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How can I represent the calculation in this image mathematically?

enter image description here

For example, with the discrete convolution (and Fourier Transform?),

$$(f * g)[n]\ \stackrel{\mathrm{def}}{=}\ \sum_{m=-\infty}^\infty f[m]\, g[n - m]$$

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  • $\begingroup$ Not your exact question, but similar: dsp.stackexchange.com/a/7847/35 $\endgroup$
    – datageist
    Apr 7, 2013 at 5:29
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    $\begingroup$ What is your question? The operation in the movie is a convolution with a particular mask. You already state how this is typically written. The mask is your $g$. $\endgroup$ Apr 7, 2013 at 18:32
  • $\begingroup$ This operation is already $\mathcal{O}(N)$, where $N$ is the number of pixels (i.e. cells); its complexity can't be improved, because every pixel must be visited at least once. The FFT -- or any other Poisson solver -- could come in if you wanted to invert the operation, as the stencil is simply the classic $5$-point stencil for $-\Delta$. $\endgroup$
    – Ben
    Jun 7, 2013 at 16:31

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From the computational point of view, each element of the final matrix (resulting image) is equal to a operation between the around elements in the initial matrix (input image) and the respective elements of the kernel, so:

$ final_{i,j} = \\ initial_{r(i-1),r(j-1)} OP kernel_{1,1} + initial_{r(i),r(j-1)} OP kernel_{2,1} + initial_{r(i+1),r(j-1)} OP kernel_{3,1} + \\ initial_{r(i-1),r(j)} OP kernel_{1,2} + initial_{r(i),r(j)} OP kernel_{2,2} + initial_{r(i+1),r(j)} OP kernel_{3,2} + \\ initial_{r(i-1),r(j+1)} OP kernel_{1,3} + initial_{r(i),r(j+1)} OP kernel_{2,3} + initial_{r(i+1),r(j+1)} OP kernel_{3,3} \\ $

where the $r(x)$ represents the integer division between the x and the matrix side.

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