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cvt_size.html
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<html>
<head>
<title>
CVT_SIZE - CVT with Prescribed Cell Sizes
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
CVT_SIZE <br> CVT with Prescribed Cell Sizes
</h1>
<hr>
<p>
<b>CVT_SIZE</b>
is a FORTRAN90 library which
attempts to compute a
(weighted) centroidal Voronoi tessellation (CVT) in which each cell
has a desired prescribed size.
</p>
<p>
Normally, given arbitrary initial data and a constant density,
the CVT would comprise cells of roughly equal area. This program
modifies the CVT iteration, allowing the user to specify
a relative cell size for each cell. Thus, if the region
has area 100, and we make a tessellation of two cells, and
we specify weights of 1 and 3, we would hope for the cells
to be computed with areas of 25 and 75 units respectively.
</p>
<p>
Internally, the desired cell sizes are normalized, and
then the appropriate root is taken to account for the
dimensionality of the space. (In 2 space, you take the
square root of the normalized cell size, for instance.)
This gives you the weight to be assigned to a generator.
</p>
<p>
Then, when it is time to compute the distance for a sample
point to the center of each Voronoi cell and take the
"closest" one, we divide each cell's distance by the adjusted
weight. This has the effect of making cells with a larger
weight seem closer than they really are, and hence they
attract more sample points, and end up with a larger area.
</p>
<p>
If only a small number of generators are used, or if the
weights vary greatly in size, the effects
of local geometry and curvature come into play, and the
chosen weights produce areas that are only roughly what is
desired. Increasing the number
of cells being used seems to produce more regular behavior,
with the cells tending to have the areas requested by the user.
</p>
<p>
Some of the activity on this project was inspired by
a query from Professor Alexandru Telea, of the Department
of Mathematics and Computer Science, Technische
Universiteit Eindhoven.
</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">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/pbmlib/pbmlib.html">
PBMLIB</a>,
a FORTRAN90 library which
can create crisp and relatively small binary PPM images of the CVT regions.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Franz Aurenhammer,<br>
Voronoi diagrams -
a study of a fundamental geometric data structure,<br>
ACM Computing Surveys,<br>
Volume 23, pages 345-405, September 1991.
</li>
<li>
John Burkardt, Max Gunzburger, Janet Peterson, Rebecca Brannon,<br>
User Manual and Supporting Information for Library of Codes
for Centroidal Voronoi Placement and Associated Zeroth,
First, and Second Moment Determination,<br>
Sandia National Laboratories Technical Report SAND2002-0099,<br>
February 2002.
</li>
<li>
Qiang Du, Vance Faber, Max Gunzburger,<br>
Centroidal Voronoi Tessellations: Applications and Algorithms,<br>
SIAM Review, Volume 41, 1999, pages 637-676.
</li>
<li>
Lili Ju, Qiang Du, and Max Gunzburger,<br>
Probabilistic methods for centroidal Voronoi tessellations
and their parallel implementations,<br>
Parallel Computing,<br>
Volume 28, 2002, pages 1477-1500.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "cvt_size.f90">cvt_size.f90</a>, the source code.
</li>
<li>
<a href = "cvt_size.sh">cvt_size.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "cvt_size_prb.f90">cvt_size_prb.f90</a>, a sample problem.
To run the various cases below, you have to make changes to
the text of this file.
</li>
<li>
<a href = "cvt_size_prb.sh">cvt_size_prb.sh</a>,
commands to compile, link and run the sample problem.
</li>
<li>
<a href = "cvt_size_prb_output.txt">cvt_size_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
We were interested in watching the behavior of the Voronoi
cells over "time". We set up a problem with 10 generators,
with weights 1, 2, 3, ..., 10. We started with random initial
locations for the generators, and carried out 100 iterations
of the CVT code. We saved a
<a href = "../../data/ppmb/ppmb.html">PPMB</a> file containing
an image of each step, used the
<a href = "http://www.ijg.org/">CJPEG</a>
program to convert these to
<a href = "../../data/jpg/jpg.html">JPEG files</a> with
consecutive file names, fed that into
<a href = "http://www.apple.com/quicktime/">QuickTime Pro</a>
to create an animation, and saved the result as an MPEG-4
file.
