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Modern telescopes go to great lengths to have perfectly shaped parabolic mirrors. My question is, why go to the trouble of having a perfect mirror? Why not take a mirror roughly the right shape, and then correct for the distortion using computers?

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  • $\begingroup$ What if you "correct" wrong? Ask any professional photographer about whether it's better to have good source shot to work with or correct a blurry, poor-lit, or unfocused image in post-production. $\endgroup$
    – MonkeyZeus
    May 17 at 18:57
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correct for the distortion

An imperfect mirror does not produce a distorted image - it produces a blurry image. With light-field sensors and phase imaging, one could possibly correct for the blur, but it is much more challenging problem than normal lens distortion correction.

Distortion refers to a systematic change in how shapes are projected in an image. It results from a lens or mirror with good, accurate geometry that just does not produce a rectilinear projection.

Random imperfections in a mirror do not cause distortion. Every point in the surface of a mirror contributes to every pixel in the result image. If a single part of the mirror is at slightly wrong angle, it does not cause a distortion in one point of the image. Instead, it projects the same image at a slightly different alignment on the same sensor. (1)

In the case of a starfield, this would cause ghost images of very dim stars to appear next to the real stars. Repeat this for a thousand imperfections, and the result is just blurry dots. Deconvolution is a process that can be used to remove blurriness, but noise and other uncertainties limit its effectiveness.

(1) This may be a bit unintuitive if you think about funhouse mirrors where the image is distorted. Those work differently because they act the part of a planar mirror, where indeed each part of image is reflected by a single part of a mirror. But planar mirrors cannot form an image by themselves, instead the lens in your eye is the critical component of the image accuracy.

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    $\begingroup$ Every spot on a funhouse mirror actually reflects a pupil-sized region of the actual image, which is larger than a single point, but generally small relative to the size of the mirror. In many mirror-based cameras, however, the effective pupil size is often essentially the same as the diameter of the biggest mirror. $\endgroup$
    – supercat
    May 16 at 19:14
  • $\begingroup$ @supercat Thanks for the explanation, I had been wondering if there is a more exact way to describe the difference! $\endgroup$
    – jpa
    May 17 at 4:07
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Also see answers to

The problem is that light imaging detectors convert amplitude to power during the detection process. Phase is lost.

If you had maps of both the magnitude of the electric field and its relative phase, and had this at each wavelength of light, you could do exactly what you propose and correct in software.

In fact, you could then throw away the telescope and image the sky without it.

That's exactly how a radio telescope array works! Dozens of antennas detect waves from a patch of sky, they're all brought into a computer and for each wavelength the source shape is reconstructed from the amplitude and phase information available in the electronic signals.

But CCD detectors or photographic emulsion plates or our eyes convert the photons to other forms of energy, then average it out over time.

Phase is lost, so reconstruction is no longer possible.

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  • $\begingroup$ So why don't optical telescopes use other detection methods that do preserve phase? $\endgroup$
    – Vikki
    May 15 at 22:15
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    $\begingroup$ @Vikki-formerlySean they will some day! Visible light, say 600 nm has a frequency of 500,000,000,000,000 Hz (500,000 GHz) which is not manageable with normal electronics. In order to look at many wavelengths at the same time rather than an incredibly sharp 1 GHz wide optical bandpass, you need to digitize at this absurdly high rate and process. Basically, if the detector preserved the phase, you'd then have to either record the phase in real time or process it. If you had an extremely narrow bandpass filter (0.001 nm) and could cut the spectrum down to GHz, you could down-convert it to $\endgroup$
    – uhoh
    May 15 at 22:25
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    $\begingroup$ @Vikki-formerlySean an electronic signal in GHz range and then process normally. That's roughly the kind of thing I'd asked about in Has optical interferometry been done at radio frequency using heterodyning with a laser in a nonlinear material? $\endgroup$
    – uhoh
    May 15 at 22:26
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    $\begingroup$ @Vikki-formerlySean There's work on this, but it's hard: scientificamerican.com/article/… $\endgroup$
    – John Doty
    May 16 at 2:23
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    $\begingroup$ @asdfex there is no such thing as completely coherent nor completely incoherent light. It's a continuum; lasers are a lot more coherent than light bulbs, but light bulbs have non-zero coherence and lasers are not perfectly coherent. Then you have to address longitudinal and transverse coherence separately. There is no such thing as "normal light". There's just photons. As long as you read through what I wrote carefully and note the filter bandwidths I specified in each case, you absolutely can heterodyne filtered light with a laser and down-convert. $\endgroup$
    – uhoh
    May 16 at 18:20
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The image which is recorded by CCDs are the convolution of the true image and a point spread function. If the PSF is not a nice function, such as this one, for example [source]:

enter image description here

then (I think) it is hard to deconvolve the detected image from the PSF to get the true image. For example, you don't necesserily know the exact PSF if the telescope is only "roughly" the right shape, so you don't know what to deconvolve.

