Aruco markers with openCv, get the 3d corner coordinates?

Question

I am detecting a printed Aruco marker using opencv 3.2:

aruco::estimatePoseSingleMarkers(corners, markerLength, camMatrix, distCoeffs, rvecs,tvecs);
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Answer 1

First, let's assume that we only have one marker given with side = 2 * half_side.

Second, aruco::detectMarker returns the relative position of the camera in the marker's world. Thus, I assume that you are looking for the coordinates of the corners in camera's world.

Then, in marker's space:

     [ half_side ]      [     0     ]
E  = [     0     ], F = [ half_side ]
     [     0     ]      [     0     ]

where the center O of the square has coordinate tvec (in camera's world) and rotation mat of the marker rot_mat is computed by cv::Rodrigues(rvec,rot_mat).

Now, using the pinhole camera model, the relation between coordinates of a point P in cam's world and marker's world is:

[P_x_cam]             [P_x_marker]
[P_y_cam] = rot_mat * [P_y_marker] + tvec
[P_z_cam]             [P_z_marker]    

for example, the center O, which is [0,0,0] in marker's world, is tvec in cam's world.

So, the coordinates of E in cam's world are:

[E_x_cam]             [half_side]
|E_y_cam| = rot_mat * |    0    | + tvec
[E_z_cam]             [    0    ] 

Magically, it is the sum of rot_mat's first column multiplied by half_size and tvec. Similarly, the coodinates of F is rot_mat's second column multiplied by half_size and tvec.

Now, the corners can be computed, for example

C - O = (E - O) + (F - O), B - O = (E - O) - (F - O)

where E-O is exactly rot_mat's first column multiplied by half_size.

With all that in mind, we can compose the function:

vector<Point3f> getCornersInCameraWorld(double side, Vec3d rvec, Vec3d tvec){

     double half_side = side/2;


     // compute rot_mat
     Mat rot_mat;
     Rodrigues(rvec, rot_mat);

     // transpose of rot_mat for easy columns extraction
     Mat rot_mat_t = rot_mat.t();

     // the two E-O and F-O vectors
     double * tmp = rot_mat_t.ptr<double>(0);
     Point3f camWorldE(tmp[0]*half_side,
                       tmp[1]*half_side,
                       tmp[2]*half_side);

     tmp = rot_mat_t.ptr<double>(1);
     Point3f camWorldF(tmp[0]*half_side,
                       tmp[1]*half_side,
                       tmp[2]*half_side);

     // convert tvec to point
     Point3f tvec_3f(tvec[0], tvec[1], tvec[2]);

     // return vector:
     vector<Point3f> ret(4,tvec_3f);

     ret[0] +=  camWorldE + camWorldF;
     ret[1] += -camWorldE + camWorldF;
     ret[2] += -camWorldE - camWorldF;
     ret[3] +=  camWorldE - camWorldF;

     return ret;
}

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Note 1: I hate that SO doesn't have MathJax

Note 2: there must be some faster implementation which I don't know of.

Answer 2

A python implementation I wrote for the above described rotation of marker corners using rvec and tvec as returned from cv2.aruco.estimatePoseSingleMarkers(). Thanks @Quang Hoang for the detailed explanation.

import numpy as np

# rotate a markers corners by rvec and translate by tvec if given
# input is the size of a marker.
# In the markerworld the 4 markercorners are at (x,y) = (+- markersize/2, +- markersize/2)
# returns the rotated and translated corners and the rotation matrix
def rotate_marker_corners(rvec, markersize, tvec = None):

    mhalf = markersize / 2.0
    # convert rot vector to rot matrix both do: markerworld -> cam-world
    mrv, jacobian = cv2.Rodrigues(rvec)

    #in markerworld the corners are all in the xy-plane so z is zero at first
    X = mhalf * mrv[:,0] #rotate the x = mhalf
    Y = mhalf * mrv[:,1] #rotate the y = mhalf
    minusX = X * (-1)
    minusY = Y * (-1)

    # calculate 4 corners of the marker in camworld. corners are enumerated clockwise
    markercorners = []
    markercorners.append(np.add(minusX, Y)) #was upper left in markerworld
    markercorners.append(np.add(X, Y)) #was upper right in markerworld
    markercorners.append(np.add( X, minusY)) #was lower right in markerworld
    markercorners.append(np.add(minusX, minusY)) #was lower left in markerworld
    # if tvec given, move all by tvec
    if tvec is not None:
        C = tvec #center of marker in camworld
        for i, mc in enumerate(markercorners):
            makercorners[i] = np.add(C,mc) #add tvec to each corner
    #print('Vec X, Y, C, dot(X,Y)', X,Y,C, np.dot(X,Y)) # just for debug
    markercorners = np.array(markercorners,dtype=np.float32) # type needed when used as input to cv2
    return markercorners, mrv

'''
Copyright 2019 Marco Noll, Garmin International Inc. Licensed under the Apache 
License, Version 2.0 (the "License"); you may not use this file except in compliance 
with the License. You may obtain a copy of the License at

http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software distributed 
under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR 
CONDITIONS OF ANY KIND, either express or implied. See the License for the specific 
language governing permissions and limitations under the License.
'''

Answer 3

Building upon @Quang's answer, C# code for transforming any point to camera coordinates. Of course it needs R and t vectors, so you're going to need a marker in order to get them.

private Point3d GetWorldPoint(Point3d input, Vec3d rvec, Vec3d tvec)
{
    var rot_mat = new Mat();

    Cv2.Rodrigues(MatOfDouble.FromArray(rvec.Item0, rvec.Item1, rvec.Item2), rot_mat);

    var pointProject = (rot_mat * MatOfDouble.FromArray(input.X, input.Y, input.Z)).ToMat();
    return tvec + new Point3d(pointProject.Get<double>(0, 0), pointProject.Get<double>(0, 1), pointProject.Get<double>(0, 2));
}