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Metric rectification from orthogonal lines

perform metric rectification of an image. Given at least 5 pairs of image of orthogonal lines, the script finds the image of the dual conic to circular points and computes the homography that brings it back to its canonical form

Luca Magri
Politecnico di Milano
2020

clear;
close all;
img = imread('images/img1.JPG');

figure;
imshow(img);
numConstraints = 10; %>=5
hold all;
fprintf('Draw 5 pairs of orthogonal segments\n');
count = 1;
A = zeros(numConstraints,6);

% select pairs of orthogonal segments
while (count <=numConstraints)
    figure(gcf);
    title(['Draw ', num2str(numConstraints),' pairs of orthogonal segments: step ',num2str(count) ]);
    col = 'rgbcmykwrgbcmykw';
    segment1 = drawline('Color',col(count));
    segment2 = drawline('Color',col(count));

    l = segToLine(segment1.Position);
    m = segToLine(segment2.Position);

    % each pair of orthogonal lines gives rise to a constraint on the image
    % of the dual conic of principal points imDCCP
    % [l(1)*m(1),0.5*(l(1)*m(2)+l(2)*m(1)),l(2)*m(2),0.5*(l(1)*m(3)+l(3)*m(1))
    %  0.5*(l(2)*m(3)+l(3)*m(2)), l(3)*m(3)]*v = 0
    % store the constraints in a matrix A
    A(count,:) = [l(1)*m(1),0.5*(l(1)*m(2)+l(2)*m(1)),l(2)*m(2),...
        0.5*(l(1)*m(3)+l(3)*m(1)),  0.5*(l(2)*m(3)+l(3)*m(2)), l(3)*m(3)];
    count = count+1;
end

Draw 5 pairs of orthogonal segments

Compute the imDCCP image of the dual conic to circular points

[~,~,v] = svd(A); %
sol = v(:,end); %sol = (a,b,c,d,e,f)  [a,b/2,d/2; b/2,c,e/2; d/2 e/2 f];
imDCCP = [sol(1)  , sol(2)/2, sol(4)/2;...
    sol(2)/2, sol(3)  , sol(5)/2;...
    sol(4)/2, sol(5)/2  sol(6)];

compute the rectifying homography

[U,D,V] = svd(imDCCP);
D(3,3) = 1;
A = U*sqrt(D);

C = [eye(2),zeros(2,1);zeros(1,3)];
min(norm(A*C*A' - imDCCP),norm(A*C*A' + imDCCP))

H = inv(A); % rectifying homography
min(norm(H*imDCCP*H'./norm(H*imDCCP*H') - C./norm(C)),norm(H*imDCCP*H'./norm(H*imDCCP*H') + C./norm(C)))

ans =

   1.8299e-08


ans =

   1.8299e-08

tform = projective2d(H');
J = imwarp(img,tform);

figure;
imshow(J);

Published with MATLAB® R2021a