I have found an interesting paper on a digital image registration algorithm. There are many equations in the paper that I only understand partially, but there is a particular one I would like to understand better (at the top of page 4).
$(T^{(-1)}+w)^{(-1)}=T \circ(Id + w \circ T)^{(-1)}$
Where $T$ is a non-parametric transformations that map every point $x$ in image $I$ to $T(x)$ in image $J$. We can find $T^{(-1)}$ iteratively: $T^{(-1)}_{k+1}(x)=(T^{(-1)}_k\circ w_k)(x)$ where $w$ is also a transformation called the adjustment field.
My questions are:
1: What are the steps needed to arrive from the LHS of the equation to the RHS?
2: What are the necessary assumptions for this equation to hold?
Just by looking at the equation, my guess is that something like this might have happened:
$(T^{(-1)}+w)^{(-1)}=T\circ T^{(-1)} \circ (T^{(-1)}+w)^{(-1)}$
$T\circ T^{(-1)} \circ (T^{(-1)}+w)^{(-1)} =T\circ (T^{(-1)}\circ T+w \circ T)^{(-1)}=T \circ(Id + w \circ T)^{(-1)}$
But I don't know why would it be possible to pass $T^{(-1)}$ through the parenthesis or why would the function composition be distributive with addition, or even if this is really the way this equation was done.