Image reconstruction

The concept

Image reconstruction

What happens when an optical system projects an image of a source?

Divide that source up into $N$ pixels. They need not even be arranged in a rectangular pattern. Put them in a spiral of that suits your fancy.

Now look at the projected image and sample it at $N$ different places. Every one of these samples will represent some combination of light from the original $N$ pixels.

If the optical system was "perfect" (i.e., idealized geometric optics) and the grid of pixels at the source and the image were a perfect match, the mapping would be 1:1. Each pixel in the image would get light from exactly one source pixels. The image would be a flawless, sharp image.

But optical systems are not like that. If for now other reason, it's because of the wave nature of light, but they blur the image. This is especially true for the SGL: it is a very blurry lens.

So each of those $N$ image pixels contain a combination of light from all the $N$ source pixels. Do we know what combination? We do! We have a robust mathematical description of the SGL as a lens, so we know exactly how it combines light at various wavelengths.

That means that for the $N$ source pixels, we have a (linear, actually) system of $N$ equations. This is solvable in principle!

And this is what image reconstruction amounts to: "devoncolution", applying the transformation performed by the lens in reverse to the image, to recover a sharp version of the source.

There is a very important caveat, however. If we picked the wrong pixels, the system of equations may be degenerate or nearly so. This means that when we do deconvolution, even tiny errors become significant. The practical consequence is that deconvolution amplifies noise at the expense of signal.

This is bad news, because we have large amounts of noise present: the light we see filters through the much brighter solar corona. The noise added by the corona can only be mitigated by collecting more light. This requires more time, which is problematic considering that our target is a moving target, a moving exoplanet. In any case, we want to get meaningful results in weeks, months, or maybe a couple of years, tops, not decades, centuries or millennia. For this reason, it is very important for us to understand the magnitude of noise and plan our observations accordingly.











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