468 lines
16 KiB
C
468 lines
16 KiB
C
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// Copyright (c) 1997-2001 ETH Zurich (Switzerland).
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// All rights reserved.
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//
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// This file is part of CGAL (www.cgal.org).
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// You can redistribute it and/or modify it under the terms of the GNU
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// General Public License as published by the Free Software Foundation,
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// either version 3 of the License, or (at your option) any later version.
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//
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// Licensees holding a valid commercial license may use this file in
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// accordance with the commercial license agreement provided with the software.
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//
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// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
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// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
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//
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// $URL$
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// $Id$
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// SPDX-License-Identifier: GPL-3.0+
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//
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//
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// Author(s) : Kaspar Fischer <fischerk@inf.ethz.ch>
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#ifndef CGAL_CGAL_APPROX_MIN_ELL_D_H
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#define CGAL_CGAL_APPROX_MIN_ELL_D_H
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#include <CGAL/license/Bounding_volumes.h>
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#include <cstdlib>
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#include <cmath>
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#include <vector>
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#include <iostream>
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#include <CGAL/Simple_cartesian.h>
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#include <CGAL/Approximate_min_ellipsoid_d/Khachiyan_approximation.h>
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namespace CGAL {
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template<class Traits>
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class Approximate_min_ellipsoid_d
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// An instance of class Approximate_min_ellipsoid_d represents an
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// eps-approximation of the smallest enclosing ellipsoid of the
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// point set P in R^d, that is, an ellipsoid E satisfying
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//
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// (i) E contains P,
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//
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// (ii) the volume of E is at most (1+eps) times larger than the
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// volume of the smallest enclosing ellipsoid of P.
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{
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public: // types:
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typedef typename Traits::FT FT;
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typedef typename Traits::Point Point;
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typedef typename Traits::Cartesian_const_iterator Cartesian_const_iterator;
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private:
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typedef std::vector<Point> Point_list;
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typedef typename Traits::Cartesian_const_iterator C_it;
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protected: // member variables:
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Traits tco; // traits class object
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const Point_list P; // the input points in R^d
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int d; // dimension of the input points
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const double eps; // the desired epsilon
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double e_eps; // (see below)
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// We obtain our eps-approximating ellipsoid by embedding the input
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// points P into R^{d+1} by mapping p to (p,1). Then we compute in
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// R^{d+1} an (1+e_eps)-approximating centrally symmetric ellipsoid E for
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// the embedded points from which the desired (1+eps)-approximating
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// ellipsoid of the original points can be obtained by projection. The
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// following variable E represents the e_eps-approximating centrally
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// symmetric ellipsoid in R^{d+1}:
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Khachiyan_approximation<true,Traits> *E;
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// When the input points do not affinely span the whole space
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// (i.e., if dim(aff(P)) < d), then the smallest enclosing
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// ellipsoid of P has no volume in R^d and so the points are
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// called "degnerate" (see is_degenerate()) below.
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// As discussed below (before (*)), the centrally symmetric ellipsoid
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// E':= sqrt{(1+a_eps)(d+1)} E contains (under exact arithmetic) the
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// embedded points. (Here, a_eps is the value obtained by
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// achieved_epsilon().) Denote by M the defining matrix of E; we
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// then have
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//
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// p^T M p <= alpha,
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//
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// for all p in E and alpha = (1+a_eps)(d+1). Since this is equivalent
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// to
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//
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// p^T M/alpha p <= 1, (***)
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//
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// we see that E' = sqrt{alpha} E has M/alpha as its defining
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// matrix. Consequently, when we return M in the routines
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// defining_matrix(), defining_vector(), and defining_scalar() below,
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// these numbers need to be scaled (by the user) with the factor
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// 1/alpha. (We do not perform the scaling ourselves because we cannot do
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// it exactly in double arithmetic.)
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public: // construction & destruction:
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template<typename InputIterator>
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Approximate_min_ellipsoid_d(double eps,
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InputIterator first,InputIterator last,
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const Traits& traits = Traits())
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// Given a range [first,last) of n points P, constructs an
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// (1+eps)-approximation of the smallest enclosing ellipsoid of P.
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: tco(traits), P(first,last), eps(eps),
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has_center(false), has_axes(false)
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{
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CGAL_APPEL_LOG("appel",
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"Entering Approximate_min_ellpsoid_d." << std::endl);
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// fetch ambient dimension:
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d = tco.dimension(P[0]);
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CGAL_APPEL_ASSERT(d >= 2 || eps >= 0.0);
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// The ellipsoid E produced by Khachiyan's algorithm has the
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// property that E':= sqrt{(1+e_eps) (d+1)} E contains all
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// embedded points e_P and has volume bounded by
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//
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// vol(E') <= (1+e_eps)^{(d+1)/2} vol(E_emb*), (*)
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//
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// where E_emb* is the smallest centrally symmetric enclosing
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// ellipsoid of the embedded points e_P. By requiring that (1+eps)
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// <= (1+e_eps)^{(d+1)/2}, we get
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//
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// e_eps <= (1+eps)^{2/(d+1)} - 1 (**)
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//
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// and with this e_eps, we have vol(E') <= (1+eps) vol(E_emb*).
