Why equate an identity matrix to a transformation matrix in OpenCV - c++

Question

In opencv and C++, if I have previously found a transformation matrix between 2 images, why do I need to do this
Mat affineTransform=getAffineTransform(coordinates_1.data(),coordinates_2.data()) ;
Mat perspectiveTransform=Mat::eye(3,3,CV_64FC1);
for(unsigned int y=0; y<2; ++y){
for(unsigned int x=0; x<3; ++x){
perspectiveTransform.at<double>(y,x) = affineTransform.at<double>(y,x);
}
instead of applying the transformation matrix to an image directly. I understand the meaning of Mat::eye() but why go through all this?
Note: originalTranformationMatrix is a Mat object and the found transformation matrix is a 3 x 3 matrix

Answer 1

Affine transform has following form:
(a, b, c)
(d, e, f)
It transform point (x,y) in following way:
x_new = a*x + b*y + c;
y_new = d*x + e*y + f;
Perspective transform has following form:
(a, b, c)
(d, e, f)
(g, h, 1)
It transform point (x,y) in following way:
z = g*x + h*y + 1;
x_new = (a*x + b*y + c)/z;
y_new = (d*x + e*y + f)/z;
That means that if you want to define perspective transform that does only affine transformation, it should be:
(a, b, c)
(d, e, f)
(0, 0, 1)
And this is exactly what your code does. First it creates matrix:
(1, 0, 0)
(0, 1, 0)
(0, 0, 1)
And than replaces first two rows with rows of affine transformation. By the way, it can be done in cleaner way without loops:
perspectiveTransform(Rect(0,0,3,2)) = affineTransform.clone();