I hesitate to speculate on whether it would make the images blurrier or noisier. I don't see why either would be the case. Mathematically, there must exist a perfect inverse filter that would correct the image without any distortion at all, right? Or is that wrong? (Set aside for the moment the the inevitable fact that we could at best only approximate it and it might be extremely compute-intensive.) Wouldn't you be able to produce images without visible spikes that are very nearly as good, if not better than the raw images?
It might require sending precise photon counts for each pixel to Earth though. I expect any compression algorithm is going to cement the distortion in a way that's mathematically unrecoverable.
Well, the photon counts themselves are noisy, Poisson distributed. When you apply a de-convolution filter (to undo any kind of blurriness / light not ending up where it should but near where it should), in the frequency domain this is a high pass filter, it amplifies high frequencies i.e. shot noise. This limits how much you can "enhance" out of focus images, for example.
As far as computational issues go, light is linear, the sensor should be calibrated to be linear, so assuming the sensors are not saturating or the like, the effect of the aperture is a convolution (assuming a single spectral band for simplicity). In frequency domain, that is a multiplication. So you convert the image to the frequency domain, and multiply with the inverse of that, and convert back to spatial domain. That is your basic de-convolution. Not very computationally expensive. Also, doesn't work right if the sensor is saturating (that would break linearity). That can be done as quickly as 2 fast Fourier transforms.
If the sensor is saturating, I guess the solution would be to treat it as a nonlinear optimization problem, solve for the "true" image given the model of those diffraction spikes, and clamping. Also shouldn't be too computationally expensive, probably no more than minutes per image on a GPU. You could just use (slightly nonlinear) least squares. I think it should also increase noise because it would still be fundamentally a high pass filter.
There's more clever things you could do accounting for the noise, you could e.g. compute the most probable input image given some prior probability distribution over input images and photon noise. That tends to be done with neural networks these days, because neural networks are able to represent a prior probability distribution over images. However that would not be very useful scientifically because it could outright hallucinate details that aren't there.
edit: I don't expect the results to be good, though. The parts of image obscured by a diffraction spike have the shot noise from the light of the obscured object, and the shot noise from the diffraction spike. That intrinsically results in loss of information.
The standard deviation of Poisson distribution is proportional to the square root of the mean. Let's suppose you have mean of 100 photons from the distant object on a pixel, plus mean of 800 photons from the diffraction spike. The total is 9x greater, the noise is 3x greater, that noise will stay there even if you subtract the mean value of the spike. In this example instead of standard deviation of something like 10% of the value as it would be without the spike, you get 30% where the spike was subtracted. Looks noisier visually.