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// An argument by Khachiyan (see "Rounding of polytopes in the
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// real number model of computation", eq. (4.1)) then guarantees
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// that the intersection E_int of E' with the hyperplane { (x,y)
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// in R^{d+1} | y = 1} is an ellipsoid satisfying vol(E_int) <=
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// (1+eps) vol(E*), where E* is the smallest enclosing ellipsoid
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// of the original points P. According to (**) we thus set e_eps
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// to (a lower bound on) (1+eps)^{2/(d+1)} - 1:
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FPU_CW_t old = FPU_get_and_set_cw(CGAL_FE_TOWARDZERO); // round to zero
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e_eps = std::exp(2.0/(d+1)*std::log(1.0+eps))-1.0;
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FPU_set_cw(old); // restore
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// rounding mode
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// Find e (1+e_eps)-approximation for the embedded points. This
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// only works when the points affinely span R^{d+1}.
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E = new Khachiyan_approximation<true,Traits>(d, static_cast<int>(P.size()),tco);
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const bool is_deg = !E->add(P.begin(),P.end(),e_eps);
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// debugging:
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CGAL_APPEL_ASSERT(is_deg == E->is_degenerate());
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CGAL_APPEL_LOG("appel",
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" Input points are " << (is_deg? "" : "not ") <<
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"degnerate." << std::endl);
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if (is_deg)
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find_lower_dimensional_approximation();
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CGAL_APPEL_LOG("appel",
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"Leaving Approximate_min_ellipsoid_d." << std::endl);
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}
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~Approximate_min_ellipsoid_d()
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{
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// dispose of approximation:
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if (E != static_cast<Khachiyan_approximation<true,Traits> *>(0))
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delete E;
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}
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public: // access:
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unsigned int number_of_points() const
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// Returns the number of points, i.e., |P|.
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{
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return P.size();
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}
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bool is_empty() const
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// Returns true iff the approximate ellipsoid is empty (which
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// implies degeneracy). This is the case iff the number of input
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// points was zero at construction time.
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{
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return P.size() == 0;
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}
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private: // access:
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bool is_degenerate() const
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// Returns true iff the approximate ellipsoid is degenerate, i.e.,
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// iff the dimension of the affine hull of S doesn't match the
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// dimension of the ambient space.
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{
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return E->is_degenerate();
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}
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public: // access:
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// Here's how the routines defining_matrix(), defining_vector(),
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// and defining_scalar() are implemented. From (***) we know that
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// the ellipsoid E' = sqrt{alpha} E
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//
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// (a) encloses all embedded points (p,1), p in P,
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// (b) has defining matrix M/alpha, i.e.,
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//
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// E' = { x | x^T M/alpha x <= 1 },
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//
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// where alpha = (1+a_eps)(d+1) with a_eps the return value
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// of achieved_epsilon().
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//
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// The ellipsoid E* we actuallly want is the intersection of E' with
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// the hyperplane { (y,z) in R^{d+1} | y = 1}. Writing
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//
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// [ M' m ] [ y ]
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// M = [ m^T nu ] and x = [ 1 ] (*****)
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//
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// we thus obtain
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//
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// x^T M/alpha x = y^T y alpha/M + 2/alpha y^Tm + nu/alpha.
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//
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// It follows
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//
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// E* = { y | y^T M'/alpha y + 2/alpha y^Tm + (nu/alpha-1) <= 0 }. (****)
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//
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// This is what the routines defining_matrix(), defining_vector(),
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// and defining_scalar() implement.
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bool is_full_dimensional() const
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// Returns !is_degenerate().
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{
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return !is_degenerate();
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}
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FT defining_matrix(int i,int j) const
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// Returns the entry M(i,j) of the symmetric matrix M in the
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// representation
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//
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// E* = { x | x^T M x + x^T m + mu <= 0 }
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//
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// of the computed approximation. More precisely, the routine does not
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// return M(i,j) but the number (1+achieved_epsilon())*(d+1)*M(i,j).
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//
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// Precondition: !is_degenerate() && 0<=i<d && 0<=j<d
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{
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CGAL_APPEL_ASSERT(!is_degenerate() && 0<=i && i<d && 0<=j && j<d);
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return E->matrix(i,j);
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}
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FT defining_vector(int i) const
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// Returns the entry m(i) of the vector m in the representation
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//
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// E* = { x | x^T M x + x^T m + mu <= 0 }
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//
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// of the computed approximation. More precisely, the routine does not
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// return m(i) but the number (1+achieved_epsilon())*(d+1)*m(i).
