olliess
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Senior Member
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Posts: 1,349

Re: The whole question of lens sharpness...

Alphoid wrote:

My comment was, specifically, any blur caused by the lens. Diffraction, limited depth-of-field, antialiasing filter, resolution limits of the sensor, etc. are not caused by the lens. You would have those even with an ideal lens.

Of course diffraction and defocus blur are "caused" by the lens" (even an ideal lens). They are certainly a motivation for much of the real-world work on image deconvolution, and I would argue that a "perfect" characterization of the PSF would have to include diffraction at the very least.

Let's leave things which transform the focal plane (tilt, field curvature, and the like) out of the discussion for now. Those one cannot correct for, but it's not clear whether they are of any relevance to sharpness in photography (subjects are rarely flat).

This would also seem to imply that subjects are rarely contained exactly in the plane of focus, hence defocus blur would be relevant after all. But I agree, let's leave some of the extra complications aside for now.

You will get back, in your notation, O(x,y)+(P^-1 * N)(x,y). This is the exact original image, with a transformed version of the noise. In practice, you end up increasing high-frequency noise, but with a typical lens, to an extent that does not matter at low ISO. Note that this, specifically, undoes any blur caused by the lens, as in my original claim.

Yes, in principle you have removed some of the blur caused by the lens, but you have also added an unknown amount of extra noise. Since you cannot show unambiguously which new information is restored signal and which is added noise, it's kind of a bogus claim. It would be like claiming you've successfully removed all of the cruft from the ceiling of the Cistine Chapel, but you just aren't sure how much of Michelangelo's paint you've removed in the process.

Even you take away the noise, then you're still left with something that looks just like the 2-d heat equation. Thus if you are guaranteed an unique inverse for the blur problem, then it seems to imply that you are also guaranteed solutions to the backward heat equation.

This is not correct. The systems are sort-of-similar in that they sort-of-blur things, but the math is different (hence, my counterexample). If one is not invertable, it does not guarantee that the other is not.

Well, since both inverse problems are ill-posed for essentially the same mathematical reasons, I'm not sure how you can claim this.

That said, I do believe (and the key word is believe -- we're slightly outside of my domain of expertise for heat equation)

If you are outside your domain of expertise (what is your expertise btw, if you don't mind my asking), then why are you so sure that the problems are mathematically different?

that the 2d heat equation is invertable (and a quick Google search brings up papers which believe the same).

A quick Google Scholar search brings up papers on both topics which immediately discuss why both the backwards heat equation and the image deconvolution problem are fundamentally ill-posed, and then talks about all the clever ways people are trying to come up with clever (and practical) workarounds.

How does entropy relate? I believe you have a misunderstanding of entropy. If you give me an exact state of any physical system (classical or quantum -- but I'll ignore quantum for the purposes of this discussion, since it will only complicate things), I can simulate it backwards to get the state at any point in the past.

And what the ill-posedness of the backward heat equation, negative diffusion processes, and image deconvolution tell you is that in general you may not be able to find solutions to simulate backwards in time.

Thermodynamics just tells me that I cannot physically bring it back to that state without increasing entropy elsewhere. Total entropy in the world increases.

You may find interesting some of the discussion in the literature about the relationship between the ill-posedness of the backwards heat equation (and other similar processes) and the arrow of time.

Anyway, let me respond to the more specific points about the edge effects in a following post, because the topic is somewhat separate from the discussion above.