Additional context: $H = |δ^2f / δx_iδx_j|$ is the Hessian matrix. $(3)$
From my previous question: What are the functionality of δ symbol and $δr^T$?,
I got a few questions:
I have read more about the Hessian Matrix and I got a general idea that it can be used to see rate of change in a function by using 2nd order partial derivatives. But from the link above, I see a function $f(x_1, x_2, ..., x_n)$. From what I understand is that we are dealing with multiple variables in one dimension. However, if I am to use this in image processing which I got rows and columns, for instance, $x_1$ to $x_n$ and $y_1$ to $y_n$, how should I approach Hessian Matrix in this case? From the additional context $(3)$ above and from wikipedia itself, can I simply "plug in" $x_n$ to $x_j$ and $y_n$ to $x_i$, correct? I just want to be very sure here. This is how it should be, yes?
In the equation (1) in the snapshot, what is the definition for gradient $∇_0$ in this case? I have tried read it from here, but I am not sure what should I be looking for.
From my previous question, I see that the letter T stands for Transpose in Matrices, however, from the description above in the image, I understand that $r_0$ is just a point and $δr$ is a small rate of change to $r_0$. Shouldn't rate of change $δr$ be just a scalar? How can we transpose $δr$ in this case? Or am I missing something here?
If I appear to make any mistake above, please correct me.