Android & OpenCV: Homography to Camera Pose considering Camera Intrinsics and Backprojection

Question

Libs: OpenCV Target: Android (OpenCV4Android)

I try to compute the Homography of a world plane (e.g. monitor screen) to get the camera pose, transform it and reproject the points back for tracking tasks. I'm using OpenCVs findHomography() / getPerspectiveTransform() to get the homography. The reprojection of the points using perspectiveTransform() (as explained here: http://docs.opencv.org/doc/tutorials/features2d/feature_homography/feature_homography.html) which works pretty well. The "screenPoints" are the world coordinates of the monitor edges (using the aspect ratio and a z-value of 0) and the "imagePoints" are the x/y-coordinates of the screen edges in the image.

Mat homography = org.opencv.imgproc.Imgproc.getPerspectiveTransform(screenPoints, imagePoints);

I have the camera calibration matrix (I have used the matlab calibration toolbox) and I found a hint (in the comments @ https://dsp.stackexchange.com/questions/2736/step-by-step-camera-pose-estimation-for-visual-tracking-and-planar-markers) for considering the camera parameters in the homography.

H' = K^-1 * H

(H' - Homography-Matrix considering camera calibration, H - Homography-Matrix, K^-1 - inverse camera calibration matrix).

Mat intrinsicInverse = new Mat(3, 3, CvType.CV_32FC1);
Core.invert(intrinsic, intrinsicInverse);
intrinsicInverse.convertTo(intrinsicInverse, CvType.CV_32FC1);          
homography.convertTo(homography, CvType.CV_32FC1);
// compute H respect the intrinsics
Core.gemm(intrinsicInverse, homography, 1, new Mat(), 0, homography);

My next step ist to compute the camera pose from homography as decribed here Computing camera pose with homography matrix based on 4 coplanar points.

Since im trying to do this on Android i had to port the C++ Code to Java:

private Mat cameraPoseFromHomography(Mat h) {
    Log.d("DEBUG", "cameraPoseFromHomography: homography " + matToString(h));

    Mat pose = Mat.eye(3, 4, CvType.CV_32FC1);  // 3x4 matrix, the camera pose
    float norm1 = (float) Core.norm(h.col(0));
    float norm2 = (float) Core.norm(h.col(1));
    float tnorm = (norm1 + norm2) / 2.0f;       // Normalization value

    Mat normalizedTemp = new Mat();
    Core.normalize(h.col(0), normalizedTemp);
    normalizedTemp.convertTo(normalizedTemp, CvType.CV_32FC1);
    normalizedTemp.copyTo(pose.col(0));

    Core.normalize(h.col(1), normalizedTemp);
    normalizedTemp.convertTo(normalizedTemp, CvType.CV_32FC1);    
    normalizedTemp.copyTo(pose.col(1));

    Mat p3 = pose.col(0).cross(pose.col(1));
    p3.copyTo(pose.col(2));

    Mat temp = h.col(2);
    double[] buffer = new double[3];
    h.col(2).get(0, 0, buffer);
    pose.put(0, 3, buffer[0] / tnorm);
    pose.put(1, 3, buffer[1] / tnorm);
    pose.put(2, 3, buffer[2] / tnorm);

    return pose;
}

I can't check if the code is doing the right thing but it's running. At this point I assume to have the full camera pose considering the camera calibration.

As described here http://opencv.willowgarage.com/documentation/python/calib3d_camera_calibration_and_3d_reconstruction.html#rodrigues2, the reprojection of a 3D-Point is just

p = K * CP * P

(p - 2D-Position, K - calibration matrix, CP - camera pose, P - 3D-Point)

    Core.gemm(intrinsic, cameraPosition, 1, new Mat(), 0, vec4t);
    Core.gemm(vec4t, point, 1, new Mat(), 0, result);

The result is far away from the source image positions of the screen edges. But I can identify all three edges by its relative differences - so it might be just some factor which is wrong.

It's the first time I'm doing such a Computer Vision task and it's possible I did some basically wrong. I have the "Multiple View Geometry" book from Zisserman and I read all related parts - but to be honest - I didn't get most of it.

UPDATE:

Found a bug in my camera matrix - the implementation above is just working fine!

Answer 1

Get it to work on another way. Instead of using findHomography()/getP erspectiveTransform() i found another function called solvePnP() which returns the camera pose based on world and images points and an intrinsic camera matrix.

Using that function in combination with the projectPoints() method - i was able to reproject the 3d points back to the image.

In case of the screen edges there are placed on the right spot in the image.

UPDATE:

I found a bug in my implementation - my camera intrinsic matrix was wrong. The camera pose from homography implementation above is working for me!

Answer 2

The relationship between Homography in calibrated case (H) and uncalibrated case (H') is

H′=𝐾𝐻𝐾^(−1), where K is intrinsic matrix of camera.