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copied!<p>What was missing from the code above, was that the camera matrices also need to be transformed by the normalizing transformation. Below is an example that works. Notice also the difference in structure of the normalizing transformations. In the question's example the scaling factor is put in the last element. Below it's multiplied with all the elements, such that the last element is 1.</p> <pre><code> function testTriangulationDavid() close all clear all P = [ -1.81066 0.00000 0.00000 0.00000; 0.00000 2.41421 0.00000 0.00000; 0.00000 0.00000 1.00000 1.50000 ]; Q = [ -0.36118 0.00000 0.00000 0.00000 0.00000 2.00875 1.33916 0.00000 0.00000 -0.55470 0.83205 1.80278 ]; % homogenous 3D coordinates Pts = [ -0.2 -0.2 -0.2 -0.2 0.2 0.2 0.2 0.2; -0.2 -0.2 0.2 0.2 -0.2 -0.2 0.2 0.2; -0.2 0.2 -0.2 0.2 -0.2 0.2 -0.2 0.2; 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ]; numPts = length(Pts(1,:)); % project points PPtsHom = P*Pts; QPtsHom = Q*Pts; % normalize for homogenous scaling for i = 1:3 PPtsHom(i, :) = PPtsHom(i, :) ./ PPtsHom(3, :); QPtsHom(i, :) = QPtsHom(i, :) ./ QPtsHom(3, :); end % 2D cartesian coordinates PPtsCart = PPtsHom(1:2,:); QPtsCart = QPtsHom(1:2,:); % compute normalizing transform % calc centroid centroidPPtsCart = mean(PPtsCart,2); centroidQPtsCart = mean(QPtsCart,2); % calc mean distance to centroid normsPPtsCart = zeros(1, numPts); normsQPtsCart = zeros(1, numPts); for i = 1:numPts normsPPtsCart(1,i) = norm(PPtsCart(:,i) - centroidPPtsCart); normsQPtsCart(1,i) = norm(QPtsCart(:,i) - centroidQPtsCart); end mdcPPtsCart = mean(normsPPtsCart); mdcQPtsCart = mean(normsQPtsCart); % setup transformation scaleP = sqrt(2)/mdcPPtsCart; scaleQ = sqrt(2)/mdcQPtsCart; tP = [ scaleP 0 -scaleP*centroidPPtsCart(1); 0 scaleP -scaleP*centroidPPtsCart(2); 0 0 1]; tQ = [ scaleQ 0 -scaleQ*centroidQPtsCart(1); 0 scaleQ -scaleQ*centroidQPtsCart(2); 0 0 1]; % transform points PPtsHom = tP * PPtsHom; QPtsHom = tQ * QPtsHom; % normalize for homogenous scaling for i = 1:3 PPtsHom(i, :) = PPtsHom(i, :) ./ PPtsHom(3, :); QPtsHom(i, :) = QPtsHom(i, :) ./ QPtsHom(3, :); end % 2D cartesian coordinates PPtsCart = PPtsHom(1:2,:); QPtsCart = QPtsHom(1:2,:); % transform cameras P = tP * P; Q = tQ * Q; % triangulating points TriangulatedPoints = zeros(4,numPts); for i = 1:numPts A = [ PPtsCart(1, i) * P(3, :) - P(1, :); PPtsCart(2, i) * P(3, :) - P(2, :); QPtsCart(1, i) * Q(3,:) - Q(1,:); QPtsCart(2, i) * Q(3,:) - Q(2,:) ] return; [U S V] = svd(A); TriangulatedPoints(:, i) = V(:, end); end for i = 1:4 TriangulatedPoints(i, :) = TriangulatedPoints(i, :) ./ TriangulatedPoints(4, :); end figure() scatter3(TriangulatedPoints(1, :), TriangulatedPoints(2, :), TriangulatedPoints(3, :)); title('Triangulated Points'); axis equal; </code></pre>

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