452 lines
22 KiB
HTML
452 lines
22 KiB
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
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<html><head>
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<meta http-equiv="Content-Language" content="en-us">
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<meta http-equiv="Content-Type" content="text/html; charset=windows-1252"><title>Voronoi Main</title>
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<meta http-equiv="content-type" content="text/html; charset=utf-8"></head><body>
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<table style="margin: 0pt; padding: 0pt; width: 100%;" border="0" cellpadding="0" cellspacing="0">
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<tbody>
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<tr>
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<td style="background-color: rgb(238, 238, 238);" nowrap="1" valign="top">
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<div style="padding: 5px;" align="center"> <img src="images/boost.png" border="0" height="86" width="277"><a title="www.boost.org home page" tabindex="2" style="border: medium none ;" href="http://www.boost.org/"> </a></div>
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<div style="margin: 5px;">
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<h3 class="navbar">Contents</h3>
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<ul>
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<li><a href="index.htm">Boost.Polygon Main Page</a></li>
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<li><a href="gtl_design_overview.htm">Design Overview</a></li>
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<li><a href="gtl_isotropy.htm">Isotropy</a></li>
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<li><a href="gtl_coordinate_concept.htm">Coordinate Concept</a></li>
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<li><a href="gtl_interval_concept.htm">Interval Concept</a></li>
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<li><a href="gtl_point_concept.htm">Point Concept</a></li>
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<li><a href="gtl_segment_concept.htm">Segment Concept</a></li>
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<li><a href="gtl_rectangle_concept.htm">Rectangle Concept</a></li>
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<li><a href="gtl_polygon_90_concept.htm">Polygon 90 Concept</a></li>
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<li><a href="gtl_polygon_90_with_holes_concept.htm">Polygon 90
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With Holes Concept</a></li>
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<li><a href="gtl_polygon_45_concept.htm">Polygon 45 Concept</a></li>
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<li><a href="gtl_polygon_45_with_holes_concept.htm">Polygon 45
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With Holes Concept</a></li>
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<li><a href="gtl_polygon_concept.htm">Polygon Concept</a></li>
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<li><a href="gtl_polygon_with_holes_concept.htm">Polygon With
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Holes Concept</a></li>
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<li><a href="gtl_polygon_90_set_concept.htm">Polygon 90 Set
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Concept</a></li>
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<li><a href="gtl_polygon_45_set_concept.htm">Polygon 45 Set
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Concept</a></li>
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<li><a href="gtl_polygon_set_concept.htm">Polygon Set Concept</a></li>
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<li><a href="gtl_connectivity_extraction_90.htm">Connectivity
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Extraction 90</a></li>
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<li><a href="gtl_connectivity_extraction_45.htm">Connectivity
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Extraction 45</a></li>
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<li><a href="gtl_connectivity_extraction.htm">Connectivity
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Extraction</a></li>
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<li><a href="gtl_property_merge_90.htm">Property Merge 90</a></li>
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<li><a href="gtl_property_merge_45.htm">Property Merge 45</a></li>
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<li><a href="gtl_property_merge.htm">Property Merge</a></li>
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<li>Voronoi Main Page </li>
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<li><a href="voronoi_benchmark.htm">Voronoi Benchmark</a></li>
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<li><a href="voronoi_builder.htm">Voronoi Builder</a> </li>
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<li><a href="voronoi_diagram.htm">Voronoi Diagram</a></li>
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</ul>
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<h3 class="navbar">Other Resources</h3>
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<ul>
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<li><a href="GTL_boostcon2009.pdf">GTL Boostcon 2009 Paper</a></li>
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<li><a href="GTL_boostcon_draft03.pdf">GTL Boostcon 2009
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Presentation</a></li>
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<li><a href="analysis.htm">Performance Analysis</a></li>
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<li><a href="gtl_tutorial.htm">Layout Versus Schematic Tutorial</a></li>
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<li><a href="gtl_minkowski_tutorial.htm">Minkowski Sum Tutorial</a></li>
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<li><a href="voronoi_basic_tutorial.htm">Voronoi Basic Tutorial</a></li>
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<li><a href="voronoi_advanced_tutorial.htm">Voronoi Advanced
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Tutorial</a></li>
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</ul>
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</div>
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<h3 class="navbar">Polygon Sponsor</h3>
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<div style="padding: 5px;" align="center"> <img src="images/intlogo.gif" border="0" height="51" width="127"><a title="www.adobe.com home page" tabindex="2" style="border: medium none ;" href="http://www.adobe.com/"> </a></div>
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</td>
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<td style="padding-left: 10px; padding-right: 10px; padding-bottom: 10px;" valign="top" width="100%"><!-- End Header --> <br>
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<h1>THE BOOST.POLYGON VORONOI LIBRARY </h1>
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<img style="width: 900px; height: 300px;" alt="" src="images/voronoi3.png"><br>
