How to correctly use cv::triangulatePoints()

Question

I am trying to triangulate some points with OpenCV and I found this cv::triangulatePoints() function. The problem is that there is almost no documentation or examples of it.

I have some doubts about it.

1. What method does it use? I've making a small research about triangulations and there are several methods (Linear, Linear LS, eigen, iterative LS, iterative eigen,...) but I can't find which one is it using in OpenCV.

2. How should I use it? It seems that as an input it needs a projection matrix and 3xN homogeneous 2D points. I have them defined as std::vector<cv::Point3d> pnts, but as an output it needs 4xN arrays and obviously I can't create a std::vector<cv::Point4d> because it doesn't exist, so how should I define the output vector?

For the second question I tried: cv::Mat pnts3D(4,N,CV_64F); and cv::Mat pnts3d;, neither seems to work (it throws an exception).

Answer 1

1.- The method used is Least Squares. There are more complex algorithms than this one. Still it is the most common one, as the other methods may fail in some cases (i.e. some others fails if points are on plane or on infinite).

The method can be found in Multiple View Geometry in Computer Vision by Richard Hartley and Andrew Zisserman (p312)

2.-The usage:

cv::Mat pnts3D(1,N,CV_64FC4);
cv::Mat cam0pnts(1,N,CV_64FC2);
cv::Mat cam1pnts(1,N,CV_64FC2);

Fill the 2 chanel point Matrices with the points in images.

cam0 and cam1 are Mat3x4 camera matrices (intrinsic and extrinsic parameters). You can construct them by multiplying A*RT, where A is the intrinsic parameter matrix and RT the rotation translation 3x4 pose matrix.

cv::triangulatePoints(cam0,cam1,cam0pnts,cam1pnts,pnts3D);

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NOTE: pnts3D NEEDs to be a 4 channel 1xN cv::Mat when defined, throws exception if not, but the result is a cv::Mat(4,N,cv_64FC1) matrix. Really confusing, but it is the only way I didn't got an exception.

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UPDATE: As of version 3.0 or possibly earlier, this is no longer true, and pnts3D can also be of type Mat(4,N,CV_64FC1) or may be left completely empty (as usual, it is created inside the function).

Answer 2

A small addition to @Ander Biguri's answer. You should get your image points on a non-undistorted image, and invoke undistortPoints() on the cam0pnts and cam1pnts, because cv::triangulatePoints expects the 2D points in normalized coordinates (independent from the camera) and cam0 and cam1 should be only [R|t^T] matricies you do not need to multiple it with A.

Answer 3

Thanks to Ander Biguri! His answer helped me a lot. But I always prefer the alternative with std::vector, I edited his solution to this:

std::vector<cv::Point2d> cam0pnts;
std::vector<cv::Point2d> cam1pnts;
// You fill them, both with the same size...

// You can pick any of the following 2 (your choice)
// cv::Mat pnts3D(1,cam0pnts.size(),CV_64FC4);
cv::Mat pnts3D(4,cam0pnts.size(),CV_64F);

cv::triangulatePoints(cam0,cam1,cam0pnts,cam1pnts,pnts3D);

So you just need to do emplace_back in the points. Main advantage: you do not need to know the size N before start filling them. Unfortunately, there is no cv::Point4f, so pnts3D must be a cv::Mat...

Answer 4

I tried cv::triangulatePoints, but somehow it calculates garbage. I was forced to implement a linear triangulation method manually, which returns a 4x1 matrix for the triangulated 3D point:

Mat triangulate_Linear_LS(Mat mat_P_l, Mat mat_P_r, Mat warped_back_l, Mat warped_back_r)
{
    Mat A(4,3,CV_64FC1), b(4,1,CV_64FC1), X(3,1,CV_64FC1), X_homogeneous(4,1,CV_64FC1), W(1,1,CV_64FC1);
    W.at<double>(0,0) = 1.0;
    A.at<double>(0,0) = (warped_back_l.at<double>(0,0)/warped_back_l.at<double>(2,0))*mat_P_l.at<double>(2,0) - mat_P_l.at<double>(0,0);
    A.at<double>(0,1) = (warped_back_l.at<double>(0,0)/warped_back_l.at<double>(2,0))*mat_P_l.at<double>(2,1) - mat_P_l.at<double>(0,1);
    A.at<double>(0,2) = (warped_back_l.at<double>(0,0)/warped_back_l.at<double>(2,0))*mat_P_l.at<double>(2,2) - mat_P_l.at<double>(0,2);
    A.at<double>(1,0) = (warped_back_l.at<double>(1,0)/warped_back_l.at<double>(2,0))*mat_P_l.at<double>(2,0) - mat_P_l.at<double>(1,0);
    A.at<double>(1,1) = (warped_back_l.at<double>(1,0)/warped_back_l.at<double>(2,0))*mat_P_l.at<double>(2,1) - mat_P_l.at<double>(1,1);
    A.at<double>(1,2) = (warped_back_l.at<double>(1,0)/warped_back_l.at<double>(2,0))*mat_P_l.at<double>(2,2) - mat_P_l.at<double>(1,2);
    A.at<double>(2,0) = (warped_back_r.at<double>(0,0)/warped_back_r.at<double>(2,0))*mat_P_r.at<double>(2,0) - mat_P_r.at<double>(0,0);
    A.at<double>(2,1) = (warped_back_r.at<double>(0,0)/warped_back_r.at<double>(2,0))*mat_P_r.at<double>(2,1) - mat_P_r.at<double>(0,1);
    A.at<double>(2,2) = (warped_back_r.at<double>(0,0)/warped_back_r.at<double>(2,0))*mat_P_r.at<double>(2,2) - mat_P_r.at<double>(0,2);
    A.at<double>(3,0) = (warped_back_r.at<double>(1,0)/warped_back_r.at<double>(2,0))*mat_P_r.at<double>(2,0) - mat_P_r.at<double>(1,0);
    A.at<double>(3,1) = (warped_back_r.at<double>(1,0)/warped_back_r.at<double>(2,0))*mat_P_r.at<double>(2,1) - mat_P_r.at<double>(1,1);
    A.at<double>(3,2) = (warped_back_r.at<double>(1,0)/warped_back_r.at<double>(2,0))*mat_P_r.at<double>(2,2) - mat_P_r.at<double>(1,2);
    b.at<double>(0,0) = -((warped_back_l.at<double>(0,0)/warped_back_l.at<double>(2,0))*mat_P_l.at<double>(2,3) - mat_P_l.at<double>(0,3));
    b.at<double>(1,0) = -((warped_back_l.at<double>(1,0)/warped_back_l.at<double>(2,0))*mat_P_l.at<double>(2,3) - mat_P_l.at<double>(1,3));
    b.at<double>(2,0) = -((warped_back_r.at<double>(0,0)/warped_back_r.at<double>(2,0))*mat_P_r.at<double>(2,3) - mat_P_r.at<double>(0,3));
    b.at<double>(3,0) = -((warped_back_r.at<double>(1,0)/warped_back_r.at<double>(2,0))*mat_P_r.at<double>(2,3) - mat_P_r.at<double>(1,3));
    solve(A,b,X,DECOMP_SVD);
    vconcat(X,W,X_homogeneous);
    return X_homogeneous;
}

the input parameters are two 3x4 camera projection matrices and a corresponding left/right pixel pair (x,y,w).