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I have an elementary function $p:\mathbb{R}^2\rightarrow\mathbb{R}$ which (locally) represents an image. It's a polynomial, and its the result of the following 2D convolution:

$$p=f\star G\star \mathbf{1}_{A}$$

where $G(x,y)$ is a 2D Gaussian, and $\mathbf{1}_A$ 'box' function:

$$G(x,y)=e^{-\frac{(x^2+y^2)}{2\sigma^2}}, \quad\mathbf{1}_A(x,y)=1 \text{ for }x \text{ and } y < a, 0\text{ otherwise.}$$

Is there a closed form for a function with which I can convolve $p$, which results in $f$?

Currently I am 'avoiding' the use of such a filter by performing Fourier deconvolution, but mathematically this opens a huge can of worms.

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The FT of the convolutional inverse is the point-wise inverse in the FD. So the convolutional inverse of a gaussian might have numerical problems. The FT of a gaussian is a gaussian. Its point-wise inverse thus can be large in many places.

For a box function, the situation doesn't look all that good either. Its FT is a variant on $\operatorname{sinc}$, which has many zeroes and loci of low magnitude.

You can certainly determine the convolutional inverse of a gaussian, but the inverse of a box function is not even a function. Convolving with a box will cancel any function that is periodic with a period multiple of the edge length.

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The problem with deconvolution is that you amplify quantization noise and other stuff and the inverse might not even have a stable representation. Look up "Wiener filtering": it is least-square error inverse filtering with a signal and noise model and can usually be done easily enough in the frequency domain.

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