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I'm trying to find fast ways to evaluate the following integral defined over points $\mathbf{x} \in \mathbb{R}^2$:

$$\Phi(\mathbf{x}) = \frac{1}{2\pi\sigma^2(\mathbf{x})}\int d^2y\, \exp\left(-\frac{|\mathbf{x}-\mathbf{y}|^2}{2\sigma^2(\mathbf{x})}\right)\phi(\mathbf{y}).$$

Here $\phi(\mathbf{x})$ is a smooth function, which one might assume represents an image. This almost looks like a Gaussian blur, except for the fact that the standard deviation $\sigma(\mathbf{x})$ depends on the coordinate $\mathbf{x}$. Because this equation isn't a convolution, I can't directly use FFT based methods.

Are there fast methods to evaluate such integrals numerically?

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  • $\begingroup$ Probably depends on the specific function for std deviation. What is it? $\endgroup$
    – Davey
    Commented 2 days ago
  • $\begingroup$ $\sigma(\mathbf{x})$ is a smooth function. It can be made periodic, but apart from that I want to keep it general. $\endgroup$
    – B215826
    Commented 2 days ago
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    $\begingroup$ This seems like it could be accomplished in scale space en.wikipedia.org/wiki/Scale_space where $\sigma^2(\mathbf x)$ is a 2D surface "slice" of the volume of the scale space. $\endgroup$
    – Davey
    Commented 2 days ago
  • $\begingroup$ @Davey very interesting. I think this is what people have used for this sort of problems before, e.g, this Python module that does something similar. $\endgroup$
    – B215826
    Commented 2 days ago
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    $\begingroup$ This is implemented in DIPlib for 2D and 3D images, not for 1D. It is a brute-force implementation, but is quite efficient. diplib.org/diplib-docs/… $\endgroup$
    – Cris Luengo
    Commented 2 days ago

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