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The Voronoi extension of the Boost.Polygon library provides
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functionality to construct a <a href="voronoi_diagram.htm">Voronoi
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diagram</a>
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of a set of points and linear segments in 2D space with the following
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set of
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limitations:<br>
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<ul>
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<li>coordinates of the input points and endpoints of the
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input segments
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should be of the integer type;</li>
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<li>input segments should not overlap
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except their endpoints.</li>
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</ul>
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While the first restriction is permanent (it
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allows to give the exact warranties about the output precision and
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algorithm execution flow),
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the second one may be resolved using the Boost.Polygon <a href="gtl_segment_concept.htm">segment utils</a>.
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The strong sides of the
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library and main benefits comparing to the other implementations are
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discussed in the following paragraphs.<br>
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<h2>Fully Functional with Segments</h2>
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There are just a few implementations of the Voronoi diagram
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construction
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algorithm that can
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handle input data sets that contain linear segment, even considering
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the commercial
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libraries.
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Support of the
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segments allows to discretize any input geometry (sampled
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floating-point coordinates can be scaled and snapped to the integer
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grid): circle, ellipse,
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parabola. This functionality allows to compute
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the medial axis transform of the arbitrary set of input geometries,
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with direct applications in the computer vision
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projects.
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<h2>Robustness and Efficiency</h2>
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Robustness issues can be divided onto the two main categories: memory
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management
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issues and numeric stability issues. The implementation avoids the
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first type of the issues using pure STL data structures, thus there is
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no
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presence of the new operator in the code. The second category of
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the problems is
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resolved using the multiprecision geometric
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predicates.
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Even for the commercial libraries, usage of such predicates
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results in a vast performance slowdown. The Voronoi implementation
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overcomes this by avoiding the multiprecision
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computations in the 95% of the cases and
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uses the efficient, floating-point based predicates. Such preciates
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don't
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produce the correct result always, however the library embeds the
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relative
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error arithmetic apparatus to identify such situations and switch
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to the
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higher precision predicates when appropriate. As the result, the
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implementation has a solid performance comparing to the other known
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libraries (more details in the <a href="voronoi_benchmark.htm">benchmarks</a>).<br>
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<h2>Precision of the Output Structures </h2>
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The Voronoi implementation guaranties, that the relative error of the
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coordinates of the output
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geometries is at most 64 machine epsilons (6
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bits of mantissa, for the IEEE-754 floating-point type), while on
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average it's slightly lower. This means, that the precision of the
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output
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geometries can be increased simply by using a floating-point type with
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the larger mantissa. The practical point of this statements is
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explained in the following table:<br>
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<table style="text-align: left; width: 100%;" border="1" cellpadding="2" cellspacing="2">
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<tbody>
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<tr>
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<td style="vertical-align: top;">Output Coordinate Type </td>
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<td style="vertical-align: top;">Output Coordinate Value </td>
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<td style="vertical-align: top;">Max Absolute Error </td>
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<td style="vertical-align: top;">Precise Value Range </td>
