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<html>
<head>
<title>
CVT_DEMO - Interactive Graphic CVT Demonstrations
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
CVT_DEMO <br> Interactive Graphic CVT Demonstrations
</h1>
<hr>
<p>
<b>CVT_DEMO</b>
is a directory of MATLAB programs which
use interaction and graphics to demonstrate properties of
the CVT (Centroidal Voronoi Tessellation) calculation.
</p>
<p>
As you watch the CVT develop, for instance, you will be able to
guess something about the angle of intersection between the boundaries of the
region, and the edges of Voronoi regions.
</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>CVT_DEMO</b> is available in
<a href = "../../m_src/cvt_demo/cvt_demo.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/ccvt_box/ccvt_box.html">
CCVT_BOX</a>,
a MATLAB program which
constructs a modified CVT in which some points are forced to
lie on the boundary.
</p>
<p>
<a href = "../../m_src/ccvt_reflect/ccvt_reflect.html">
CCVT_REFLECT</a>,
a MATLAB program which
tries to construct a modified CVT in which some points are forced to
lie on the boundary, using a reflection idea.
</p>
<p>
<a href = "../../m_src/cvt_1d_lloyd/cvt_1d_lloyd.html">
CVT_1D_LLOYD</a>,
a MATLAB program which
computes an N-point Centroidal Voronoi Tessellation (CVT)
within the interval [0,1], under a uniform density.
</p>
<p>
<a href = "../../m_src/cvt_1d_nonuniform/cvt_1d_nonuniform.html">
CVT_1D_NONUNIFORM</a>,
a MATLAB library which
allows the user to watch the evolution of a CVT computed over
a 1D interval with a nonuniform density.
</p>
<p>
<a href = "../../m_src/cvt_1d_sampling/cvt_1d_sampling.html">
CVT_1D_SAMPLING</a>,
a MATLAB program which
computes an N-point Centroidal Voronoi Tessellation (CVT)
within the interval [0,1], under a uniform density,
using sampling to estimate the Voronoi regions.
</p>
<p>
<a href = "../../m_src/cvt_2d_sampling/cvt_2d_sampling.html">
CVT_2D_SAMPLING</a>,
a MATLAB program which
computes an N-point Centroidal Voronoi Tessellation (CVT)
within the unit square [0,1]x[0,1], under a uniform density,
using sampling to estimate the Voronoi regions.
</p>
<p>
<a href = "../../m_src/cvt_dataset/cvt_dataset.html">
CVT_DATASET</a>,
a MATLAB program which
creates a CVT dataset.
</p>
<p>
<a href = "../../m_src/discrete_pdf_sample_2d/discrete_pdf_sample_2d.html">
DISCRETE_PDF_SAMPLE_2D</a>,
a MATLAB program which
demonstrates how to construct a Probability Density Function (PDF)
from a table of sample data, and then to use that PDF to create new samples.
</p>
<p>
<a href = "../../m_src/lcvt/lcvt.html">
LCVT</a>,
a MATLAB library which
computes a "Latinized" Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../m_src/test_triangulation/test_triangulation.html">
TEST_TRIANGULATION</a>,
a MATLAB library which
defines the geometry of a number of sample regions.
</p>
<p>
<a href = "../../m_src/voronoi_plot/voronoi_plot.html">
VORONOI_PLOT</a>,
a MATLAB program which
plots the Voronoi neighborhoods of points using L1, L2, LInfinity
or arbitrary LP norms;
</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, Number 3, 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,<br>
Volume 41, Number 4, December 1999, pages 637-676.
</li>
</ol>
</p>
<hr>
<h2 align = "center">
CVT_CIRCLE_NONUNIFORM
</h2>
<p>
<b>CVT_CIRCLE_NONUNIFORM</b> is a MATLAB function for computing a CVT
in a circle with a nonuniform density. The spatial region is the unit circle,
but a nonuniform density is applied to that region. The density is implicitly defined
by the sampling routine, which returns many more points near
the circumference than near the center of the circle. The result
is that, during the CVT iteration, the generators arrange themselves
more densely towards the outer part of the circle.
<blockquote>
[ p, t ] = cvt_circle_nonuniform ( n, sample_num, delaunay_display )
</blockquote>
<ul>
<li>
<b>N</b>, the number of generators.
</li>
<li>
<b>SAMPLE_NUM</b>, the number of sample points per generator.
</li>
<li>
<b>DELAUNAY_DISPLAY</b>, 0 to hide, 1 to show the Delaunay triangulation.
</li>
</ul>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "cvt_circle_nonuniform.m">cvt_circle_nonuniform.m</a>,
the MATLAB function.
</li>
<li>
<a href = "circle_nonuniform.png">circle_nonuniform.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of 1000
sample points.
</li>
<li>
<a href = "cvt_circle_nonuniform.png">cvt_circle_nonuniform.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of the
Voronoi diagram of for 200 generators after several CVT iterations.
