Alphoid
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Veteran Member
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Posts: 5,333

Re: The whole question of lens sharpness...

olliess wrote:

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),

For the purposes of this discussion, for signal/image/audio processing, my expertise goes well beyond everything we've talked about. I can make statements with very high levels of confidence there. I have a basic level of knowledge about thermodynamics -- on the order of what a physics major would learn in an undergraduate course on thermodynamics and statistical physics, as well as a scattering of more advanced relevant topics from graduate-level physics (Lagrangian mechanics, quantum mechanics, etc.). I have a deep understanding of concepts like entropy, but I do not have the applied experience of e.g. a numerically trying to invert the heat equation. You can assume a level of knowledge about information theory somewhere in between the two, and sufficient for this discussion (I'd actually love to see you try to make the information theory entropy argument you alluded to; I believe I could take that one apart).

Yourself?

then why are you so sure that the problems are mathematically different?

- I can read math. I can come up with counterexamples, and places where intuition for one differs from the intuition for the other. I gave one example, but there are many others, and in both directions. The level of 'blur' caused by solving forward a heat equation is much greater than you would find in any sane optical system. The steady-state solution of a heat equation is a harmonic function. Harmonic functions are painfully hard to invert/extrapolate from (I'd encourage you to try to find something resembling a harmonic function in an optical system). There's the octagon example I gave before. I can give a hundred others.
- You are making statements based on the heat equation which do not apply to image deconvolution.

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.

The image deconvolution problem is not ill-posed. It becomes ill-posed in a few circumstances:

- Unknown PSF (blind de-convolution)
- Extreme levels of blur (band-limited PSF, zeros in the PSF, or nearly band-limited).

In addition, it becomes impractical if there are high levels of noise, either from the sensor or from quantization. Here, the proper term isn't 'ill-posed,' but that's a technicality. If you give me a image taken with a modern camera sensor and a modern lens at ISO100, you will not have too much noise or quantization, and the level of blur will not be such that you are just missing information.

If you give me an image where you're trying to correct for unknown atmospheric blur, or a $37 million space telescope with an incorrectly ground lens, a Holga, or an ISO6400 image, all bets are off. That's where the research papers kick in that talk about ill-posed problems, and much fancier algorithms.

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.

You were about to get to how entropy fit in.... I'm still waiting. So far, the best I've gotten is to use it as a fancy word to mean 'time.'

1) The operation of masking the image with a fixed frame (e.g., a rectangular windowing) is equivalent to convolution in the frequency domain. Since the Fourier transform of a rectangle has infinite support, some of the variance below the Nyquist limit must be spread beyond the Nyquist limit. Do you agree?

I am not sure. I don't believe I agree, but I think I may be misunderstanding what you're trying to say. There are several things which I'm finding ambiguous (e.g. What operation in the optical chain corresponds to 'masking the image with a fixed frame?'). Can you write out the above being slightly more verbose/specific?

The lens blurs the original image; let's assume for simplicity it's exactly a convolution with a fixed PSF. Since the frame you capture is finite, some information that should have been within the frame has now been blurred outside the frame and is lost. Also, sources that should have been completely outside the frame can contaminate the captured frame.

In the spatial domain, your captured image (I) is the convolution of the PSF (P) and the original image (O), windowed by a rectangular window (R):

I = R . (O * P).

I agree with the mathematics. I am not trying to correct all errors caused by the lens -- just the blur -- not the contrast. Loosely defined, sharpness can be thought of as (depending on who you ask):

- The 10%-90% rise time of the PSF.
- The -3dB of the lens' MTF plot
- Etc.

For any modern lens, the part of the PSF responsible for the blur is a just a few pixels. I can take an inverse of the PSF, approximate it as a fairly small FIR filter. After filtering the image, I can crop the few pixels around the edges -- little enough as to not be noticable.

As an aside, the PSF for the Hubble Space Telescope was not particularly small.

True that. I would not be making the same claims in an astronomy forum. There, the related problems do, indeed, run into numerical instability.

Furthermore, no matter how "small" (in physical units) your PSF is, it will become relatively "large" with increasing resolution.

True too. I might stop making these claims when in a decade, when we see 200MP sensors in our cameras.

To sort of repeat the argument made in my previous post, it's meaningless to say you've "undone" the lens blur when you've also added an unspecified amount of noise which may or may not obscure the gains of your inversion.

Key is 'may or may not.' If you're base image is at a reasonable ISO, you're using a lens made in the past decade that cost more than $100, and a modern sensor with a reasonable number of bits, and generally the type of equipment and settings you'd see people in this forum using for the vast majority of their photos, this falls clearly in the 'may not' camp.