We can express 2D convolution between $f(m,n)$ and $h(m,n)$ as following
\begin{align} g(m,n) &= \displaystyle \sum_{k_1=-\infty}^{\infty} \sum_{k_2=-\infty}^{\infty} f_{\!_{k_1,k_2}}h(m-k_1,n-k_2)\tag{1}\\ &=f(m,n)*h(m,n)\tag{2} \end{align} where \begin{equation} h(m,n) = \begin{cases} h_{m,n} & {0\leq m \leq L_r, 0\leq n \leq L_c}\\ 0 & \text{otherwise} \end{cases} \end{equation}
and
\begin{equation} f(m,n) = f_{m,n} \forall {-\infty\leq m \leq \infty, -\infty\leq n \leq \infty} \end{equation}
From (1), it seems as if the value of g at location (m,n) is equal to the double sum. But (2) is more like the input-output relationship that can be represent as. here $f(m,n)$ and $g(m,n)$ represent the entire 2D signals.
Can someone please figure out where I am making a mistake