</p>
<p>
<a href = "../../data/mp4/cvt_size_movie.html">
Click HERE to see the CVT movie</a>.
</p>
<p>
We ran the code on a variety of different sets of data.
<ul>
<li>
We ran a problem called
<a href = "cvt_1111133333/cvt_1111133333.html">
cvt_1111133333</a>
with 10 generators, with weights 1, 1, 1, 1, 1, 3, 3, 3, 3, 3.
</li>
<li>
We ran a problem called
<a href = "cvt_123456789x/cvt_123456789x.html">
cvt_123456789x</a>
with 10 generators, with weights 1, 2, 3, ..., 9, 10.
</li>
<li>
We ran a problem called
<a href = "cvt_1to37/cvt_1to37.html">
cvt_1to37</a>
with 10 generators, with weights 1, 5, 9, ..., 37.
</li>
<li>
We ran a problem called
<a href = "cvt_1to80/cvt_1to80.html">
cvt_1to80</a>
with 80 generators, with weights 1 through 80.
</li>
<li>
We ran a problem called
<a href = "cvt_1to317/cvt_1to317.html">
cvt_1to80</a>
with 80 generators, with weights 1, 5, 9, ..., 317.
</li>
<li>
We ran a problem called
<a href = "cvt_1to1024/cvt_1to1024.html">
cvt_1to1024</a>
with 11 generators, with weights 1, 2, 4, ..., 1024.
</li>
</ul>
</p>
<p>
We made a sequence of calculations giving half the cells weight 1
and half weight 4. Increasing the number of generators seemed to
show regular behavior.
<ul>
<li>
<a href = "cvt_halfandhalf10/cvt_halfandhalf10.html">
cvt_halfandhalf10</a> uses 10 generators.
</li>
<li>
<a href = "cvt_halfandhalf20/cvt_halfandhalf20.html">
cvt_halfandhalf20</a> uses 20 generators.
</li>
<li>
<a href = "cvt_halfandhalf40/cvt_halfandhalf40.html">
cvt_halfandhalf40</a> uses 40 generators.
</li>
<li>
<a href = "cvt_halfandhalf80/cvt_halfandhalf80.html">
cvt_halfandhalf80</a> uses 80 generators.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ANGLE_TO_RGB</b> returns a color on the perimeter of the color hexagon.
</li>
<li>
<b>CH_IS_DIGIT</b> returns .TRUE. if a character is a decimal digit.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>CVT_SIZE_ITERATION</b> takes one step of the CVT size iteration.
</li>
<li>
<b>DIGIT_INC</b> increments a decimal digit.
</li>
<li>
<b>DIGIT_TO_CH</b> returns the character representation of a decimal digit.
</li>
<li>
<b>FILE_NAME_INC</b> generates the next filename in a series.
</li>
<li>
<b>FIND_CLOSEST_SIZE</b> finds the Voronoi cell generator closest to a point X.
</li>
<li>
<b>GENERATOR_INIT</b> initializes the Voronoi cell generators.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>HEXCOL</b> returns a color on the perimeter of the color hexagon.
</li>
<li>
<b>I4_LOG_2</b> returns the integer part of the logarithm base 2 of |I|.
</li>
<li>
<b>I4_TO_ANGLE</b> maps integers to points on a circle.
</li>
<li>
<b>I4_UNIFORM</b> returns a scaled pseudorandom I4.
</li>
<li>
<b>R82VEC_DIST_L2</b> returns the L2 distance between a pair of real 2-vectors.
</li>
<li>
<b>RANDOM_INITIALIZE</b> initializes the FORTRAN 90 random number seed.
</li>
<li>
<b>REGION_PLOT_PPMB</b> makes a binary PPM plot of the CVT regions.
</li>
<li>
<b>REGION_SAMPLER</b> returns a sample point in the physical region.
</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 12 November 2006.
</i>
<!-- John Burkardt -->
</body>
</html>