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//
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// Precondition: !is_degenerate() && 0<=i<d
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{
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CGAL_APPEL_ASSERT(!is_degenerate() && 0<=i && i<d);
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return FT(2)*E->matrix(d,i); // Note: if FT is double, the
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// multiplication by 2.0 is exact.
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}
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FT defining_scalar() const
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// Returns the number mu in the representation
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//
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// E* = { x | x^T M x + x^T m + mu <= 0}
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//
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// of the computed approximation. More precisely, the routine does not
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// return mu but the number (1+achieved_epsilon())*(d+1)*(mu+1).
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//
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// Precondition: !is_degenerate()
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{
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CGAL_APPEL_ASSERT(!is_degenerate());
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return E->matrix(d,d);
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}
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double achieved_epsilon() const
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// Returns the approximation ratio; more precisely, this returns a
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// number r such that the computed ellipsoid is a (1+r)-approximation
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// of MEL(P).
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//
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// Precondition: !is_degenerate()
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{
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CGAL_APPEL_ASSERT(!is_degenerate());
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// From (*) we known that Khachian's algorithm produces a
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// centrally-symmetric ellipsoid in R^{d+1} fulfilling
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//
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// vol(E') <= (1+k_eps)^{(d+1)/2} vol(E_emb*),
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//
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// where k_eps = E->exact_epsilon(). The projecting argument
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// mentioned after (**) says that these approximation ratio also
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// applies to the projected (d-dimensional) ellipsoids, and so the
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// actual approximation ratio we obtain is
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//
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// ratio:= (1+k_eps)^{(d+1)/2}.
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//
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// So all we need to do is compute the smallest (let's say: a small)
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// double number larger or equal to ratio.
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//
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// Todo: make the calculation below more stable, numerically?
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const double k_eps = E->exact_epsilon();
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FPU_CW_t old = FPU_get_and_set_cw(CGAL_FE_UPWARD); // round up
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const double sum = 1.0 + k_eps;
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double tmp = sum;
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for (int i=0; i<d; ++i)
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tmp *= sum;
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const double eps = std::sqrt(tmp)-1.0;
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FPU_set_cw(old); // restore
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CGAL_APPEL_ASSERT(eps >= 0.0);
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return eps;
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}
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Traits traits() const
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{
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return tco;
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}
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int dimension() const
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{
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return d;
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}
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public: // miscellaneous:
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bool is_valid(bool verbose) const
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// Returns true if and only if the computed ellipsoid is indeed an
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// approximate ellipsoid, that is ... Todo.
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{
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return E->is_valid(verbose);
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}
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public: // miscellaneous 2D/3D support:
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typedef std::vector<double>::const_iterator Center_coordinate_iterator;
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typedef std::vector<double>::const_iterator Axes_lengths_iterator;
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typedef std::vector<double>::const_iterator
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Axes_direction_coordinate_iterator;
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Center_coordinate_iterator center_cartesian_begin()
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// Returns a STL random-access iterator pointing to the first of the d
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// Cartesian coordinates of the computed ellipsoid's center. The center
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// described in this way is a floating-point approximation to the
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// ellipsoid's exact center; no guarantee is given w.r.t. the involved
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// relative error.
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//
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// Precondition: !is_degenerate()
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{
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CGAL_APPEL_ASSERT(!is_degenerate());
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if (!has_center)
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compute_center();
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return center_.begin();
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}
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||
|
|
|
||
|
|
Center_coordinate_iterator center_cartesian_end()
|
||
|
|
// Returns the past-the-end iterator corresponding to
|
||
|
|
// center_cartesian_begin().
|
||
|
|
//
|
||
|
|
// Precondition: !is_degenerate()
|
||
|
|
{
|
||
|
|
CGAL_APPEL_ASSERT(!is_degenerate());
|
||
|
|
if (!has_center)
|
||
|
|
compute_center();
|
||
|
|
|
||
|
|
return center_.end();
|
||
|
|
}
|
||
|
|
|
||
|
|
Axes_lengths_iterator axes_lengths_begin()
|
||
|
|
// Returns a STL random-access iterator to the first of the d lengths of
|
||
|
|
// the computed ellipsoid's axes. The d lengths are floating-point
|
||
|
|
// approximations to the exact axes-lengths of the computed ellipsoid; no
|
||
|
|
// guarantee is given w.r.t. the involved relative error. (See also method
|
||
|
|
// axes_direction_cartesian_begin().) The elements of the iterator are
|
||
|
|
// sorted descending.
