How to simplify multiple addition and convolution operations into one convolution kernel
Let's rearrange
\begin{align}
x(n,m) * \left[ f_1(n,m) + f_2(n,m) + f_3(n,m)\right]
&amp;= x(n,m) + kx(n,m) - x(n,m) * kg(n,m)\\
x(n,m) * \left[ f_1(n,m) + f_2(n,m) \right]
&amp;= x(n,m) + kx(n,m)\\
&amp;=(1+k)x(n,m)
\end{align}
It directly follows from the linearity of convolution that $f_1+f_2=(1+k)\delta_{n,m}^{(N \times M)}$, where $\delta^{(N\times M)}$ is the Kronecker delta of the same size as your input $x$, i.e. an all-zero matrix with a 1 in the $0,0$ element; the neutral element for 2D-convolution.

So, it's possible to calculate what the sum of $f_1 + f_2$ is, but not the individual $f_1$ and $f_2$; $f_1= \delta^{(N\times M)}, f_2 = k\delta^{(N\times M)}$ is as correct as
$f_1=\begin{pmatrix}1234 &amp; 1 &amp; \cdots &amp;1\\0 &amp; 0 &amp; \cdots &amp;0\\0 &amp; 0 &amp; \cdots &amp;0\\\end{pmatrix},f_2=\begin{pmatrix}-1233+k &amp; -1 &amp; \cdots &amp;-1\\0 &amp; 0 &amp; \cdots &amp;0\\0 &amp; 0 &amp; \cdots &amp;0\\\end{pmatrix}$.