Hall 2019_03_04 Calculations

Camera Pose

Calculate camera coordinates from world coordinates using extrinsics

We can solve for window coordinates as follows.

The camera location is obtained from MATLAB 

   cam = -T*R'/12

The camera z-axis is obtained from

or in MATLAB 

   kn = R(1:3,3)';

Look-Down Angle


The yellow line illustrates the camera z-axis. It can be calculated from the extrinsics as 
   kn = R(1:3,3)'
The blue line is parallel to the floor. We can obtain that by dropping the third array element. 
   kb = [kn(1:2) 0]

We cam get the look-down angle in radius by first making kn and kb into unit vectors, and then finding 

   down = acos(dot(kb,kn))
or 
    down = asin(cross(kb,kn))

Compare this value to the angle calculated from the picture of the camera.

Alternative Calculation of Look Direction

Calculate the vector between the camera location and the projected location of the principal point shown by the magenta asterisk.

Calculate the vector between the camera location and the projected location of the principal point, shown by the blue line.

We can also obtain the look-down angle. First calculate the horizontal distance 

  dist = norm(pploc - cam(1:2))

The lookdown angle can then be calculated in radians from 

   down = atan(cam(3)/dist)

This should agree with previous values.

Locate Pose for Second Image

We assume that we moved the camera and captured another image - without altering the height or orientation of the camera.

We basically need to determine the translation and rotation needed to superimpose known features, such as the checkerboard origin and vector along the Y-axis.

We transform the second image using the tform obtained for the first image. Then we calculate the required translation and rotation angle. 

tpts2 = transformPointsForward(tf,imgPoints(:,:,nfile));
diff = (tpts2(1,:)-tpts1(1,:))/40/12;
 
kt = tpts2(8,:)-tpts2(1,:);
phi = atan2(kt(1),kt(2));
Rf = [cos(phi) sin(phi); -sin(phi) cos(phi)];
 
cam2 = (cam-diff)*Rf; 
kb2 = 4*kb*Rf;

Results are shown as follows

   

Camera Pose by Homography vs. Extrinsics

The red dot shows the second camera pose determined by homography. The green dot was detrmined by extrinsics.

Maintained by John Loomis, last updated 10 March 2019