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fmatrix.m
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fmatrix.m
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%FMATRIX Estimate fundamental matrix
%
% F = FMATRIX(P1, P2, OPTIONS) is the fundamental matrix (3x3) that
% relates two sets of corresponding points P1 (2xN) and P2 (2xN) from
% two different camera views.
%
% Notes::
% - The points must be corresponding, no outlier rejection is performed.
% - Contains a RANSAC driver, which means it can be passed to ransac().
% - F is a rank 2 matrix, that is, it is singular.
%
% Reference::
% Hartley and Zisserman,
% 'Multiple View Geometry in Computer Vision',
% page 270.
%
% Author::
% Based on fundamental matrix code by
% Peter Kovesi,
% School of Computer Science & Software Engineering,
% The University of Western Australia,
% http://www.csse.uwa.edu.au/,
%
% See also RANSAC, HOMOGRAPHY, EPILINE, EPIDIST.
function [F,resid] = fmatrix(p1, p2)
% RANSAC integration
if isstruct(p1)
F = ransac_driver(p1);
return;
end
if numcols(p1) < 7
error('must be at least 7 corresponding points');
end
% get data from passed arrays into homogeneous p1 and p2
%
% if 6xN assume that data is conditioned
X = p1;
if numrows(X) == 6
p1 = X(1:3,:);
p2 = X(4:6,:);
C1 = eye(3,3);
C2 = eye(3,3);
else
% data is not conditioned
if numrows(X) == 4
p1 = X(1:2,:);
p2 = X(3:4,:);
elseif numrows(p1) == 2,
if nargin < 2,
error('must pass uv1 and uv2');
end
p1 = X;
if numcols(p1) ~= numcols(p2),
error('must have same number of points in each set');
end
if numrows(p1) ~= numrows(p2),
error('p1 and p2 must have same number of rows')
end
end
% make data homogeneous
p1 = e2h(p1);
p2 = e2h(p2);
% and condition it
C1 = vgg_conditioner_from_pts(p1);
C2 = vgg_conditioner_from_pts(p2);
p1 = vgg_condition_2d(p1, C1);
p2 = vgg_condition_2d(p2, C2);
end
if numcols(p1) == 7,
% special case of 7 points
Fvgg = vgg_F_from_7pts_2img(p1, p2);
if isempty(Fvgg)
F = [];
else
% Store the (potentially) 3 solutions in a cell array
Nsolutions = size(Fvgg, 3);
for n = 1:Nsolutions
F{n} = Fvgg(:,:,n);
end
end
resid = 0;
return;
else
% normal case
x1 = p1(1,:)';
y1 = p1(2,:)';
x2 = p2(1,:)';
y2 = p2(2,:)';
% linear estimate
A = [x1.*x2 y1.*x2 x2 x1.*y2 y1.*y2 y2 x1 y1 ones(size(x1))];
[U,S,V] = svd(A);
f = V(:,end);
F = reshape(f, 3, 3)';
% enforce the rank 2 constraint
[U,S,V] = svd(F);
S(3,3) = 0;
F = U * S * V';
% check the residuals
d = fdist(F, p1, p2);
resid = max(d);
if nargout < 2,
fprintf('maximum residual %.4g\n', resid);
end
% decondition the result
F = C2' * F * C1;
end
end
%----------------------------------------------------------------------------------
% out = fmatrix(ransac)
%
% ransac.cmd string what operation to perform
% 'size'
% 'condition'
% 'decondition'
% 'valid'
% 'estimate'
% 'error'
% ransac.debug logical display what's going on
% ransac.X 6xN data to work on
% ransac.t 1x1 threshold
% ransac.theta 3x3 estimated quantity to test
% ransac.misc cell private data for deconditioning
%
% out.s 1x1 sample size
% out.X 6xN conditioned data
% out.misc cell private data for conditioning
% out.inlier 1xM list of inliers
% out.valid logical if data is valid for estimation
% out.theta 3x3 estimated quantity
%----------------------------------------------------------------------------------
function out = ransac_driver(ransac)
cmd = ransac.cmd;
if ransac.debug
fprintf('RANSAC command <%s>\n', cmd);
end
switch cmd
case 'size'
% return sample size
% 7 is technically possible but results are not so good...
out.s = 8;
case 'condition'
if numrows(ransac.X) == 4
p1 = ransac.X(1:2,:);
p2 = ransac.X(3:4,:);
p1 = e2h(p1);
p2 = e2h(p2);
elseif numrows(ransac.X) == 6
p1 = ransac.X(1:3,:);
p2 = ransac.X(3:6,:);
end
% condition the point data
C1 = vgg_conditioner_from_pts(p1);
C2 = vgg_conditioner_from_pts(p2);
p1 = vgg_condition_2d(p1, C1);
p2 = vgg_condition_2d(p2, C2);
out.X = [p1; p2];
out.misc = {C1, C2};
case 'decondition'
F = ransac.theta;
misc = ransac.misc;
C1 = misc{1}; C2 = misc{2};
out.theta = C2' * F * C1;
case 'valid'
out.valid = true;
case 'error'
% [bestInliers, bestF] = funddist(F, x, t);
[out.inliers, out.theta] = funddist(ransac.theta, ransac.X, ransac.t);
case 'estimate'
[out.theta, out.resid] = fmatrix(ransac.X);
otherwise
error('bad RANSAC command')
end
end
%--------------------------------------------------------------------------
% Function to evaluate the first order approximation of the geometric error
% (Sampson distance) of the fit of a fundamental matrix with respect to a
% set of matched points as needed by RANSAC. See: Hartley and Zisserman,
% 'Multiple View Geometry in Computer Vision', page 270.
%
% Note that this code allows for F being a cell array of fundamental matrices of
% which we have to pick the best one. (A 7 point solution can return up to 3
% solutions)
% Copyright (c) 2004-2005 Peter Kovesi
% School of Computer Science & Software Engineering
% The University of Western Australia
% 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.
% February 2004 Original version
% August 2005 Distance error function changed to match changes in RANSAC
function [bestInliers, bestF] = funddist(F, x, t);
x1 = x(1:3,:); % Extract x1 and x2 from x
x2 = x(4:6,:);
if iscell(F) % We have several solutions each of which must be tested
nF = length(F); % Number of solutions to test
bestF = F{1}; % Initial allocation of best solution
ninliers = 0; % Number of inliers
for k = 1:nF
d = fdist(F{k}, x1, x2);
inliers = find(abs(d) < t); % Indices of inlying points
if length(inliers) > ninliers % Record best solution
ninliers = length(inliers);
bestF = F{k};
bestInliers = inliers;
end
end
else % We just have one solution
d = fdist(F, x1, x2);
bestInliers = find(abs(d) < t); % Indices of inlying points
bestF = F; % Copy F directly to bestF
end
end
function d = fdist(F, x1, x2)
x2tFx1 = zeros(1,length(x1));
for n = 1:length(x1)
x2tFx1(n) = x2(:,n)'*F*x1(:,n);
end
Fx1 = F*x1;
Ftx2 = F'*x2;
% Evaluate distances
d = x2tFx1.^2 ./ ...
(Fx1(1,:).^2 + Fx1(2,:).^2 + Ftx2(1,:).^2 + Ftx2(2,:).^2);
end