A mathematical way to represent an image kernel?

Question

How to represent the calculation in this image mathematically?

For example: With the discrete convolution and Fourier Transform.

It tries to do a calculation on the original image (image $A$/input) and save the result of the calculation in image $B$/output. It tries to multiply the pixel you are dealing with, with $5$ and it tries to multiply one pixel at the left, one pixel at the right, one pixel above of it and one pixel under it with $-1$ and adds the value of all the $5$ pixels you just multiplied together. After that, it will save the value of the new pixel in picture $B$. This happens to every single pixel in the image. Since there are no surrounding pixel on the corners, it simply takes them from somewhere else.

Which makes more sense?

Solution A:

$c(i,j) = \displaystyle\sum_{k_1 \in \Bbb N} \displaystyle\sum_{k_2 \in \Bbb N} a(k_1, k_2) b(i - k_1, j - k_2)$ where $a$ and $b$ are your matrices.

Supposing $a$ is your picture, by setting $b(0,0)=5$, $b(±1,0)=−1$, $b(0,±1)=−1$ and $b=0$ elsewhere.

Solution B:

$$ (f \ast g)(x,y) = \sum_{i=-\infty}^\infty \sum_{j=-\infty}^\infty f(i,j) g(x-i,y-j). $$ $f$ is the filter, $g$ is the image, and $f \ast g$ is the filtered image. We have $f(0,0) = 5$, $f(-1,0) = f(1,0) = f(0,-1) = f(0,1) = -1$, and $f(i,j) = 0$ otherwise.




Note to answerer: Please try not to overcomplicate your answer...

Answer 1

Let $A$ be an $m\times n$ matrix representing the input image, and $K$ be an $s\times t$ matrix representing the kernel, with $s$ and $t$ odd (since we need it to have a central entry). We will index $A$ with $$(i,j)\in \{1,\dotsc,m\}\times\{1,\dotsc,n\}$$ and we will index $K$ with $$(h,k)\in\left\{-\sigma, \dotsc, \sigma\right\}\times\left\{-\tau, \dotsc,\tau\right\}$$ where $\sigma=\left\lfloor \frac{s}{2}\right\rfloor$, $\tau=\left\lfloor \frac{t}{2}\right\rfloor$, and with $\lfloor \cdot \rfloor$ we mean the floor function. We do this so that the central entry of $K$ will have index $(h,k)=(0,0)$.

To take care of the edge entries we add two rows and two columns to $A$ by repeating the first and last row and column. That is, we define the extra entries $$ \begin{align} A(0,0)&:=A(1,1) & A(0,j)&:=A(1,j) & A(0,n+1)&:=A(1,n)\\ A(i,0)&:=A(i,1) & & & A(i,n+1)&:=A(i,n) \\ A(m+1,0)&:=A(m+1,1) & A(m+1,j)&:=A(m,j) & A(m+1,n+1)&:=A(m,n) \end{align} $$ This corresponds to the "extension" edge handling method. You can easily change those extra rows and columns to adapt to other methods.

Then for any fixed $(i,j)\in\{1,\dotsc,m\}\times\{1,\dotsc,n\}$ the entry $(i,j)$ of $K*A$ will be $$ (K*A)(i,j) = \sum_{h=-\sigma}^{\sigma}\sum_{k=-\tau}^{\tau} K(h,k)\;A(i+h,j+k) $$