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1 | <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> |
2 | <html><head> | |
3 | <meta http-equiv="Content-Language" content="en-us"> | |
4 | <meta http-equiv="Content-Type" content="text/html; charset=windows-1252"><title>Voronoi Main</title> | |
5 | <meta http-equiv="content-type" content="text/html; charset=utf-8"> | |
6 | <meta http-equiv="content-type" content="text/html; charset=utf-8"> | |
7 | <meta http-equiv="content-type" content="text/html; charset=utf-8"> | |
8 | <meta http-equiv="content-type" content="text/html; charset=utf-8"></head><body> | |
9 | <table style="margin: 0pt; padding: 0pt; width: 100%;" border="0" cellpadding="0" cellspacing="0"> | |
10 | <tbody> | |
11 | <tr> | |
12 | <td style="background-color: rgb(238, 238, 238);" nowrap="1" valign="top"> | |
13 | <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> | |
14 | <div style="margin: 5px;"> | |
15 | <h3 class="navbar">Contents</h3> | |
16 | <ul> | |
17 | <li><a href="index.htm">Boost.Polygon Main Page</a></li> | |
18 | <li><a href="gtl_design_overview.htm">Design Overview</a></li> | |
19 | <li><a href="gtl_isotropy.htm">Isotropy</a></li> | |
20 | <li><a href="gtl_coordinate_concept.htm">Coordinate Concept</a></li> | |
21 | <li><a href="gtl_interval_concept.htm">Interval Concept</a></li> | |
22 | <li><a href="gtl_point_concept.htm">Point Concept</a></li> | |
23 | <li><a href="gtl_segment_concept.htm">Segment Concept</a></li> | |
24 | <li><a href="gtl_rectangle_concept.htm">Rectangle Concept</a></li> | |
25 | <li><a href="gtl_polygon_90_concept.htm">Polygon 90 Concept</a></li> | |
26 | <li><a href="gtl_polygon_90_with_holes_concept.htm">Polygon 90 With Holes Concept</a></li> | |
27 | <li><a href="gtl_polygon_45_concept.htm">Polygon 45 Concept</a></li> | |
28 | <li><a href="gtl_polygon_45_with_holes_concept.htm">Polygon 45 With Holes Concept</a></li> | |
29 | <li><a href="gtl_polygon_concept.htm">Polygon Concept</a></li> | |
30 | <li><a href="gtl_polygon_with_holes_concept.htm">Polygon With Holes Concept</a></li> | |
31 | <li><a href="gtl_polygon_90_set_concept.htm">Polygon 90 Set Concept</a></li> | |
32 | <li><a href="gtl_polygon_45_set_concept.htm">Polygon 45 Set Concept</a></li> | |
33 | <li><a href="gtl_polygon_set_concept.htm">Polygon Set Concept</a></li> | |
34 | <li><a href="gtl_connectivity_extraction_90.htm">Connectivity Extraction 90</a></li> | |
35 | <li><a href="gtl_connectivity_extraction_45.htm">Connectivity Extraction 45</a></li> | |
36 | <li><a href="gtl_connectivity_extraction.htm">Connectivity Extraction</a></li> | |
37 | <li><a href="gtl_property_merge_90.htm">Property Merge 90</a></li> | |
38 | <li><a href="gtl_property_merge_45.htm">Property Merge 45</a></li> | |
39 | <li><a href="gtl_property_merge.htm">Property Merge</a></li> | |
40 | <li>Voronoi Main Page </li> | |
41 | <li><a href="voronoi_benchmark.htm">Voronoi Benchmark</a></li> | |
42 | <li><a href="voronoi_builder.htm">Voronoi Builder</a> </li> | |
43 | <li><a href="voronoi_diagram.htm">Voronoi Diagram</a></li> | |
44 | </ul> | |
45 | <h3 class="navbar">Other Resources</h3> | |
46 | <ul> | |
47 | <li><a href="GTL_boostcon2009.pdf">GTL Boostcon 2009 Paper</a></li> | |
48 | <li><a href="GTL_boostcon_draft03.pdf">GTL Boostcon 2009 Presentation</a></li> | |
49 | <li><a href="analysis.htm">Performance Analysis</a></li> | |
50 | <li><a href="gtl_tutorial.htm">Layout Versus Schematic Tutorial</a></li> | |
51 | <li><a href="gtl_minkowski_tutorial.htm">Minkowski Sum Tutorial</a></li> | |
52 | <li><a href="voronoi_basic_tutorial.htm">Voronoi Basic Tutorial</a></li> | |
53 | <li><a href="voronoi_advanced_tutorial.htm">Voronoi Advanced Tutorial</a></li> | |
54 | </ul> | |
55 | </div> | |
56 | <h3 class="navbar">Polygon Sponsor</h3> | |
57 | <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> | |
58 | </td> | |
59 | <td style="padding-left: 10px; padding-right: 10px; padding-bottom: 10px;" valign="top" width="100%"><!-- End Header --> <br> | |
60 | <h1>THE BOOST POLYGON VORONOI EXTENSIONS</h1> | |
61 | <img style="width: 900px; height: 300px;" alt="" src="images/voronoi3.png"><br> | |
62 | The Voronoi extensions of the Boost Polygon library provide functionality to | |
63 | construct a <a href="voronoi_diagram.htm">Voronoi diagram</a> of a set of points | |
