% RQ3 RQ decomposition of 3x3 matrix
%
% Usage: [R,Q] = rq3(A)
%
% Argument: A - 3 x 3 matrix
% Returns: R - Upper triangular 3 x 3 matrix
% Q - 3 x 3 orthonormal rotation matrix
% Such that R*Q = A
%
% The signs of the rows and columns of R and Q are chosen so that the diagonal
% elements of R are +ve.
%
% See also: DECOMPOSECAMERA
% Follows algorithm given by Hartley and Zisserman 2nd Ed. A4.1 p 579
% Copyright (c) 2010 Peter Kovesi
% Centre for Exploration Targeting
% School of Earth and Environment
% The University of Western Australia
% peter.kovesi at 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.
%
% October 2010
% February 2014 Incorporated modifications suggested by Mathias Rothermel to
% avoid potential division by zero problems
function [R,Q] = rq3(A)
if ~all(size(A)==[3 3])
error('A must be 3x3');
end
eps = 1e-10;
% Find rotation Qx to set A(3,2) to 0
A(3,3) = A(3,3) + eps;
c = -A(3,3)/sqrt(A(3,3)^2+A(3,2)^2);
s = A(3,2)/sqrt(A(3,3)^2+A(3,2)^2);
Qx = [1 0 0; 0 c -s; 0 s c];
R = A*Qx;
% Find rotation Qy to set A(3,1) to 0
R(3,3) = R(3,3) + eps;
c = R(3,3)/sqrt(R(3,3)^2+R(3,1)^2);
s = R(3,1)/sqrt(R(3,3)^2+R(3,1)^2);
Qy = [c 0 s; 0 1 0;-s 0 c];
R = R*Qy;
% Find rotation Qz to set A(2,1) to 0
R(2,2) = R(2,2) + eps;
c = -R(2,2)/sqrt(R(2,2)^2+R(2,1)^2);
s = R(2,1)/sqrt(R(2,2)^2+R(2,1)^2);
Qz = [c -s 0; s c 0; 0 0 1];
R = R*Qz;
Q = Qz'*Qy'*Qx';
% Adjust R and Q so that the diagonal elements of R are +ve
for n = 1:3
if R(n,n) < 0
R(:,n) = -R(:,n);
Q(n,:) = -Q(n,:);
end
end