</li>
</ul>
</p>
<hr>
<h2 align = "center">
CVT_ELLIPSE_UNIFORM
</h2>
<p>
<b>CVT_ELLIPSE_UNIFORM</b> is a MATLAB function which works the same
as the code for the square, except that
we are now working in a more interesting region, namely, the
ellipse defined by <b>x'*A*x<=36</b> with A = [ 9, 6; 6, 20 ];
The only tricky thing we have to do is write a routine that
returns sample points from the region.
The calling sequence is:
<blockquote>
[ p, t ] = cvt_ellipse_uniform ( n, sample_num, delaunay_display )
</blockquote>
<ul>
<li>
<b>N</b>, the number of generators.
</li>
<li>
<b>SAMPLE_NUM</b>, the number of sample points per generator.
</li>
<li>
<b>DELAUNAY_DISPLAY</b>, 0 to hide, 1 to show the Delaunay triangulation.
</li>
</ul>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "cvt_ellipse_uniform.m">cvt_ellipse_uniform.m</a>,
the MATLAB function.
</li>
<li>
<a href = "ellipse_uniform.png">ellipse_uniform.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of 1000
sample points.
</li>
<li>
<a href = "cvt_ellipse_uniform.png">cvt_ellipse_uniform.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of the
Voronoi diagram of the points after several CVT iterations.
</li>
</ul>
</p>
<hr>
<h2 align = "center">
CVT_SQUARE_UNIFORM
</h2>
<p>
<b>CVT_SQUARE_UNIFORM</b> is a MATLAB function which works in a
square with a uniform density. The function starts from a random initial
configuration. Another step of the CVT iteration is carried out
each time the user hits RETURN. The MATLAB routines <b>trimesh</b>
and <b>voronoi</b> are used to simultaneously display the
Voronoi diagram and Delaunay triangulation associated with the
generators. Moreover, the nearest-neighbor search and
centroid computation are carried out efficiently using MATLAB's
Delaunay search and rapid array accumulation. The calling
sequence is:
<blockquote>
[ p, t ] = cvt_square_uniform ( n, sample_num, delaunay_display )
</blockquote>
<ul>
<li>
<b>N</b>, the number of generators.
</li>
<li>
<b>SAMPLE_NUM</b>, the number of sample points per generator.
</li>
<li>
<b>DELAUNAY_DISPLAY</b>, 0 to hide, 1 to show the Delaunay triangulation.
</li>
</ul>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "cvt_square_uniform.m">cvt_square_uniform.m</a>,
the MATLAB function.
</li>
<li>
<a href = "cvt_square_uniform.png">cvt_square_uniform.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of the
Voronoi diagram of the points after several CVT iterations.
</li>
</ul>
</p>
<hr>
<h2 align = "center">
CVT_SQUARE_PDF_DISCRETE
</h2>
<p>
<b>CVT_SQUARE_PDF_DISCRETE</b> is a MATLAB function which works the same
as the code for the uniform square, except that
we are now working with a density function defined by discrete data.
In this case, the density is defined by a 20 x 20 grid of sample
density values. The calling sequence is:
<blockquote>
[ p, t ] = cvt_square_pdf_discrete ( n, sample_num, delaunay_display )
</blockquote>
<ul>
<li>
<b>N</b>, the number of generators.
</li>
<li>
<b>SAMPLE_NUM</b>, the number of sample points per generator.
</li>
<li>
<b>DELAUNAY_DISPLAY</b>, 0 to hide, 1 to show the Delaunay triangulation.
</li>
</ul>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "cvt_square_pdf_discrete.m">cvt_square_pdf_discrete.m</a>,
the MATLAB function.
</li>
<li>
<a href = "square_pdf_discrete.png">square_pdf_discrete.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of 1000
sample points.
</li>
<li>
<a href = "cvt_square_pdf_discrete.png">cvt_square_pdf_discrete.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of the
Voronoi diagram of the points after 25 CVT iteration.
</li>
</ul>
</p>
<hr>
<h2 align = "center">
CVT_TRIANGLE_UNIFORM
</h2>
<p>
<b>CVT_TRIANGLE_UNIFORM</b> is a MATLAB function which generates a CVT
in a triangle with a uniform density. The calling sequence is:
<blockquote>
[ p, t ] = cvt_triangle_uniform ( n, sample_num, delaunay_display )
</blockquote>
<ul>
<li>
<b>N</b>, the number of generators.
</li>
<li>
<b>SAMPLE_NUM</b>, the number of sample points per generator.
</li>
<li>
<b>DELAUNAY_DISPLAY</b>, 0 to hide, 1 to show the Delaunay triangulation.
</li>
</ul>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "cvt_triangle_uniform.m">cvt_triangle_uniform.m</a>,
the MATLAB function.
</li>
<li>
<a href = "cvt_triangle_uniform.png">cvt_triangle_uniform.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of the
Voronoi diagram of the points after several CVT iterations.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 16 July 2009.
</i>
<!-- John Burkardt -->
</body>
</html>