b> determines the area ratio quality measure.
</li>
<li>
<b>BANDWIDTH_MESH:</b> bandwidth of finite element mesh.
</li>
<li>
<b>BETA_MEASURE</b> determines the pointset quality measure BETA.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer ( kind = 4 ) value of a base 10 digit.
</li>
<li>
<b>CHI_MEASURE</b> determines the pointset quality measure CHI.
</li>
<li>
<b>D_MEASURE</b> determines the pointset quality measure D.
</li>
<li>
<b>DGE_DET</b> computes the determinant of a square matrix in DGE storage.
</li>
<li>
<b>DIAEDG</b> chooses a diagonal edge.
</li>
<li>
<b>DTRIS2</b> constructs a Delaunay triangulation of 2D vertices.
</li>
<li>
<b>E_MEASURE</b> determines the pointset quality measure E.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>FIND_CLOSEST</b> finds the nearest R point to each S point.
</li>
<li>
<b>GAMMA_MEASURE</b> determines the pointset quality measure GAMMA.
</li>
<li>
<b>GET_INPUT_FILE_NAME</b> gets the input file name.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>H_MEASURE</b> determines the pointset quality measure H.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of integer ( kind = 4 ) division.
</li>
<li>
<b>I4_WRAP</b> forces an I4 to lie between given limits by wrapping.
</li>
<li>
<b>I4VEC_INDICATOR</b> sets an I4VEC to the indicator vector.
</li>
<li>
<b>I4VEC_PRINT</b> prints an I4VEC.
</li>
<li>
<b>LAMBDA_MEASURE</b> determines the pointset quality measure LAMBDA.
</li>
<li>
<b>LRLINE</b> determines if a point is left of, right or, or on a directed line.
</li>
<li>
<b>MU_MEASURE</b> determines the pointset quality measure MU.
</li>
<li>
<b>NU_MEASURE</b> determines the pointset quality measure NU.
</li>
<li>
<b>PERM_INV</b> inverts a permutation "in place".
</li>
<li>
<b>POINTSET_SPACING</b> determines the minimum spacing between points in the set.
</li>
<li>
<b>Q_MEASURE</b> determines the triangulated pointset quality measure Q.
</li>
<li>
<b>R0_MEASURE</b> determines the pointset quality measure R0.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R82VEC_PERMUTE</b> permutes an R82 vector in place.
</li>
<li>
<b>R82VEC_SORT_HEAP_INDEX_A</b> does an indexed heap ascending sort of an R82VEC.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>R8MAT_IN_01</b> is TRUE if the entries of an R8MAT are in the range [0,1].
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8MAT_UNIFORM_01</b> returns a unit pseudorandom R8MAT.
</li>
<li>
<b>R8VEC_NORMAL_01</b> returns a unit pseudonormal R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>RADIUS_MAXIMUS</b> finds the biggest possible nonintersecting sphere.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>SAMPLE_HYPERCUBE_UNIFORM</b> returns sample points in the unit hypercube.
</li>
<li>
<b>SAMPLE_SPHERE_UNIFORM</b> samples points inside the unit sphere.
</li>
<li>
<b>SPHERE_MEASURE</b> determines the pointset quality measure S.
</li>
<li>
<b>SPHERE_VOLUME_ND</b> computes the volume of a sphere in ND.
</li>
<li>
<b>SWAPEC</b> swaps diagonal edges until all triangles are Delaunay.
</li>
<li>
<b>TAU_MEASURE</b> determines the pointset quality measure TAU.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>VBEDG</b> determines which boundary edges are visible to a point.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 01 February 2010.
</i>
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