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</tr>
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<tr>
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<td style="vertical-align: top;">double (53 bit mantissa) </td>
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<td style="vertical-align: top;">1 </td>
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<td style="vertical-align: top;">2<sup>-53</sup> * 2<sup>6</sup>
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= 2<sup>-47</sup></td>
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<td style="vertical-align: top;">1 <20> 2<sup>-47</sup></td>
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</tr>
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<tr>
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<td style="vertical-align: top;">double (53 bit mantissa) </td>
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<td style="vertical-align: top;">2<sup>31</sup> </td>
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<td style="vertical-align: top;">2<sup>-53</sup> * 2<sup>31</sup>
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* 2<sup>6</sup> = 2<sup>-16</sup></td>
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<td style="vertical-align: top;">2<sup>31</sup> <20> 2<sup>-16</sup></td>
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</tr>
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<tr>
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<td style="vertical-align: top;">long double (64 bit
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mantissa)</td>
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<td style="vertical-align: top;">1 </td>
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<td style="vertical-align: top;">2<sup>-64</sup> * 2<sup>6</sup>
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= 2<sup>-58</sup></td>
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<td style="vertical-align: top;">1 <20> 2<sup>-58</sup></td>
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</tr>
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<tr>
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<td style="vertical-align: top;">long double (64 bit
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mantissa) </td>
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<td style="vertical-align: top;">2<sup>31</sup></td>
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<td style="vertical-align: top;">2<sup>-64</sup> * 2<sup>31</sup>
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* 2<sup>6</sup> = 2<sup>-27</sup></td>
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<td style="vertical-align: top;">2<sup>31</sup> <20> 2<sup>-27</sup></td>
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</tr>
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</tbody>
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</table>
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Detailed description of the absolute and relative errors evaluation can
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be found in the article: <a href="http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html">"What
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Every Computer Scientist Should Know About Floating-Point Arithmetic"</a><a href="http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html"></a>.<br>
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<br>
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During the finalization step the implementation unites the Voronoi
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vertices
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whose both
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coordinates are situated within the relative error range equal to 128
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machine epsilons and removes any Voronoi edges between those. This is
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the only case, that might cause differences between the algorithm
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output
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topology and theoretically precise one, and practically means the
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following: for the Voronoi diagram of a set of solid bodies inside the
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Solar System (radius 2<sup>42</sup> metres) and the long double (64 bit
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mantissa) output coordinate type the maximum absolute error within the
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Solar System rectangle will be equal to 2<sup>-64</sup> * 2<sup>42</sup>
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* 2<sup>6</sup> = 2<sup>-18</sup> metres; as the result, vertices with
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both coordinates that are within 2<sup>-18</sup> metres (8
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micrometres or the size of a bacteria) will be considered
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equal and united.<br>
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<h2>Simple Interface </h2>
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The <a href="../../../boost/polygon/voronoi.hpp">boost/polygon/</a><a href="../../../boost/polygon/voronoi.hpp">voronoi.hpp</a>
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library header defines the following static functions to integrate the
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Voronoi library
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functionality with the Boost.Polygon interfaces:<br>
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<br>
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<table style="text-align: left; width: 100%;" border="1" cellpadding="2" cellspacing="2">
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<tbody>
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<tr>
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<td style="font-family: Courier New,Courier,monospace;">template
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<typename Point, typename VB><br>
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size_t <span style="font-weight: bold;">insert</span>(const Point
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&point, VB *vb) </td>
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<td>Inserts a point into the Voronoi builder data structure.<br>
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Point type should model the point concept.<br>
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Returns index of the inserted site. </td>
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</tr>
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<tr>
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<td style="font-family: Courier New,Courier,monospace;">template
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<typename PointIterator, typename VB><br>
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void <span style="font-weight: bold;">insert</span>(PointIterator
