<p>The integral in the Fredholm equation has limits from <span class="math-container">$- \infty$</span> to <span class="math-container">$\infty$</span>.  When one or both of the limits are finite, it is called the Wiener-Hopf function. Astronomers have more use for the latter.</p>
<p>Astronomers commonly use this function as used in general signal processing where it is called the Wiener filter.  This Fourier filter minimizes the mean square error between the observed signal with noise and the real signal. The Wiener filter is applied in an iterative process for the deconvolution of images in an algorithm called <a href="https://en.m.wikipedia.org/wiki/Richardson%E2%80%93Lucy_deconvolution" rel="nofollow noreferrer">Lucy Deconvolution</a>.</p>
<p>It is also used to help fill in gaps in maps, like to fill in the galactic plane in maps of the microwave background or to fill gaps in the galaxy velocity field when reconstructing the density map from the velocity field.</p>
<p>See <a href="https://en.m.wikipedia.org/wiki/Wiener_deconvolution" rel="nofollow noreferrer">Wiener Deconvolution</a> in Wikipedia.  Here it is defined using Laplace transforms, but astronomers seem to prefer using Fourier transforms.</p>

CC BY-SA 4.0
2024-06-29T02:01:04.500Z
Thanks for your answer! It's left me puzzled but in a good way, now I'll have to figure out what this all means :-)
uhoh
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2024-06-28T15:55:50.927Z
2024-06-28T16:26:46.623Z
2024-06-28T16:26:46.623Z
eshaya
<p>The Fredholm equation of the first kind (<a href="https://mathworld.wolfram.com/FredholmIntegralEquationoftheFirstKind.html" rel="nofollow noreferrer">Wolfram Mathworld</a> and <a href="https://en.wikipedia.org/wiki/Fredholm_integral_equation#Equation_of_the_first_kind" rel="nofollow noreferrer">Wikipedia</a>) written as (Wolfram's):</p>
<blockquote>
<p><span class="math-container">$$f(x) = \int_a^b K(x, t) \ \phi(t) \ dt$$</span></p>
<p>where <span class="math-container">$K(x, t)$</span> is the kernel and <span class="math-container">$\phi(t)$</span> is an unknown function to be solved for (Arfken 1985, p. 865).</p>
<p>If the kernel is of the special form <span class="math-container">$K(x - t)$</span> and the limits are infinite so that the equation becomes</p>
<p><span class="math-container">$$f(x) = \int_{-\infty}^{\infty} K(x - t) \ \phi(t) \ dt$$</span></p>
<p>then the solution (assuming the relevant transforms exist) is given by</p>
<p><span class="math-container">$$\phi(x) = \int_{-\infty}^{\infty} \frac{F_x[f(x)](k)}{F_x[K(x)](k)} \ \exp(2 i \pi k x) \ dk$$</span></p>
<p>where <span class="math-container">$F_x$</span> is the Fourier transforms operator (Arfken 1985, pp. 875 and 877).</p>
</blockquote>
<p>This I can understand; the Fourier transform of a convolution of two functions is just the product of their Fourier transforms. The <span class="math-container">$\int_{-\infty}^{\infty} \exp(2 i \pi k x) \ dk$</span> is just the inverse Fourier transform, and the fraction comes from moving the Fourier transform of the kernel <span class="math-container">$K$</span> from one side to the other.</p>
<p>To me this looks like it could be a way to solve for the un-diffracted image of a galaxy <span class="math-container">$\phi$</span> starting with the diffracted image <span class="math-container">$f$</span> and say an Airy function kernel <span class="math-container">$K$</span>.</p>
<p>I suppose an array of radio telescope dishes would have a more complicated kernel.</p>
<p>While observational astronomers might under some conditions remove instrumental effects by de-convoluting an image before analysis or fitting something to it, <strong>the alternative would be to <em>convolute the model with the kernel</em> and fit that to the raw data instead.</strong></p>
<p><strong>Question:</strong> Is the Fredholm equation ever used by observational astronomers? Perhaps in solving for/removing instrumental effects in imaging?</p>

CC BY-SA 4.0
2021-12-15T23:39:00.450Z
Have you considered noise in the image...? BTW Convoluting the model with the point spread function and matching to the observations is the standard way to do this
ProfRob
CC BY-SA 4.0
2021-12-15T23:43:12.790Z
@ProfRob only in the last sixty seconds :-) My limited experience in this area isn't enough to tell me if deconvoluting the data is better or worse than convoluting the model because I only fit smooth functions. But Oh! you guys have stars, both real and spurious. Yikes!
uhoh
CC BY-SA 4.0
2021-12-15T23:50:21.420Z
@ProfRob but note that I've also discussed radio astronomy where the data is smoother and the noise not as pixelated.
uhoh
CC BY-SA 4.0
2024-06-28T04:31:33.737Z
Looks like it's being used. The question is how to define a kernel for a particular defect. It would also be interesting to look at numerical simulations of such transformations. [Integral equation models for image restoration: high accuracy methods and fast algorithms](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/85415/ip10_4_045006.pdf?sequence=1&isAllowed=y) [Strategies for the deconvolution of hypertelescope images](https://www.aanda.org/articles/aa/pdf/2012/07/aa17554-11.pdf)
ayr
<observational-astronomy><radio-astronomy><photography><image-processing>
Is the Fredholm equation ever used by observational astronomers? Perhaps in solving for/removing instrumental effects in imaging?
uhoh
# Is the Fredholm equation ever used by observational astronomers? Perhaps in solving for/removing instrumental effects in imaging?

