% RANSACFITFUNDMATRIX - fits fundamental matrix using RANSAC
%
% Usage: [F, inliers] = ransacfitfundmatrix(x1, x2, t)
%
% Arguments:
% x1 - 2xN or 3xN set of homogeneous points. If the data is
% 2xN it is assumed the homogeneous scale factor is 1.
% x2 - 2xN or 3xN set of homogeneous points such that x1<->x2.
% t - The distance threshold between data point and the model
% used to decide whether a point is an inlier or not.
% Note that point coordinates are normalised to that their
% mean distance from the origin is sqrt(2). The value of
% t should be set relative to this, say in the range
% 0.001 - 0.01
%
% Note that it is assumed that the matching of x1 and x2 are putative and it
% is expected that a percentage of matches will be wrong.
%
% Returns:
% F - The 3x3 fundamental matrix such that x2'Fx1 = 0.
% inliers - An array of indices of the elements of x1, x2 that were
% the inliers for the best model.
%
% See Also: RANSAC, FUNDMATRIX
% 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 [F, inliers] = ransacfitfundmatrix(x1, x2, t, feedback)
if ~all(size(x1)==size(x2))
error('Data sets x1 and x2 must have the same dimension');
end
if nargin == 3
feedback = 0;
end
[rows,npts] = size(x1);
if rows~=2 & rows~=3
error('x1 and x2 must have 2 or 3 rows');
end
if rows == 2 % Pad data with homogeneous scale factor of 1
x1 = [x1; ones(1,npts)];
x2 = [x2; ones(1,npts)];
end
% Normalise each set of points so that the origin is at centroid and
% mean distance from origin is sqrt(2). normalise2dpts also ensures the
% scale parameter is 1. Note that 'fundmatrix' will also call
% 'normalise2dpts' but the code in 'ransac' that calls the distance
% function will not - so it is best that we normalise beforehand.
[x1, T1] = normalise2dpts(x1);
[x2, T2] = normalise2dpts(x2);
s = 8; % Number of points needed to fit a fundamental matrix. Note that
% only 7 are needed but the function 'fundmatrix' only
% implements the 8-point solution.
fittingfn = @fundmatrix;
distfn = @funddist;
degenfn = @isdegenerate;
% x1 and x2 are 'stacked' to create a 6xN array for ransac
[F, inliers] = ransac([x1; x2], fittingfn, distfn, degenfn, s, t, feedback);
% Now do a final least squares fit on the data points considered to
% be inliers.
F = fundmatrix(x1(:,inliers), x2(:,inliers));
% Denormalise
F = T2'*F*T1;
%--------------------------------------------------------------------------
% 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)
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
x2tFx1 = zeros(1,length(x1));
for n = 1:length(x1)
x2tFx1(n) = x2(:,n)'*F{k}*x1(:,n);
end
Fx1 = F{k}*x1;
Ftx2 = F{k}'*x2;
% Evaluate distances
d = x2tFx1.^2 ./ ...
(Fx1(1,:).^2 + Fx1(2,:).^2 + Ftx2(1,:).^2 + Ftx2(2,:).^2);
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
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);
bestInliers = find(abs(d) < t); % Indices of inlying points
bestF = F; % Copy F directly to bestF
end
%----------------------------------------------------------------------
% (Degenerate!) function to determine if a set of matched points will result
% in a degeneracy in the calculation of a fundamental matrix as needed by
% RANSAC. This function assumes this cannot happen...
function r = isdegenerate(x)
r = 0;