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first, <br>
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PointIterator last,<br>
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VB
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*vb) </td>
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<td>Inserts an iterator range of points into the Voronoi
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builder data structure.<br>
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Corresponding point type should model the point concept. </td>
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</tr>
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<tr>
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<td style="font-family: Courier New,Courier,monospace;">template
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<typename Segment, typename VB><br>
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size_t <span style="font-weight: bold;">insert</span>(const Segment
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&segment, VB *vb) </td>
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<td>Inserts a segment into the Voronoi builder data
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structure.<br>
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Segment type should model the segment concept.<br>
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Returns index of the inserted site. </td>
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</tr>
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<tr>
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<td style="font-family: Courier New,Courier,monospace;">template
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<typename SegmentIterator, typename VB><br>
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void <span style="font-weight: bold;">insert</span>(SegmentIterator
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first,<br>
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SegmentIterator last,<br>
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VB
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*vb) </td>
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<td>Inserts an iterator range of segments into the Voronoi
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builder data structure.<br>
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Corresponding segment type should model the segment concept. </td>
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</tr>
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<tr>
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<td style="font-family: Courier New,Courier,monospace;">template
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<typename PointIterator, typename VD><br>
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void <span style="font-weight: bold;">construct_voronoi</span>(PointIterator
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first,<br>
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PointIterator last,<br>
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VD *vd) </td>
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<td>Constructs the Voronoi diagram of a set of points.<br>
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Corresponding point type should model the point concept.<br>
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Complexity: O(N * log N), memory usage: O(N), where N is the total number of input points.<br>
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</td>
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</tr>
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<tr>
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<td style="font-family: Courier New,Courier,monospace;">template
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<typename SegmentIterator, typename VD><br>
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void <span style="font-weight: bold;">construct_voronoi</span>(SegmentIterator
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first,<br>
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SegmentIterator last,<br>
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VD *vd) </td>
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<td>Constructs the Voronoi diagram of a set of segments.<br>
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Corresponding segment type should model the segment concept.<br>
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Complexity: O(N * log N), memory usage: O(N), where N is the total number of input segments.<br>
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</td>
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</tr>
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<tr>
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<td style="font-family: Courier New,Courier,monospace;">template
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<typename PointIterator,<br>
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typename
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SegmentIterator,<br>
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typename VD><br>
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void <span style="font-weight: bold;">construct_voronoi</span>(PointIterator
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p_first,<br>
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PointIterator p_last,<br>
|
|||
|
|
|
|||
|
|
SegmentIterator s_first,<br>
|
|||
|
|
|
|||
|
|
SegmentIterator s_last,<br>
|
|||
|
|
|
|||
|
|
VD *vd) </td>
|
|||
|
|
<td>Constructs the Voronoi
|
|||
|
|
diagram of a set of points and segments.<br>
|
|||
|
|
Corresponding point type should model the point concept.<br>
|
|||
|
|
Corresponding segment type should model the segment concept.<br>
|
|||
|
|
Complexity: O(N* log N), memory usage: O(N), where N is the total number of input points and segments.<br>
|
|||
|
|
</td>
|
|||
|
|
</tr>
|
|||
|
|
</tbody>
|
|||
|
|
</table>
|
|||
|
|
<br>
|
|||
|
|
The
|
|||
|
|
following two lines of code construct the Voronoi diagram of a set of
|
|||
|
|
points (as
|
|||
|
|
long as the corresponding input geometry type satisfies the
|
|||
|
|
Boost.Polygon <a href="gtl_design_overview.htm">concept model</a>):<br>
|
|||
|
|
<br>
|
|||
|
|
<span style="font-family: Courier New,Courier,monospace;">voronoi_diagram<double>
|
|||
|
|
vd;</span><br style="font-family: Courier New,Courier,monospace;">
|
|||
|
|
<span style="font-family: Courier New,Courier,monospace;">construct_voronoi(points.begin(),
|
|||
|
|
points.end(), &vd);</span><br>
|
|||
|
|
<br>
|
|||
|
|
The library provides the clear interfaces to associate the user data
|
|||
|
|
with the
|
|||
|
|
output geometries and efficiently traverse
|
|||
|
|
the
|
|||
|
|
Voronoi graph.
|
|||
|
|
More details on those topics are covered in the <a href="voronoi_basic_tutorial.htm">basic Voronoi tutorial</a>. Advanced
|
|||
|
|
usage of the library with the configuration of the coordinate
|
|||
|
|
types is explained in the <a href="voronoi_advanced_tutorial.htm">advanced
|
|||
|
|
Voronoi tutorial</a>.
