sl.h

// -*- c++ -*-

// Copyright (C) 2005,2009 Tom Drummond (twd20@cam.ac.uk),
// Gerhard Reitmayr (gr281@cam.ac.uk)
//
// This file is part of the TooN Library.  This library is free
// software; 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 2, or (at your option)
// any later version.

// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
// GNU General Public License for more details.

// You should have received a copy of the GNU General Public License along
// with this library; see the file COPYING.  If not, write to the Free
// Software Foundation, 59 Temple Place - Suite 330, Boston, MA 02111-1307,
// USA.

// As a special exception, you may use this file as part of a free software
// library without restriction.  Specifically, if other files instantiate
// templates or use macros or inline functions from this file, or you compile
// this file and link it with other files to produce an executable, this
// file does not by itself cause the resulting executable to be covered by
// the GNU General Public License.  This exception does not however
// invalidate any other reasons why the executable file might be covered by
// the GNU General Public License.

#ifndef TOON_INCLUDE_SL_H
#define TOON_INCLUDE_SL_H

#include <TooN/TooN.h>
#include <TooN/helpers.h>
#include <TooN/gaussian_elimination.h>
#include <TooN/determinant.h>
#include <cassert>

namespace TooN {

template <int N, typename P> class SL;
template <int N, typename P> std::istream & operator>>(std::istream &, SL<N, P> &);

/// represents an element from the group SL(n), the NxN matrices M with det(M) = 1.
/// This can be used to conveniently estimate homographies on n-1 dimentional spaces.
/// The implementation uses the matrix exponential function @ref exp for
/// exponentiation from an element in the Lie algebra and LU to compute an inverse.
/// 
/// The Lie algebra are the NxN matrices M with trace(M) = 0. The N*N-1 generators used
/// to represent this vector space are the following:
/// - diag(...,1,-1,...), n-1 along the diagonal
/// - symmetric generators for every pair of off-diagonal elements
/// - anti-symmetric generators for every pair of off-diagonal elements
/// This choice represents the fact that SL(n) can be interpreted as the product
/// of all symmetric matrices with det() = 1 times SO(n).
/// @ingroup gTransforms
template <int N, typename Precision = DefaultPrecision>
class SL {
    friend std::istream & operator>> <N,Precision>(std::istream &, SL &);
public:
    static const int size = N;          ///< size of the matrices represented by SL<N>
    static const int dim = N*N - 1;     ///< dimension of the vector space represented by SL<N>

    /// default constructor, creates identity element
    SL() : my_matrix(Identity) {}

    /// exp constructor, creates element through exponentiation of Lie algebra vector. see @ref SL::exp.
    template <int S, typename P, typename B>
    SL( const Vector<S,P,B> & v ) { *this = exp(v); }

    /// copy constructor from a matrix, coerces matrix to be of determinant = 1
    template <int R, int C, typename P, typename A>
    SL(const Matrix<R,C,P,A>& M) : my_matrix(M) { coerce(); }

    /// returns the represented matrix
    const Matrix<N,N,Precision> & get_matrix() const { return my_matrix; }
    /// returns the inverse using LU
    SL inverse() const { return SL(*this, Invert()); }

    /// multiplies to SLs together by multiplying the underlying matrices
    template <typename P>
    SL<N,typename Internal::MultiplyType<Precision, P>::type> operator*( const SL<N,P> & rhs) const { return SL<N,typename Internal::MultiplyType<Precision, P>::type>(*this, rhs); }

    /// right multiplies this SL with another one
    template <typename P>
    SL operator*=( const SL<N,P> & rhs) { *this = *this*rhs; return *this; }

    /// exponentiates a vector in the Lie algebra to compute the corresponding element
    /// @arg v a vector of dimension SL::dim
    template <int S, typename P, typename B>
    static inline SL exp( const Vector<S,P,B> &);

    inline Vector<N*N-1, Precision> ln() const ;

    /// returns one generator of the group. see SL for a detailed description of 
    /// the generators used.
    /// @arg i number of the generator between 0 and SL::dim -1 inclusive
    static inline Matrix<N,N,Precision> generator(int);

private:
    struct Invert {};
    SL( const SL & from, struct Invert ) {
        const Matrix<N> id = Identity;
        my_matrix = gaussian_elimination(from.my_matrix, id);
    }
    SL( const SL & a, const SL & b) : my_matrix(a.get_matrix() * b.get_matrix()) {}

    void coerce(){
        using std::abs;
        Precision det = determinant(my_matrix);
        assert(abs(det) > 0);
        using std::pow;
        my_matrix /= pow(det, 1.0/N);
    }

