Quality Measures for Pointsets in M dimensions
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
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;
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
QUALITY is available in
a C++ version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
a MATLAB program which
reads a file of N points in M dimensions and computes its diaphony, a measure
of point dispersion.
a MATLAB library which
includes some quality measures for
Qualitative Measures for Initial Meshes,
International Journal of Numerical Methods in Engineering,
Volume 47, 2000, pages 887-906.
D P Hardin, Edward Saff,
Discretizing Manifolds via Minimum Energy Points,
Notices of the AMS,
Volume 51, Number 10, November 2004, pages 1186-1194.
Per-Olof Persson, Gilbert Strang,
A Simple Mesh Generator in MATLAB,
Volume 46, Number 2, pages 329-345, June 2004.
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,
Last revised on 15 February 2019.