I am trying to understand and implement least square matching for images. However, I am having trouble with the theoretical part. In the Gruen's Paper, I understood up to obtaining solution vector on page 5. However, I do not understand the what is $P$ here and how the function $g(x,y)$ is updated iteratively. Also I found another document about the subject, which is simplified version. Again here, I do not understand how do they update the second image, since $\Delta$ and $V$ variables are not explained in the equation $V + B\Delta = f$.
Can anyone help me to understand this subject?
I'll just write what I think is happening in the second paper you linked.
Let $g(x,y)$ and $h(x,y)$ be our image data, which are known. We write the transformation of $h$ to $g$ via the following model: $$ g(x,y) = k_1h(a_1x+a_2y+a_3,b_1x+b_2y+b_3) + k_2 $$ so the transformation has a number of unknown parameters, which we can write as: $$ \eta = [a_1,a_2,a_3,b_1,b_2,b_3,k_1,k_2] $$ Let the initial parameter vector be given by: $ \eta_0 = [1,0,0,0,1,0,1,0] $.
This can be rewritten as: $$ F(\eta) = g(x,y)-k_1h(a_1x+a_2y+a_3,b_1x+b_2y+b_3) - k_2 = 0 $$ Unfortunately, this is quite difficult to solve, so instead we linearize it.
Recall that the linear approximation (i.e. first order Taylor approximation expanded about $\eta_0$) to $F$ can be written as: $$ F_\mathcal{L}(\eta) = F(\eta_0) + (\eta-\eta_0)^T\nabla_\eta F $$ In terms of the second paper you linked, it seems that $B:=\nabla_\eta F$, $v:=F(\eta_0)$, $\Delta := \eta-\eta_0$, and $f:=F_\mathcal{L}(\eta)$.