(Asked by: username_0 on 2021-12-15)

The Fredholm equation of the first kind ([Wolfram Mathworld](https://mathworld.wolfram.com/FredholmIntegralEquationoftheFirstKind.html) and [Wikipedia](https://en.wikipedia.org/wiki/Fredholm_integral_equation#Equation_of_the_first_kind)) written as (Wolfram's):

> $$f(x) = \int\_a^b K(x, t) \ \phi(t) \ dt$$
>
> where $K(x, t)$ is the kernel and $\phi(t)$ is an unknown function to be solved for (Arfken 1985, p. 865).
>
> If the kernel is of the special form $K(x - t)$ and the limits are infinite so that the equation becomes
>
> $$f(x) = \int\_{-\infty}^{\infty} K(x - t) \ \phi(t) \ dt$$
>
> then the solution (assuming the relevant transforms exist) is given by
>
> $$\phi(x) = \int\_{-\infty}^{\infty} \frac{F\_x[f(x)](k)}{F\_x[K(x)](k)} \ \exp(2 i \pi k x) \ dk$$
>
> where $F\_x$ is the Fourier transforms operator (Arfken 1985, pp. 875 and 877).

This I can understand; the Fourier transform of a convolution of two functions is just the product of their Fourier transforms. The $\int\_{-\infty}^{\infty} \exp(2 i \pi k x) \ dk$ is just the inverse Fourier transform, and the fraction comes from moving the Fourier transform of the kernel $K$ from one side to the other.

To me this looks like it could be a way to solve for the un-diffracted image of a galaxy $\phi$ starting with the diffracted image $f$ and say an Airy function kernel $K$.

I suppose an array of radio telescope dishes would have a more complicated kernel.

While observational astronomers might under some conditions remove instrumental effects by de-convoluting an image before analysis or fitting something to it, **the alternative would be to *convolute the model with the kernel* and fit that to the raw data instead.**

**Question:** Is the Fredholm equation ever used by observational astronomers? Perhaps in solving for/removing instrumental effects in imaging?

**Tags:** observational-astronomy, radio-astronomy, photography, image-processing
> **Comments:**
> **username_1**: Have you considered noise in the image...? BTW Convoluting the model with the point spread function and matching to the observations is the standard way to do this  
> **username_0**: @ProfRob only in the last sixty seconds :-) My limited experience in this area isn't enough to tell me if deconvoluting the data is better or worse than convoluting the model because I only fit smooth functions. But Oh! you guys have stars, both real and spurious. Yikes!  
> **username_0**: @ProfRob but note that I've also discussed radio astronomy where the data is smoother and the noise not as pixelated.  
> **username_2**: Looks like it's being used. The question is how to define a kernel for a particular defect. It would also be interesting to look at numerical simulations of such transformations. [Integral equation models for image restoration: high accuracy methods and fast algorithms](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/85415/ip10_4_045006.pdf?sequence=1&isAllowed=y) [Strategies for the deconvolution of hypertelescope images](https://www.aanda.org/articles/aa/pdf/2012/07/aa17554-11.pdf)  

---

## 1 answers

### Answer by username_3 on 2024-06-28. Score: 3 | Accepted answer

The integral in the Fredholm equation has limits from $- \infty$ to $\infty$. When one or both of the limits are finite, it is called the Wiener-Hopf function. Astronomers have more use for the latter.

Astronomers commonly use this function as used in general signal processing where it is called the Wiener filter. This Fourier filter minimizes the mean square error between the observed signal with noise and the real signal. The Wiener filter is applied in an iterative process for the deconvolution of images in an algorithm called [Lucy Deconvolution](https://en.m.wikipedia.org/wiki/Richardson%E2%80%93Lucy_deconvolution).

It is also used to help fill in gaps in maps, like to fill in the galactic plane in maps of the microwave background or to fill gaps in the galaxy velocity field when reconstructing the density map from the velocity field.

See [Wiener Deconvolution](https://en.m.wikipedia.org/wiki/Wiener_deconvolution) in Wikipedia. Here it is defined using Laplace transforms, but astronomers seem to prefer using Fourier transforms.

> **Comments:**
> **username_0**: Thanks for your answer! It's left me puzzled but in a good way, now I'll have to figure out what this all means :-)  

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