I've got coordinates of 4 points in 2D that form a rectangle and their coordinates after a perspective transformation has been applied.
The perspective transformation is calculated in homogeneous coordinates and defined by a 3x3 matrix M. If the matrix is not known, how can I calculate it from the given points?
The calculation for one point would be:
| M11 M12 M13 | | P1.x | | w*P1'.x |
| M21 M22 M23 | * | P1.y | = | w*P1'.y |
| M31 M32 M33 | | 1 | | w*1 |
To calculate all points simultaneously I write them together in one matrix A and analogously for the transformed points in a matrix B:
| P1.x P2.x P3.x P4.x |
A = | P1.y P2.y P3.y P4.y |
| 1 1 1 1 |
So the equation is M*A=B and this can be solved for M in MATLAB by M = B/A or M = (A'\B')'.
But it's not that easy. I know the coordinates of the points after transformation, but I don't know the exact B, because there is the factor w and it's not necessary 1 after a homogeneous transformation. Cause in homogeneous coordinates every multiple of a vector is the same point and I don't know which multiple I'll get.
To take account of these unknown factors I write the equation as M*A=B*W
where W is a diagonal matrix with the factors w1...w4 for every point in B on the diagonal. So A and B are now completely known and I have to solve this equation for M and W.
If I could rearrange the equation into the form x*A=B or A*x=B where x would be something like M*W I could solve it and knowing the solution for M*W would maybe be enough already. However despite trying every possible rearrangement I didn't managed to do so. Until it hit me that encapsulating (M*W) would not be possible, since one is a 3x3 matrix and the other a 4x4 matrix. And here I'm stuck.
Also M*A=B*W does not have a single solution for M, because every multiple of M is the same transformation. Writing this as a system of linear equations one could simply fix one of the entries of M to get a single solution. Furthermore there might be inputs that have no solution for M at all, but let's not worry about this for now.
What I'm actually trying to achieve is some kind of vector graphics editing program where the user can drag the corners of a shape's bounding box to transform it, while internally the transformation matrix is calculated.
And actually I need this in JavaScript, but if I can't even solve this in MATLAB I'm completely stuck.