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  • $\begingroup$ it should be possible to determine the PSF, if you know what the result should look like. A laser pointer shined from an airplane should suffice as a calibration pattern? $\endgroup$ May 16 at 16:49
  • $\begingroup$ I think you are probably right. $\endgroup$
    – zabop
    May 16 at 19:58
  • $\begingroup$ Deconvolution is extremely error-prone operation, any noise will corrupt resulting data exponentially. It can be somewhat done in space, where there are no air disturbances and when mirror temperature is kept stable; on Earth this is not viable solution. Sure deconvolution is used in astronomy image processing to remove diffraction and other image distortions, which cannot be avoided by making better mirrors :) $\endgroup$
    – Arvo
    May 18 at 8:59
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    $\begingroup$ This image shows the problem very well - one of the main measures taken from these images is the amount of light received from the star (it's brightness). As can be seen in the image, that light has spread over a wide area and so it has to be summed up to get the total. However, you can also see it is now covering other objects and separating out which light came from which object is hard and introduces uncertainties. In a perfect world with perfect mirrors and no atmosphere, each object would be contained in a single pixel and so easily measurable. $\endgroup$ May 18 at 10:25
  • $\begingroup$ @JonathanTwite In the perfect world you describe there would still be point spread functions, because the diffraction limit of a finite-sized mirror. So each object would not be "contained in a single pixel" (unless you made your pixels absurdly large, in which case you'd have many different objects in each pixel...). $\endgroup$ May 20 at 10:47
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The short answer is that blurring destroys information, and no amount of correction after the fact can bring it back.

A slightly less short answer: As others have mentioned, the blur caused by imperfections in the mirror is a convolution of the true image with the "point spread function" created by the imperfections.

Convolution of the image is mathematically equivalent to multiplying the Fourier transform of the image with the Fourier transform of the point spread function. Wherever the transform of that point-spread function is zero or close to it, the image information at the corresponding frequencies is destroyed.

The point-spread function created by random imperfections looks like a Gaussian distribution. The Gaussian distribution has the interesting property that its Fourier transform is also a Gaussian, centered on 0-frequency with bandwidth inversely proportional to the width of the point-spread function.

In other words, the point-spread function is a low-pass filter. It destroys the high-frequency information in the image, where all the fine detail lives. The wider the point spread, the worse it is.

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They do corrections in computers. But it's not a perfect thing. Just how you can't tell the difference between 1 + 2 + 1 and 1.3 + 1.4 + 1.3, the equations we solve don't have just one solution. They have many. So we have to do guesses in the algorithm. Given that science is exploring that which is just outside of our knowledge, its useful to get actual raw data rather than having to manipulate it first.

I think my favorite example of this is the famous black hole picture from 2019. It was, of course, a computer reconstruction of the data, using software from a team headed by Katie Bouman. She has a wonderful TED talk on the technology where she points out the challenges with biasing algorithms to detect what we expect to see. One of the tools they relied on to gain confidence that the algorithm wasn't just showing them what they expected to see was to train the algorithm on pictures of her friends and faculty. If the resulting image processed after training on these pictures was basically the same as the one from generated from processing after training on pictures of known stellar objects, they were reasonably confident it wasn't bias.

But its always easier to just have better raw data.