    /// these constants indicate which parts of the parameter vector 
    /// map to which generators
    ///{
    static const int COUNT_DIAG = N - 1;
    static const int COUNT_SYMM = (dim - COUNT_DIAG)/2;
    static const int COUNT_ASYMM = COUNT_SYMM;
    static const int DIAG_LIMIT = COUNT_DIAG;
    static const int SYMM_LIMIT = COUNT_SYMM + DIAG_LIMIT;
    ///}

    Matrix<N,N,Precision> my_matrix;
};

template <int N, typename Precision>
template <int S, typename P, typename B>
inline SL<N, Precision> SL<N, Precision>::exp( const Vector<S,P,B> & v){
    SizeMismatch<S,dim>::test(v.size(), dim);
    Matrix<N,N,Precision> t(Zeros);
    for(int i = 0; i < dim; ++i)
        t += generator(i) * v[i];
    SL<N, Precision> result;
    result.my_matrix = TooN::exp(t);
    return result;
}

template <int N, typename Precision>
inline Vector<N*N-1, Precision> SL<N, Precision>::ln() const {
    const Matrix<N> l = TooN::log(my_matrix);
    Vector<SL<N,Precision>::dim, Precision> v;
    Precision last = 0;
    for(int i = 0; i < DIAG_LIMIT; ++i){    // diagonal elements
        v[i] = l(i,i) + last;
        last = l(i,i);
    }
    for(int i = DIAG_LIMIT, row = 0, col = 1; i < SYMM_LIMIT; ++i) {    // calculate symmetric and antisymmetric in one go
        // do the right thing here to calculate the correct indices !
        v[i] = (l(row, col) + l(col, row))*0.5;
        v[i+COUNT_SYMM] = (-l(row, col) + l(col, row))*0.5;
        ++col;
        if( col == N ){
            ++row;
            col = row+1;
        }
    }
    return v;
}

template <int N, typename Precision>
inline Matrix<N,N,Precision> SL<N, Precision>::generator(int i){
    assert( i > -1 && i < dim );
    Matrix<N,N,Precision> result(Zeros);
    if(i < DIAG_LIMIT) {                // first ones are the diagonal ones
        result(i,i) = 1;
        result(i+1,i+1) = -1;
    } else if(i < SYMM_LIMIT){          // then the symmetric ones
        int row = 0, col = i - DIAG_LIMIT + 1;
        while(col > (N - row - 1)){
            col -= (N - row - 1); 
            ++row;
        }
        col += row;
        result(row, col) = result(col, row) = 1;
    } else {                            // finally the antisymmetric ones
        int row = 0, col = i - SYMM_LIMIT + 1;
        while(col > N - row - 1){
            col -= N - row - 1; 
            ++row;
        }
        col += row;
        result(row, col) = -1;
        result(col, row) = 1;
    }
    return result;
}

template <int S, typename PV, typename B, int N, typename P>
Vector<N, typename Internal::MultiplyType<P, PV>::type> operator*( const SL<N, P> & lhs, const Vector<S,PV,B> & rhs ){
    return lhs.get_matrix() * rhs;
}

template <int S, typename PV, typename B, int N, typename P>
Vector<N, typename Internal::MultiplyType<PV, P>::type> operator*( const Vector<S,PV,B> & lhs, const SL<N,P> & rhs ){
    return lhs * rhs.get_matrix();
}

template<int R, int C, typename PM, typename A, int N, typename P> inline
Matrix<N, C, typename Internal::MultiplyType<P, PM>::type> operator*(const SL<N,P>& lhs, const Matrix<R, C, PM, A>& rhs){
    return lhs.get_matrix() * rhs;
}

template<int R, int C, typename PM, typename A, int N, typename P> inline
Matrix<R, N, typename Internal::MultiplyType<PM, P>::type> operator*(const Matrix<R, C, PM, A>& lhs, const SL<N,P>& rhs){
    return lhs * rhs.get_matrix();
}

template <int N, typename P>
std::ostream & operator<<(std::ostream & out, const SL<N, P> & h){
    out << h.get_matrix();
    return out;
}

template <int N, typename P>
std::istream & operator>>(std::istream & in, SL<N, P> & h){
    in >> h.my_matrix;
    h.coerce();
    return in;
}

};

#endif