下面是一个Matlab函数，它实现了多视图几何中描述的三角测量方法。它需要一个立方体的8个角点（OriginalPoints）并将它们投影在两个屏幕上。然后屏幕点在CameraPoints和ProjectorPoints中。目标是使用直接线性变换从这两个视图重建原始点。不幸的是结果是胡言乱语。
有another topic here on SO，从中可以采用A行的标准化，但这并没有改善结果。
三个一般问题：
% 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');
答案 0 :(得分：1)
上面代码中缺少的是摄像机矩阵也需要通过规范化转换进行转换。下面是一个有效的例子。还要注意归一化变换的结构差异。在问题的例子中，缩放因子放在最后一个元素中。在它下面乘以所有元素，这样最后一个元素就是1.
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;