|
||
|
|
//
|
||
|
|
// Precondition: !is_degenerate() && (d==2 || d==3)
|
||
|
|
{
|
||
|
|
CGAL_APPEL_ASSERT(!is_degenerate() && (d==2 || d==3));
|
||
|
|
if (!has_axes)
|
||
|
|
compute_axes_2_3();
|
||
|
|
|
||
|
|
return lengths_.begin();
|
||
|
|
}
|
||
|
|
|
||
|
|
Axes_lengths_iterator axes_lengths_end()
|
||
|
|
// Returns the past-the-end iterator corresponding to
|
||
|
|
// axes_lengths_begin().
|
||
|
|
//
|
||
|
|
// Precondition: !is_degenerate() && (d==2 || d==3)
|
||
|
|
{
|
||
|
|
CGAL_APPEL_ASSERT(!is_degenerate() && (d==2 || d==3));
|
||
|
|
if (!has_axes)
|
||
|
|
compute_axes_2_3();
|
||
|
|
|
||
|
|
return lengths_.end();
|
||
|
|
}
|
||
|
|
|
||
|
|
Axes_direction_coordinate_iterator axis_direction_cartesian_begin(int i)
|
||
|
|
// Returns a STL random-access iterator pointing to the first of the d
|
||
|
|
// Cartesian coordinates of the computed ellipsoid's i-th axis direction
|
||
|
|
// (i.e., unit vector in direction of the ellipsoid's i-th axis). The
|
||
|
|
// direction described by this iterator is a floating-point approximation
|
||
|
|
// to the exact axis direction of the computed ellipsoid; no guarantee is
|
||
|
|
// given w.r.t. the involved relative error. An approximation to the
|
||
|
|
// length of axis i is given by the i-th entry of axes_lengths_begin().
|
||
|
|
//
|
||
|
|
// Precondition: !is_degenerate() && (d==2 || d==3) && (0 <= i < d)
|
||
|
|
{
|
||
|
|
CGAL_APPEL_ASSERT(!is_degenerate() && (d==2 || d==3) &&
|
||
|
|
0 <= i && i < d);
|
||
|
|
if (!has_axes)
|
||
|
|
compute_axes_2_3();
|
||
|
|
|
||
|
|
return directions_[i].begin();
|
||
|
|
}
|
||
|
|
|
||
|
|
Axes_direction_coordinate_iterator axis_direction_cartesian_end(int i)
|
||
|
|
// Returns the past-the-end iterator corresponding to
|
||
|
|
// axis_direction_cartesian_begin().
|
||
|
|
//
|
||
|
|
// Precondition: !is_degenerate() && (d==2 || d==3) && (0 <= i < d)
|
||
|
|
{
|
||
|
|
CGAL_APPEL_ASSERT(!is_degenerate() && (d==2 || d==3) &&
|
||
|
|
0 <= i && i < d);
|
||
|
|
if (!has_axes)
|
||
|
|
compute_axes_2_3();
|
||
|
|
|
||
|
|
return directions_[i].end();
|
||
|
|
}
|
||
|
|
|
||
|
|
public: // internal members for 2D/3D axis/center computation:
|
||
|
|
|
||
|
|
bool has_center, has_axes; // true iff the center or axes-directions and
|
||
|
|
// -lengths, respectively, have already been
|
||
|
|
// computed
|
||
|
|
std::vector<double> center_;
|
||
|
|
std::vector<double> lengths_;
|
||
|
|
std::vector< std::vector<double> > directions_;
|
||
|
|
std::vector<double> mi; // contains M^{-1} (see (*****)
|
||
|
|
// above) iff has_center is true;
|
||
|
|
// mi[i+d*j] is entry (i,j) of
|
||
|
|
// M^{-1}
|
||
|
|
|
||
|
|
void compute_center();
|
||
|
|
void compute_axes_2_3();
|
||
|
|
void compute_axes_2(const double alpha, const double factor);
|
||
|
|
void compute_axes_3(const double alpha, const double factor);
|
||
|
|
|
||
|
|
public: // "debugging" routines:
|
||
|
|
|
||
|
|
void write_eps(const std::string& name);
|
||
|
|
// Writes the and the point set P to an EPS file. Returned is the
|
||
|
|
// id under which the file was stored (filename 'id.eps').
|
||
|
|
//
|
||
|
|
// Precondition: d==2 && !is_degenerate().
|
||
|
|
|
||
|
|
private: // internal routines:
|
||
|
|
|
||
|
|
void find_lower_dimensional_approximation(); // (does nothing right now)
|
||
|
|
};
|
||
|
|
|
||
|
|
}
|
||
|
|
|
||
|
|
#include <CGAL/Approximate_min_ellipsoid_d/Approximate_min_ellipsoid_d_impl.h>
|
||
|
|
|
||
|
|
#endif // CGAL_CGAL_APPROX_MIN_ELL_D_H
|