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// Copyright (c) 2005 INRIA Sophia-Antipolis (France). // All rights reserved. // // This file is part of CGAL (www.cgal.org). // // $URL: https://github.com/CGAL/cgal/blob/v6.1/Principal_component_analysis/include/CGAL/linear_least_squares_fitting_circles_2.h $ // $Id: include/CGAL/linear_least_squares_fitting_circles_2.h b26b07a1242 $ // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial // // Author(s) : Pierre Alliez and Sylvain Pion and Ankit Gupta #ifndef CGAL_LINEAR_LEAST_SQUARES_FITTING_CIRCLES_2_H #define CGAL_LINEAR_LEAST_SQUARES_FITTING_CIRCLES_2_H #include #include #include #include #include #include #include #include namespace CGAL { namespace internal { // Fits a line to a 2D circle 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 DiagonalizeTraits > 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::Circle_2*,// used for indirection const K&, // kernel const CGAL::Dimension_tag<2>& tag, const DiagonalizeTraits&) { // types typedef typename K::FT FT; typedef typename K::Line_2 Line; typedef typename K::Vector_2 Vector; typedef typename K::Circle_2 Circle; typedef typename CGAL::Linear_algebraCd LA; typedef typename LA::Matrix Matrix; // precondition: at least one element in the container. CGAL_precondition(first != beyond); // ::::::::::DISK::::::::::::::: // 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 circles and their combined mass FT mass = 0.0; typename DiagonalizeTraits::Covariance_matrix covariance = {{ 0., 0., 0. }}; // assemble 2nd order moment about the origin. FT temp[4] = {0.25, 0.0, 0.0, 0.25}; Matrix moment = init_matrix(2,temp); // Matrix moment = Matrix(2,true,PI); for(InputIterator it = first; it != beyond; it++) { // Now for each circle, construct the 2nd order moment about the origin. // assemble the transformation matrix. const Circle& t = *it; // defined for convenience. // FT example = CGAL::to_double(t[0].x()); FT radius = CGAL::approximate_sqrt(t.squared_radius()); FT delta[4] = {radius, 0.0, 0.0, radius}; Matrix transformation = init_matrix(2,delta); FT area = t.squared_radius(); CGAL_assertion(!CGAL::is_zero(area)); // Find the 2nd order moment for the circle wrt to the origin by an affine transformation. // Transform the standard 2nd order moment using the transformation matrix transformation = area * transformation * moment * LA::transpose(transformation); // Translate the 2nd order moment to the center of the circle. FT x0 = t.center().x(); FT y0 = t.center().y(); // and add to covariance matrix covariance[0] += transformation[0][0] + area * CGAL::square(x0); covariance[1] += transformation[0][1] + area * x0*y0; covariance[2] += transformation[1][1] + area * CGAL::square(y0); mass += area; } CGAL_assertion_msg (mass != FT(0), "Can't compute PCA of null measure."); // Translate the 2nd order moment calculated about the origin to // the center of mass to get the covariance. covariance[0] -= mass * (CGAL::square(c.x())); covariance[1] -= mass * (c.x() * c.y()); covariance[2] -= mass * (CGAL::square(c.y())); // solve for eigenvalues and eigenvectors. // eigen values are sorted in ascending order, // eigen vectors are sorted in accordance. typename DiagonalizeTraits::Vector eigen_values = {{ 0. , 0. }}; typename DiagonalizeTraits::Matrix eigen_vectors = {{ 0., 0., 0. }}; DiagonalizeTraits::diagonalize_selfadjoint_covariance_matrix (covariance, eigen_values, eigen_vectors); // check unicity and build fitting line accordingly if(eigen_values[0] != eigen_values[1]) { // regular case line = Line(c, Vector (eigen_vectors[2],eigen_vectors[3])); return (FT)1.0 - eigen_values[0] / eigen_values[1]; } 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(FT(1), FT(0))); return (FT)0.0; } } // end linear_least_squares_fitting_2 for circle set with 2D tag template < typename InputIterator, typename K, typename DiagonalizeTraits > 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::Circle_2*,// used for indirection const K&, // kernel const CGAL::Dimension_tag<1>& tag, const DiagonalizeTraits& ) { // types typedef typename K::FT FT; typedef typename K::Line_2 Line; typedef typename K::Vector_2 Vector; typedef typename K::Circle_2 Circle; typedef typename CGAL::Linear_algebraCd 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 circles and their combined mass FT mass = 0.0; typename DiagonalizeTraits::Covariance_matrix covariance = {{ 0., 0., 0. }}; // assemble 2nd order moment about the origin. FT temp[4] = {1.0, 0.0, 0.0, 1.0}; Matrix moment = init_matrix(2,temp); for(InputIterator it = first; it != beyond; it++) { // Now for each circle, construct the 2nd order moment about the origin. // assemble the transformation matrix. const Circle& t = *it; // defined for convenience. // FT example = CGAL::to_double(t[0].x()); FT radius = CGAL::approximate_sqrt(t.squared_radius()); FT delta[4] = {radius, 0.0, 0.0, radius}; Matrix transformation = init_matrix(2,delta); FT length = 2 * radius; CGAL_assertion(!CGAL::is_zero(length)); // Find the 2nd order moment for the circle wrt to the origin by an affine transformation. // Transform the standard 2nd order moment using the transformation matrix transformation = FT(0.5) * length * transformation * moment * LA::transpose(transformation); // Translate the 2nd order moment to the center of the circle. FT x0 = t.center().x(); FT y0 = t.center().y(); // and add to covariance matrix covariance[0] += transformation[0][0] + length * CGAL::square(x0); covariance[1] += transformation[0][1] + length * x0*y0; covariance[2] += transformation[1][1] + length * CGAL::square(y0); mass += length; } CGAL_assertion_msg (mass != FT(0), "Can't compute PCA of null measure."); // Translate the 2nd order moment calculated about the origin to // the center of mass to get the covariance. covariance[0] -= mass * (CGAL::square(c.x())); covariance[1] -= mass * (c.x() * c.y()); covariance[2] -= mass * (CGAL::square(c.y())); // solve for eigenvalues and eigenvectors. // eigen values are sorted in ascending order, // eigen vectors are sorted in accordance. typename DiagonalizeTraits::Vector eigen_values = {{ 0. , 0. }}; typename DiagonalizeTraits::Matrix eigen_vectors = {{ 0., 0., 0. }}; DiagonalizeTraits::diagonalize_selfadjoint_covariance_matrix (covariance, eigen_values, eigen_vectors); // check unicity and build fitting line accordingly if(eigen_values[1] != eigen_values[0]) { // regular case line = Line(c, Vector(eigen_vectors[2],eigen_vectors[3])); return (FT)1.0 - eigen_values[0] / eigen_values[1]; } 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(FT(1), FT(0))); return (FT)0.0; } } // end linear_least_squares_fitting_2 for circle set with 1D tag } // end namespace internal } //namespace CGAL #endif // CGAL_LINEAR_LEAST_SQUARES_FITTING_CIRCLES_2_H