I did read it earlier - although I struggled to understand some of it. What I took from it was that we agree (a) if the lens and sensor are balanced the whole setup is more effective than a great lens and poor sensor or poor lens and great sensor. (b) There is a point where we could add more pixels and get an improvement
but the improvement is too small to justify doing so.
I actually made a stronger statement than that.
But the area where we agree is weaker than each of us has said.
I don't understand your point about agreement. I don't see any agreement here. Don't confuse issues about reconstruction techniques with the correctness of the Nyquist sampling theorem. Everything I've said about that theorem is true.
OK. Let's do this simple thought experiment.
There is a pattern of light and dark falling on the sensor, it
does follow a simple wave form. 3 cycles from darker to lighter and back over the full width the sensor.
Nyquist tells you how many samples you need to record those three cycles without aliasing. 3 cycles per image width so Nyquist says sample 6 times. So a sensor six pixels wide will do the job. Definitely no aliasing, just as Nyquist tells us.
Now are we seeing any problem with printing the our six pixel wide image ?
I said you'll never get the reproduction perfect (except in the purely theoretical case of a perfect sensor.) You rocked up with "Infidel How dare you question the prophet Nyquist". And we've wasted hours ever since. Respectfully, if anyone has conflated reconstruction techniques and sampling theorem, it wasn't me. What have I said is Nyquist tells you six samples covers three cycle signal, with no aliasing, but doesn't tell you can use a six pixel wide sensor to reproduce a 3 cycle image
perfectly.
After that point it's "how many angels can dance on the head of a pin" territory ; I'm in the "you can always fit another angel on" camp,
Please provide your reasoning, and why Nyquist and Shannon were wrong. Remember: “Extraordinary claims require extraordinary evidence.”
To the best of my limited understanding they were correct about the number of samples needed to avoid aliasing when a sampling a signal for a given frequency.
Then why do you insist that finer pitch is better without regard for the lens, without limit?
Imagine I had the worst possible lens that could only resolve 3 line pairs per image width and work from there.
Now... if I wanted to check that loaves coming out of a bakery were the correct weight, the only way to be sure every loaf was the correct weight is to weigh every single one. But we can weigh some proportion and have a confidence level that all are OK.
If someone told me that loaf weight is function
I don't know what "loaf weight is a function" means.
and Nyquist proves that loaf weight can be perfectly assessed testing a subset of loaves, I would protest that's false. I can't even offer a Fermatesque 'I can prove it, but not in the space I have'. Much less to the person who cites Fourier and the rest to prove that they can't possibly be wrong.
You have constructed a situation to which the Nyquist sampling theorem does not apply, and now you're complaining that it doesn't apply.
No I'm not complaining, I'm saying that here is a problem domain, it relates to sampling, and if someone quoted Nyquist we'd all say they were wrong.... out of curiosity do you read to the end first ?
Or another area I work with, queues; jobs arrive at random (with a probability function) and they take a random time to service (again with a probability function). I can draw a chart of the queue length. Does Nyquist tell me how often I need to check the queue length to say what the mean and median length are over a day ?
That's just silly.
Indeed. But is a photograph more like a wave form in a signal (where Nyquist applies), or more like the chart of my queue length (where it would be silly). Because if you can show me it's the former they'll be a thack on my forehead and shout of DOH! that can be heard for miles.
There are lots of these sampling problems. If I want to know the RPM of an engine I'd expect Nyquist to help with sampling frequency. If I want to know the amount of fuel used over a race, Nyquist doesn't tell me how often to check the flow rate. I just know if I check often enough the error will be too small to worry about.
So there are cases where Nyquist applies, and where he doesn't, yes ? I think I a lot of questions we want to answer in photography fall into the "Not well formed with respect to the sampling theorem" - I can't find the words to quote verbatim. The continuously varying grey scale was one. The "how well can we separate hairs as they go out focus"
I already posted a analysis of a continuously varying gray scale. You said it doesn't have frequencies associated with it. I posted a Fourier analysis that showed that it did.
I either missed, or couldn't understand the fourier analysis but posed the question to someone else. If the frequency is very, very low, say one cycle is two sensors wide, do we get perfect reproduction with one sample per sensor ?
Proofs of where Nyquist does [not] apply? Not sure anyone has anything useful to add to the discussion. If you could prove he applies here I wouldn't understand it.
So your reaction to your unwillingness or inability to understand the sampling theorem, which requires an understanding of the frequency domain, is to reject its conclusions based on what?
As I said right at the outset: we end up in a dialogue of the deaf. You insist that the sampling theorem tells you a number of pixels required to guarantee perfect reproduction of any possible image. When I say such a thing is not possible, and its a misuse of correct theorem to say it is, you bluster to try to suggest I'm saying the theorem is wrong in general. I'm not.
while others will say there is expertise (not specifically about angels or pins) which can be used to give a number.
One is right and one is wrong. This is not something that can be settled by opinions.
OK, how many angels can dance on the head of a pin then ?
The question is not whether the Theorem is right or wrong, but whether a photograph is, in practice, a signal with a set of frequencies for which the sole question is aliasing.
You keep moving the goalposts. No one ever said the sole question was aliasing. But that's the question that applies to your assertion about needing to sample infinitely finely to get all the detail captured that a lens can lay down on the sensor.
OK. Does the theorem tell us (a) The number of samples required for perfect reproduction of the image. or (b) The number to prevent aliasing.
Last try on the infinite-resolution (or better expressed "perfect") thing.
In your terms, think of sampling a square wave. It's 100 or 0 , and what I know as the "duty cycle" is 50% (in case I'm using the term wrongly, 50% of the time at 100, 50% of the time at 0)
The frequency is 1Hz.
You sample at 2Hz and each sample is the average value over 0.1s
Now... there is a chance that the transition comes in during your 0.1 seconds, and if it is exactly in the middle you get 50/50/50/50 for all your samples. if it is a bit offset set you might get 70/30/70/30 and if it misses the boundary you get 100/0/100/0 . Simple enough to calculate the average.
Then we reduce the increase the duty cycle to 90% (so 90% at 100 and 10% at 0). So there is a high probability that all samples all fall in the 90% and we get 100/100/100/100. The pulses are happening at 1Hz but when you decompose the signal into its constituents it's not a simple 1Hz signal. The only way to detect the shorter time at 0, reliably is to sample more frequently.
And then we go to a duty cycle of 99% and so on....
Even if we make our samples 0.5 seconds at 99% we're going to see 100,98,100 which is too small a difference to count as resolved.
Now if you think of the 100% times as the stars in the example we used before and 0% times are the gaps, between as the gaps get smaller we need more samples to detect them, regardless of how far apart the stars are. As the gap tends to zero the number of samples to reliably detect it tends to infinity , and If the boundary between resolved and not resolved in the image is when the gap is zero width...
I'm tired. I'm grumpy. And I've had people shouting at me for things I haven't done while wasting your time and my own. If you have a kindergarten level explanation of why this is wrong, thanks. If we can metaphorically shake hands and say we've been talking at cross purposes because I was talking about something other than what you thought I was smashing. Otherwise you can find your own words for "James, sad to say you're too stupid to understand your own stupidity" and save yourself pages of proof.
