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POTriangulation using the direct linear transform taken directly from Hartley & Zisserman
 

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  1. POTriangulation using the direct linear transform taken directly from Hartley & Zisserman
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    <p>Below is a Matlab function that implements the triangulation method as described in Multiple View Geometry. It takes the 8 corner points of a cube (OriginalPoints) and projects them on two screens. The screen points are then in CameraPoints and ProjectorPoints. The goal is to reconstruct the original points from these two views using the Direct Linear Transform. Unfortunately the result is gibberish. </p> <p>There is <a href="https://stackoverflow.com/questions/2276445/triangulation-direct-linear-transform">another topic here on SO</a>, from which the normalization of the rows of A are taken, but that did not improve the result.</p> <p>Three general questions:</p> <ul> <li>When normalizing the data before the SVD (origin = centroid and avg distance = sqrt(2)), do I calculate the average distance over (x,y) or (x, y, w)? Inhomogeneous or homogeneous?</li> <li>After every operation, where possible, I scale the homogeneous coordinate w back to 1.0, is this correct?</li> <li>Since the camera matrices are known completely (not just up to a projective transformation) the resulting points should differ no more from the original points than a Euclidean transformation, right?</li> </ul> <pre><code> % Demos the triangulation method using the direct linear transform % algorithm function testTriangulation() close all; camWidth = 800; camHeight = 600; projectorWidth = 5080; projectorHeight = 760; % camera matrices CameraMatrix = [ -1.81066 0.00000 0.00000 0.00000; 0.00000 2.41421 0.00000 0.00000; 0.00000 0.00000 1.00000 1.50000 ]; ProjectorMatrix = [ -0.36118 0.00000 0.00000 0.00000 0.00000 2.00875 1.33916 0.00000 0.00000 -0.55470 0.83205 1.80278 ]; % generating original points and displaying them OriginalPoints = [ -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 ]; scatter3(OriginalPoints(1, :), OriginalPoints(2, :), OriginalPoints(3,:)); title('Original Points'); % projecting and normalizing camera points CameraPoints = CameraMatrix * OriginalPoints; for i = 1:3 CameraPoints(i, :) = CameraPoints(i, :) ./ CameraPoints(3,:); end % figure(); % scatter(CameraPoints(1,:), CameraPoints(2,:)); % title('Projected Camera Points'); % transforming camera points to screen coordinates camXOffset = camWidth / 2; camYOffset = camHeight / 2; CameraPoints(1,:) = CameraPoints(1,:) * camXOffset + camXOffset; CameraPoints(2,:) = CameraPoints(2,:) * camYOffset + camYOffset; figure(); scatter(CameraPoints(1,:), CameraPoints(2,:)); title('Projected Camera Points in Screen Coordinates'); % projecting and normalizing projector points ProjectorPoints = ProjectorMatrix * OriginalPoints; for i = 1:3 ProjectorPoints(i, :) = ProjectorPoints(i, :) ./ ProjectorPoints(3,:); end % figure(); % scatter(ProjectorPoints(1,:), ProjectorPoints(2,:)); % title('Projected Projector Points'); % transforming projector points to screen coordinates projectorXOffset = projectorWidth / 2; projectorYOffset = projectorHeight / 2; ProjectorPoints(1,:) = ProjectorPoints(1,:) * projectorXOffset + projectorXOffset; ProjectorPoints(2,:) = ProjectorPoints(2,:) * projectorYOffset + projectorYOffset; figure(); scatter(ProjectorPoints(1,:), ProjectorPoints(2,:)); title('Projected Projector Points in Screen Coordinates'); % normalizing camera points (i.e. origin is % the centroid and average distance is sqrt(2)). camCenter = mean(CameraPoints, 2); CameraPoints(1,:) = CameraPoints(1,:) - camCenter(1); CameraPoints(2,:) = CameraPoints(2,:) - camCenter(2); avgDistance = 0.0; for i = 1:length(CameraPoints(1,:)) avgDistance = avgDistance + norm(CameraPoints(:, i)); end avgDistance = avgDistance / length(CameraPoints(1, :)); scaleFactor = sqrt(2) / avgDistance; CameraPoints(1:2, :) = CameraPoints(1:2, :) * scaleFactor; % normalizing projector points (i.e. origin is % the centroid and average distance is sqrt(2)). projectorCenter = mean(ProjectorPoints, 2); ProjectorPoints(1,:) = ProjectorPoints(1,:) - projectorCenter(1); ProjectorPoints(2,:) = ProjectorPoints(2,:) - projectorCenter(2); avgDistance = 0.0; for i = 1:length(ProjectorPoints(1,:)) avgDistance = avgDistance + norm(ProjectorPoints(:, i)); end avgDistance = avgDistance / length(ProjectorPoints(1, :)); scaleFactor = sqrt(2) / avgDistance; ProjectorPoints(1:2, :) = ProjectorPoints(1:2, :) * scaleFactor; % triangulating points TriangulatedPoints = zeros(size(OriginalPoints)); for i = 1:length(OriginalPoints(1, :)) A = [ CameraPoints(1, i) * CameraMatrix(3, :) - CameraMatrix(1, :); CameraPoints(2, i) * CameraMatrix(3, :) - CameraMatrix(2, :); ProjectorPoints(1, i) * ProjectorMatrix(3,:) - ProjectorMatrix(1,:); ProjectorPoints(2, i) * ProjectorMatrix(3,:) - ProjectorMatrix(2,:) ]; % normalizing rows of A, as suggested in a topic on StackOverflow A(1,:) = A(1,:)/norm(A(1,:)); A(2,:) = A(2,:)/norm(A(2,:)); A(3,:) = A(3,:)/norm(A(3,:)); A(4,:) = A(4,:)/norm(A(4,:)); [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'); </code></pre>
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