matrix2quaternion.m

% MATRIX2QUATERNION - Homogeneous matrix to quaternion
%
% Converts 4x4 homogeneous rotation matrix to quaternion
%
% Usage: Q = matrix2quaternion(T)
%
% Argument:   T - 4x4 Homogeneous transformation matrix
% Returns:    Q - a quaternion in the form [w, xi, yj, zk]
%
% See Also QUATERNION2MATRIX

% Copyright (c) 2008 Peter Kovesi
% School of Computer Science & Software Engineering
% The University of Western Australia
% pk at csse uwa edu au
% http://www.csse.uwa.edu.au/
% 
% Permission is hereby granted, free of charge, to any person obtaining a copy
% of this software and associated documentation files (the "Software"), to deal
% in the Software without restriction, subject to the following conditions:
% 
% The above copyright notice and this permission notice shall be included in 
% all copies or substantial portions of the Software.
%
% The Software is provided "as is", without warranty of any kind.

function Q = matrix2quaternion(T)

    % This code follows the implementation suggested by Hartley and Zisserman    
    R = T(1:3, 1:3);   % Extract rotation part of T
    
    % Find rotation axis as the eigenvector having unit eigenvalue
    % Solve (R-I)v = 0;
    [v,d] = eig(R-eye(3));
    
    % The following code assumes the eigenvalues returned are not necessarily
    % sorted by size. This may be overcautious on my part.
    d = diag(abs(d));   % Extract eigenvalues
    [s, ind] = sort(d); % Find index of smallest one
    if d(ind(1)) > 0.001   % Hopefully it is close to 0
        warning('Rotation matrix is dubious');
    end
    
    axis = v(:,ind(1)); % Extract appropriate eigenvector
    
    if abs(norm(axis) - 1) > .0001     % Debug
        warning('non unit rotation axis');
    end
    
    % Now determine the rotation angle
    twocostheta = trace(R)-1;
    twosinthetav = [R(3,2)-R(2,3), R(1,3)-R(3,1), R(2,1)-R(1,2)]';
    twosintheta = axis'*twosinthetav;
    
    theta = atan2(twosintheta, twocostheta);
    
    Q = [cos(theta/2); axis*sin(theta/2)];