matlab - Homographic image transformation distortion issue

Question

I am trying to transform an image using a 3D transformation matrix and assuming my camera is orthonormal.

I am defining my homography using the plane-induced homography formula H=R-t*n'/d (with d=Inf so H=R) as given in Hartley and Zisserman Chapter 13.

What I am confused about is when I use a rather modest rotation, the image seems to be distorting much more than I expect (I'm sure I'm not confounding radians and degrees).

What could be going wrong here?

I've attached my code and example output.

n = [0;0;-1];
d = Inf;

im = imread('cameraman.tif');

 rotations = [0 0.01 0.1 1 10];

 for ind = 1:length(rotations)
       theta = rotations(ind)*pi/180;

       R = [ 1     0           0 ;
           0  cos(theta) -sin(theta);
           0  sin(theta)  cos(theta)];

       t = [0;0;0];

       H = R-t*n'/d;

      tform = maketform('projective',H');
      imT = imtransform(im,tform);

      subplot(1,5,ind) ;
      imshow(imT)
      title(['Rot=' num2str(rotations(ind)) 'deg']);
      axis square
 end

Answer 1

The formula H = R-t*n'/d has one assumption which is not met in your case:

This formula implies that you are using pinhole camera model with focal length=1

But in your case, for your camera to be more real and for your code to work, you should set the focal length to some positive number much greater than 1. (focal length is the distance from your camera center to the image plane)

To do this you can define a calibration matrix K which handles the focal length. You just need to change your formula to H=K R inv(K) - 1/d K t n' inv(K) in which K is a 3-by-3 identity matrix whose two first elements along the diagonal are set to the focal length (e.g. f=300). The formula can be easily derived if you assume a projective camera.

Below is the corrected version of your code, in which the angles make sense.

n = [0;0;-1];
d = Inf;

im = imread('cameraman.tif');

rotations = [0 0.01 0.1 30 60];

 for ind = 1:length(rotations)
   theta = rotations(ind)*pi/180;

   R = [ 1     0           0 ;
       0  cos(theta) -sin(theta);
       0  sin(theta)  cos(theta)];

   t = [0;0;0];

  K=[300 0    0;
        0    300 0;
        0    0    1];

  H=K*R/K-1/d*K*t*n'/K;

  tform = maketform('projective',H');
  imT = imtransform(im,tform);

  subplot(1,5,ind) ;
  imshow(imT)
  title(['Rot=' num2str(rotations(ind)) 'deg']);
  axis square
end

You can see the result in the image below:

You can also rotate the image around its center. For it to happen you should set the image plane origin to the center of the image which I think is not possible with that method of matlab (maketform). You can use the method below instead.

imT=imagehomog(im,H','c');

Note that if you use this method, you'll have to change some settings in n, d, t and R to get the appropriate result. That method can be found at: https://github.com/covarep/covarep/blob/master/external/voicebox/imagehomog.m

The result of the program with imagehomog and some changes in n, d, t , and R is shown below which seems more real.

New settings are:

n = [0 0 1]';
d = 2;
t = [1 0 0]';
R = [cos(theta),  0, sin(theta);
     0,           1,          0;
     -sin(theta), 0, cos(theta)];