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  • $\begingroup$ Yes, that's done with radio astronomy; just as I described in my answer. But the question was about correction optical mirrors and at 500,000 GHz, so the analogy to radio astronomy doesn't work. $\endgroup$
    – uhoh
    May 16 at 18:27
  • $\begingroup$ @uhoh The equations may be different, but if computerized sharpening could prevent the need for quality measurements, then we would have certainly replaced our telescopes with computers. It's still a game of imperfect information. $\endgroup$
    – Cort Ammon
    May 16 at 18:32
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    $\begingroup$ 1+3+1 = 5. 1.3+1.4+1.3 = 4. I think I can tell the difference. $\endgroup$ May 17 at 16:01
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    $\begingroup$ @WaterMolecule ... I need to stop posting math when I'm sleepy! Fixed! $\endgroup$
    – Cort Ammon
    May 17 at 16:07
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    $\begingroup$ Minor nitpick: there were actually four different teams (one with Katie Bouman) working independently on the reconstruction of the M87 black hole image, testing different methods. The final result was the combination of all four teams' work. physicstoday.scitation.org/do/10.1063/PT.6.1.20190411a/full $\endgroup$ May 20 at 10:52
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In situations where one is using a camera to look at a brightly glowing or brightly illuminated object at relatively short range, and one can observe the object for a long time without it changing, the total amount of information present in the incoming photons will be much greater than the amount of useful information one would want in a final image. Using more photons will make it possible to reduce the level of noise in a picture, but halving the amount of noise will require roughly quadrupling the number of photons to be collected. When photons are plentiful, it's possible to produce images with far less noise than when photons are scarce, but reducing noise levels requires having a lot of photons.

It's possible to construct a digital filter to undo the effects of precisely modeled optical defects or limitations that would cause images to be blurry. Unfortunately, no matter how perfectly such a filter is constructed, using one to sharpen blurry images tends to amplify noise as much as it would amplify useful signal. The more photons one can exploit to reduce noise, the less of a problem this will be. Unfortunately, in astronomical photography, it's necessary to make due with a relatively limited number of photons. While it might theoretically be possible to use a suitably constructed digital filter to sharpen images when performing astronomical photography, the purpose of using large mirrors is to collect as many photons as possible, and much of the reason for doing that is to minimize noise. If use of a filter would double the amount of noise, and the increased noise wasn't acceptable, one would need to increase the mirror area by a factor of four to collect enough photons to compensate. Conversely, if one would be willing to tolerate a factor-of-two increase in the amount of noise present in an image, that would imply the mirror was four times as big as needed to meet the noise targets.

In short, when photons are scarce, trying to use digital sharpening filters to correct blurriness caused by imperfect optics may be possible, but achieving a certain image quality with imperfect optics would require increasing the size of the optics so much that it's less practical than simply using higher quality optics to start with.

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  • $\begingroup$ I believe this answer is beside the point. The optics determine the image. But sharpening a blurry image will never and under no circumstances reveal more details which are already lost in the optics (it may make better visible small differences otherwise overlooked - but not bring back those which are not there). One may try to rescue some part with applying the inverse PSF, but that only goes so far $\endgroup$ May 17 at 10:46
  • $\begingroup$ @planetmaker: If one had absolutely precise (infinite resolution) measurements of the optical defect, and absolutely precise measurements of the image produced with the defective optics, and absolutely precise knowledge of the distances to the object being observed, then one could reconstruct the original image perfectly. The difficulty is that small uncertainties in one's measurements of the defect, the image, or--for nearby objects--the distances, often translate into larger uncertainties in the reconstructed image. Astronomical distances are large enough that uncertainties there... $\endgroup$
    – supercat
    May 17 at 14:43
  • $\begingroup$ you describe the knowledge of the PSF in your comment which then can be used to deconvolve the obtained image with. Your answer only talks about sharpening the obtained image and dealing with noise - both of which will do no good to improve the data quality. $\endgroup$ May 17 at 14:48
  • $\begingroup$ ...don't really matter, but uncertainties in the observed images are significant even before one tries to correct for optical defects. Sorry my answer failed to properly describe the type of filter I had in mind. I'll edit it and let you know when I'm done. $\endgroup$
    – supercat
    May 17 at 14:49
  • $\begingroup$ @planetmaker: Okay, tell me if that's better. My intended point was that there are times when such filtering may allow one to extract more visual information from a scene than could be resolved directly with one's optics, but the number of photons one would need to grab to achieve a given level of detail would vastly exceed the number of photons one would have to grab if using better optics. Consider, for example, using a camera whose sensor resolution is four times the highest special frequency of interest in an object which is "at infinity", but the focus... $\endgroup$
    – supercat
    May 17 at 15:01

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