64 | and linear segments in 2D space with the following set of limitations:<br> | |
65 | <ul> | |
66 | <li>Coordinates of the input points and endpoints of the input segments should have integral type. The int32 data type is supported by the default implementation. Support for the other data types (e.g. int64) could be achieved through the configuration of the coordinate type traits (<a href="voronoi_advanced_tutorial.htm">Voronoi Advanced tutorial</a>).</li> | |
67 | <li>Input points and segments should not overlap except their endpoints. This means that input point should not lie inside the input segment and input segments should not intersect except their endpoints.</li> | |
68 | </ul> | |
69 | While the first restriction is permanent (it | |
70 | allows to give the exact warranties about the output precision and | |
71 | algorithm execution flow), | |
72 | the second one may be resolved using the Boost.Polygon <a href="gtl_segment_concept.htm">segment utils</a>. | |
73 | The strong sides of the | |
74 | library and the main benefits comparing to the other implementations are | |
75 | discussed in the following paragraphs.<br> | |
76 | <h2>Fully Functional with Segments</h2> | |
77 | There are just a few implementations of the Voronoi diagram | |
78 | construction | |
79 | algorithm that can | |
80 | handle input data sets that contain linear segments, even considering | |
81 | the commercial | |
82 | libraries. | |
83 | Support of the | |
84 | segments allows to discretize any input geometry (sampled | |
85 | floating-point coordinates can be scaled and snapped to the integer | |
86 | grid): circle, ellipse, | |
87 | parabola. This functionality allows to compute | |
88 | the medial axis transform of the arbitrary set of input geometries, | |
89 | with direct applications in the computer vision | |
90 | projects. | |
91 | <h2>Robustness and Efficiency</h2> | |
92 | Robustness issues can be divided onto the two main categories: memory | |
93 | management | |
94 | issues and numeric stability issues. The implementation avoids the | |
95 | first type of the issues using pure STL data structures, thus there is | |
96 | no | |
97 | presence of the new operator in the code. The second category of | |
98 | the problems is | |
99 | resolved using the multiprecision geometric | |
100 | predicates. | |
101 | Even for the commercial libraries, usage of such predicates | |
102 | results in a vast performance slowdown. The library implementation | |
103 | overcomes this by avoiding the multiprecision | |
104 | computations in the 95% of the cases by | |
105 | using the efficient, floating-point based predicates. Such predicates | |
106 | don't | |
107 | produce the correct result always, however the library embeds the | |
108 | relative | |
109 | error arithmetic apparatus to identify such situations and switch | |
110 | to the | |
111 | higher precision predicates when appropriate. As the result, the | |
112 | implementation has a solid performance comparing to the other known | |
113 | libraries (more details in the <a href="voronoi_benchmark.htm">benchmarks</a>).<br> | |
114 | <h2>Precision of the Output Structures </h2> | |
115 | The implementation guaranties, that the relative error of the | |
116 | coordinates of the output | |
117 | geometries is at most 64 machine epsilons (6 | |
118 | bits of mantissa, for the IEEE-754 floating-point type), while on | |
119 | average it's slightly lower. This means, that the precision of the | |
120 | output | |
121 | geometries can be increased simply by using a floating-point type with | |
122 | the larger mantissa. The practical point of this statements is | |
123 | explained in the following table:<br> | |
124 | <table style="text-align: left; width: 100%;" border="1" cellpadding="2" cellspacing="2"> | |
125 | <tbody> | |
126 | <tr> | |
127 | <td style="vertical-align: top;">Output Coordinate Type </td> | |
128 | <td style="vertical-align: top;">Output Coordinate Value </td> | |
129 | <td style="vertical-align: top;">Max Absolute Error </td> | |
130 | <td style="vertical-align: top;">Precise Value Range </td> | |
131 | </tr> | |
132 | <tr> | |
133 | <td style="vertical-align: top;">double (53 bit mantissa) </td> | |
134 | <td style="vertical-align: top;">1 </td> | |
135 | <td style="vertical-align: top;">2<sup>-53</sup> * 2<sup>6</sup> | |
136 | = 2<sup>-47</sup></td> | |
137 |