|
|||
|
|
The library allows users to implement their own Voronoi diagram /
|
|||
|
|
Delaunay triangulation construction routines based on the <a href="voronoi_builder.htm">Voronoi builder API</a>.<br>
|
|||
|
|
<h2>No Third Party Dependencies </h2>
|
|||
|
|
The Voronoi extension of the Boost.Polygon library doesn't depend on
|
|||
|
|
any 3rd party code
|
|||
|
|
and contains single dependency on the Boost libraries:
|
|||
|
|
boost/cstdint.hpp.
|
|||
|
|
All the required multiprecision types and related functionality are
|
|||
|
|
encapsulated as
|
|||
|
|
part of the implementation. The library is fast to compile (3 public
|
|||
|
|
and 4 private heades), has strong cohesion between its components and
|
|||
|
|
is clearly modularized from the rest of the Boost.Polygon library, with
|
|||
|
|
the optional integration through the <a href="../../../boost/polygon/voronoi.hpp">voronoi.hpp</a> header.<br>
|
|||
|
|
<h2>Extensible for the User Provided Coordinate Types</h2>
|
|||
|
|
The implementation is coordinate type agnostic. As long
|
|||
|
|
as the user provided types satisfy the set of the requirements of the <a href="voronoi_builder.htm">Voronoi builder</a> coordinate type traits,
|
|||
|
|
no additional
|
|||
|
|
changes
|
|||
|
|
are needed
|
|||
|
|
neither to the algorithm, nor to the implementation of the predicates.
|
|||
|
|
For example, it's
|
|||
|
|
possible to
|
|||
|
|
construct the Voronoi diagram with the 256-bit integer input coordinate
|
|||
|
|
type
|
|||
|
|
and
|
|||
|
|
512-bit output floating-point type without making any changes to the
|
|||
|
|
library.<br>
|
|||
|
|
<h2>Future Development </h2>
|
|||
|
|
Below one may find the list of the main directions for the future
|
|||
|
|
development of the library.<br>
|
|||
|
|
The high-priority tasks that already have the approximate
|
|||
|
|
implementation plan
|
|||
|
|
are the following (some of those may be proposed as future GSoC
|
|||
|
|
projects):<br>
|
|||
|
|
<ul>
|
|||
|
|
<li>Implement the Delaunay triangulation data structure.<br>
|
|||
|
|
Note: only data structure needs to be implemented that properly
|
|||
|
|
processes events provided by the Voronoi builder.</li>
|
|||
|
|
<li>Implement the medial axis transform data structure.<br>
|
|||
|
|
Note: in general case the Voronoi diagram has completely the same
|
|||
|
|
geometry
|
|||
|
|
as the medial axis (they are 100% equal), however for many applications
|
|||
|
|
user is not interested in the Voronoi edges inside the hole regions.
