1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
|
// Copyright (c) 2005 INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
// You can redistribute it and/or modify it under the terms of the GNU
// General Public License as published by the Free Software Foundation,
// either version 3 of the License, or (at your option) any later version.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL: svn+ssh://scm.gforge.inria.fr/svn/cgal/branches/next/Principal_component_analysis/include/CGAL/linear_least_squares_fitting_segments_2.h $
// $Id: linear_least_squares_fitting_segments_2.h 67117 2012-01-13 18:14:48Z lrineau $
//
// Author(s) : Pierre Alliez and Sylvain Pion and Ankit Gupta
#ifndef CGAL_LINEAR_LEAST_SQUARES_FITTING_SEGMENTS_2_H
#define CGAL_LINEAR_LEAST_SQUARES_FITTING_SEGMENTS_2_H
#include <CGAL/basic.h>
#include <CGAL/Object.h>
#include <CGAL/centroid.h>
#include <CGAL/eigen_2.h>
#include <CGAL/eigen.h>
#include <CGAL/Linear_algebraCd.h>
#include <CGAL/PCA_util.h>
#include <iterator>
#include <vector>
#include <cmath>
namespace CGAL {
namespace internal {
// Fits a line to a 2D segment set.
// Returns a fitting quality (1 - lambda_min/lambda_max):
// 1 is best (zero variance orthogonally to the fitting line);
// 0 is worst (isotropic case, returns a line with horizontal
// direction by default)
template < typename InputIterator, typename K >
typename K::FT
linear_least_squares_fitting_2(InputIterator first,
InputIterator beyond,
typename K::Line_2& line, // best fit line
typename K::Point_2& c, // centroid
const typename K::Segment_2*,// used for indirection
const K&, // kernel
const CGAL::Dimension_tag<1>& tag = CGAL::Dimension_tag<1>())
{
// types
typedef typename K::FT FT;
typedef typename K::Line_2 Line;
typedef typename K::Point_2 Point;
typedef typename K::Vector_2 Vector;
typedef typename K::Segment_2 Segment;
typedef typename CGAL::Linear_algebraCd<FT> LA;
typedef typename LA::Matrix Matrix;
// precondition: at least one element in the container.
CGAL_precondition(first != beyond);
// compute centroid
c = centroid(first,beyond,K(),tag);
// assemble covariance matrix as a semi-definite matrix.
// Matrix numbering:
// 0
// 1 2
//Final combined covariance matrix for all segments and their combined mass
FT mass = 0.0;
FT covariance[3] = {0.0,0.0,0.0};
// assemble 2nd order moment about the origin.
FT temp[4] = {1.0, 0.5, 0.5, 1.0};
Matrix moment = (1.0/3.0) * init_matrix<K>(2,temp);
for(InputIterator it = first;
it != beyond;
it++)
{
// Now for each segment, construct the 2nd order moment about the origin.
// assemble the transformation matrix.
const Segment& t = *it;
// defined for convenience.
// FT example = CGAL::to_double(t[0].x());
FT delta[4] = {t[0].x(), t[1].x(),
t[0].y(), t[1].y()};
Matrix transformation = init_matrix<K>(2,delta);
FT length = std::sqrt(t.squared_length());
CGAL_assertion(length != 0.0);
// Find the 2nd order moment for the segment wrt to the origin by an affine transformation.
// Transform the standard 2nd order moment using the transformation matrix
transformation = length * transformation * moment * LA::transpose(transformation);
// add to covariance matrix
covariance[0] += transformation[0][0];
covariance[1] += transformation[0][1];
covariance[2] += transformation[1][1];
mass += length;
}
// Translate the 2nd order moment calculated about the origin to
// the center of mass to get the covariance.
covariance[0] += mass * (-1.0 * c.x() * c.x());
covariance[1] += mass * (-1.0 * c.x() * c.y());
covariance[2] += mass * (-1.0 * c.y() * c.y());
// solve for eigenvalues and eigenvectors.
// eigen values are sorted in descending order,
// eigen vectors are sorted in accordance.
std::pair<FT,FT> eigen_values;
std::pair<Vector,Vector> eigen_vectors;
// internal::eigen_symmetric_2<K>(covariance, eigen_vectors, eigen_values);
FT eigen_vectors1[4];
FT eigen_values1[2];
eigen_symmetric<FT>(covariance,2, eigen_vectors1, eigen_values1);
eigen_values = std::make_pair(eigen_values1[0],eigen_values1[1]);
eigen_vectors = std::make_pair(Vector(eigen_vectors1[0],eigen_vectors1[1]),Vector(eigen_vectors1[2],eigen_vectors1[3]));
// check unicity and build fitting line accordingly
if(eigen_values.first != eigen_values.second)
{
// regular case
line = Line(c, eigen_vectors.first);
return (FT)1.0 - eigen_values.second / eigen_values.first;
}
else
{
// isotropic case (infinite number of directions)
// by default: assemble a line that goes through
// the centroid and with a default horizontal vector.
line = Line(c, Vector(1.0, 0.0));
return (FT)0.0;
}
} // end linear_least_squares_fitting_2 for segment set with 1D tag
template < typename InputIterator, typename K >
typename K::FT
linear_least_squares_fitting_2(InputIterator first,
InputIterator beyond,
typename K::Line_2& line, // best fit line
typename K::Point_2& c, // centroid
const typename K::Segment_2*,// used for indirection
const K& k, // kernel
const CGAL::Dimension_tag<0>& tag)
{
// types
typedef typename K::Point_2 Point;
typedef typename K::Segment_2 Segment;
// precondition: at least one element in the container.
CGAL_precondition(first != beyond);
std::list<Point> points;
for(InputIterator it = first;
it != beyond;
it++)
{
const Segment& s = *it;
points.push_back(s[0]);
points.push_back(s[1]);
}
return linear_least_squares_fitting_2(points.begin(),points.end(),line,c,k,(Point*)NULL,tag);
} // end linear_least_squares_fitting_2 for segment set with 1D tag
} // end namespace internal
} //namespace CGAL
#endif // CGAL_LINEAR_LEAST_SQUARES_FITTING_SEGMENTS_2_H
|
