Imagine that I have an $\mathbb{K}^{n \times m}$ Matrix called $ A $ (It can also be an 2-dimensional image with the width of m and the height of n, with $\mathbb{K}$ being a colour-vector).
I also have a non-linear, non-affine transformation $T : \mathbb{N_0}^2 \mapsto \mathbb{R}^2 $, which one could interpret as an index-transformation function.
In this sense I mean the following:
If the function $T$ was defined as $T(i,j)=(i+2,\frac{j}{2})$, an 'application' of $T$ onto the matrix' indices would result in following matrix transformation:
A := | 3 1 |
| 4 2 |
| - - - |
A' = | 3 - 1 |
| 4 - 2 |
The result is technically not a matrix, as its indices are not from the field $\mathbb{N_0}$, but from the field $\mathbb{R}$.
As you can see the result 'matrix' has some undefined entries (marked with an -).
My question is now: how can I continuously interpolate over the matrix (e.g. Bezier or Bicubic interpolation) in order to fill the 'matrix' empty entries?
It may be easier to see the result as image with floating-point pixel indices, which I now have to fill with 'colour'.
In order to maybe better visualise the problem, I've drawn a small sketch:
HINT You can try for example the matrix
$$\bf S = \left[\begin{array}{cc}1&0\\0.5&0.5\\0&1\end{array}\right]$$
now on any $2\times 2$ matrix $\bf M$ calculate ${\bf SMS}^T$ and see what happens.