|
|||
|
|
The main point
|
|||
|
|
of this data structure is to automatically filter the Voronoi edges
|
|||
|
|
that
|
|||
|
|
belong to those areas.</li>
|
|||
|
|
<li>The Voronoi
|
|||
|
|
diagram data structure can be used to find the K nearest neighbors of
|
|||
|
|
the N
|
|||
|
|
sites in O(N*K*log(K) + N*log(N)) time. The return value would be a
|
|||
|
|
list of the k nearest neighbors for each site. </li>
|
|||
|
|
<li>Use the r-tree data structure built on top of the
|
|||
|
|
bounding rectangles around the Voronoi cells to answer the nearest
|
|||
|
|
neighbor queries in log(N) time, where N is the number of the Voronoi
|
|||
|
|
cells.<br>
|
|||
|
|
Note: there should be r-tree data structure available soon as part of
|
|||
|
|
the Boost libraries.</li>
|
|||
|
|
<li>Expose O(N) interface to retrieve the convex hull of a set
|
|||
|
|
of
|
|||
|
|
points and segments from the Voronoi builder, once the Voronoi diagram
|
|||
|
|
is
|
|||
|
|
constructed.</li>
|
|||
|
|
<li>Provide serialization utilities for the Voronoi diagram
|
|||
|
|
data structure. </li>
|
|||
|
|
</ul>
|
|||
|
|
High-priority tasks to be considered:<br>
|
|||
|
|
<ul>
|
|||
|
|
<li>Drop the restriction on the non-intersecting input
|
|||
|
|
geometries.</li>
|
|||
|
|
<li>Integrate the Voronoi diagram data structure with the
|
|||
|
|
BGL (Boost
|
|||
|
|
Graph Library).</li>
|
|||
|
|
<li>Support the other types of distance metrics.</li>
|
|||
|
|
<li>Construction of the constrained Delaunay triangulation.</li>
|
|||
|
|
<li>Support of the circular input geometries.</li>
|
|||
|
|
</ul>
|
|||
|
|
Based on the community suggestions priorities may be changed.<br>
|
|||
|
|
<h2>Theoretical Research </h2>
|
|||
|
|
The Voronoi library
|
|||
|
|
was developed as part of the Google Summer of Code 2010. The
|
|||
|
|
library was actively maintained for the last three years and involved
|
|||
|
|
the
|
|||
|
|
strong mathematical research in the field of algorithms, data
|
|||
|
|
structures,
|
|||
|
|
relative error arithmetic and numerical robustness. Nowadays one can
|
|||
|
|
often read a scientific paper, that contains non-practical
|
|||
|
|
theoretical
|
|||
|
|
results or implementation with
|
|||
|
|
benchmarks nobody else can reproduce. The opposite story is with
|
|||
|
|
the Voronoi library, that contains complete
|
|||
|
|
implementation of
|
|||
|
|
the Voronoi diagram construction algorithm and
|
|||
|
|
benchmarks one may run on
|
|||
|
|
his/her PC. Upon the community request, more documentation on the
|
|||
|
|
theoretical aspects of the implementation will be published.
|
|||
|
|
The
|
|||
|
|
authors would like to acknowledge the Steven Fortune's article <span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-weight: bold;"></span></span>"<a href="http://dl.acm.org/citation.cfm?id=10549">A Sweepline algorithm
|
|||
|
|
for Voronoi diagrams</a>", that covers fundamental ideas of the
|
|||
|
|
current implementation. </td>
|
|||
|
|
</tr>
|
|||
|
|
<tr>
|
|||
|
|
<td style="background-color: rgb(238, 238, 238);" nowrap="1"> </td>
|
|||
|
|
<td style="padding-left: 10px; padding-right: 10px; padding-bottom: 10px;" valign="top" width="100%">
|
|||
|
|
<table class="docinfo" id="table2" frame="void" rules="none">
|
|||
|
|
<colgroup> <col class="docinfo-name"><col class="docinfo-content"> </colgroup> <tbody valign="top">
|
|||
|
|
<tr>
|
|||
|
|
<th class="docinfo-name">Copyright:</th>
|
|||
|
|
<td>Copyright <20> Andrii Sydorchuk 2010-2013.</td>
|
|||
|
|
</tr>
|
|||
|
|
<tr class="field">
|
|||
|
|
<th class="docinfo-name">License:</th>
|
|||
|
|
<td class="field-body">Distributed under the Boost Software
|
|||
|
|
License, Version 1.0. (See accompanying file <tt class="literal"><span class="pre">LICENSE_1_0.txt</span></tt> or copy at <a class="reference" target="_top" href="http://www.boost.org/LICENSE_1_0.txt">
|
|||
|
|
http://www.boost.org/LICENSE_1_0.txt</a>)</td>
|
|||
|
|
</tr>
|
|||
|
|
</tbody>
|
|||
|
|
</table>
|
|||
|
|
</td>
|
|||
|
|
</tr>
|
|||
|
|
</tbody>
|
|||
|
|
</table>
|
|||
|
|
|
|||
|
|
</body></html>
|