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The Wikipedia article on the Point Spread Function (link) discusses how an imaging system can be conceptually described using linear system theory. A convolution of the PSF with the image in the spatial domain is equivalent to a multiplication in the Fourier domain.

Given $m \times n$ matrices $\bf{A}$ and $\bf{B}$, and assuming that $\bf{A}$ is an image and $\bf{B}$ is another matrix, the addition of the two matrices in the spatial domain is:

$\bf{A} + \bf{B} = \bf{C}$

However, can this addition be expressed as a convolution in the spatial domain? Could I re-write the addition equation as a convolution?

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up vote 1 down vote accepted

Convolution is used to describe not only linear, but also shift-invariant system.

For $\mathbf{A}+\mathbf{B}=\mathbf{C}$, it's not even a linear system unless $\mathbf{B}=0$ (I assume $\mathbf{A}$ is the input, $\mathbf{C}$ is the output).

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OK, thanks for pushing me in the right direction. Yes, $\bf{A}$ is the input and $\bf{C}$ is the output. So what you are saying is that it is not possible to express the sum of matrices as a convolution? – Nicholas Kinar Sep 2 '12 at 19:16
@Nicholas Kinar Yes, I think so. – chaohuang Sep 2 '12 at 19:19
Thanks, I understand this now. – Nicholas Kinar Sep 2 '12 at 19:23

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