how can I verify that a set of 4 points have undergone an affine transformation - affinetransform

Question

I have two images containing same object with 4 corners.
I have a correspondence for those 4 points between images.
Let us call these points as A,B,C,D .
I have (x,y) coordinates of A in image 1 and image 2 .
Similiarly , for points B,C,D .
My aim is to verify that if there exists an affine transformation that satisfies this coordinate change of these 4 points.
How should I proceed ?
It would be beneficial if there is a code in python or c so that I can quickly verify for many images.
My understanding is that a general affine transformation can be described by
x'=ax+by+c
y'=dx+ey+f
So there are six unknowns.
Thanks for reading.

Answer 1

Let's say your points are A=(x1,y1); B=(x2,y2); C=(x3;y3); D=(x4,y4) in image 1 and A'=(x1',y1'); B'=(x2',y2'); C'=(x3';y3'); D'=(x4',y4') in image 2.
You already figured out that finding the affine transformation requires you to find the six unknowns. To find them, just write down the system of linear equations and solve:
x1' = a x1 + b y1 + c
y1' = d x1 + e y1 + f
x2' = a x2 + b y2 + c
y2' = d x2 + e y2 + f
x3' = a x3 + b y3 + c
y3' = d x3 + e y3 + f
x4' = a x4 + b y4 + c
y4' = d x4 + e y4 + f
You can actually split this into two systems of linear equations:
x1' = a x1 + b y1 + c
x2' = a x2 + b y2 + c
x3' = a x3 + b y3 + c
x4' = a x4 + b y4 + c
and
y1' = d x1 + e y1 + f
y2' = d x2 + e y2 + f
y3' = d x3 + e y3 + f
y4' = d x4 + e y4 + f
Since you are only interested in whether there exists an affine transformation (but not in the values of a, b, c, d, e, f itself), you just have to determine if these two systems of linear equations have a solution or not.
The Rouché–Capelli theorem tells us that a system of linear equations has a solution if and only if the rank of the coefficient matrix equals the rank of the augmented matrix.
In your case, the coefficient matrix for both systems of linear equations is
| x1 y1 1 |
| x2 y2 1 |
| x3 y3 1 |
| x4 y4 1 |
The augmented matrix for the first system of linear equations is
| x1 y1 1 x1' |
| x2 y2 1 x2' |
| x3 y3 1 x3' |
| x4 y4 1 x4' |
and for the second system of linear equations it is
| x1 y1 1 y1' |
| x2 y2 1 y2' |
| x3 y3 1 y3' |
| x4 y4 1 y4' |
Now compute the ranks of these three matrices (for example with numpy.linalg.matrix_rank) and if they are all equal, there exists an affine transformation.