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rjlittlefield Site Admin
Joined: 01 Aug 2006 Posts: 20649 Location: Richland, Washington State, USA

Posted: Wed Apr 01, 2020 12:10 pm Post subject: How many photons are available to make a white pixel? 


This post is prompted by the simultaneous occurrence of two events:
1. I've recently been rereading the section of "Animal Eyes" (Michael F. Land & Dan Eric Nillson, 2nd edition, 2012) that discusses limitations imposed by available photons.
2. Recent discussions here in the forum have touched on signal noise as an important limitation in image reconstruction.
The thrust of this post is simple: I'm going to calculate how many photons are available to make a white pixel, for a 40 megapixel fullframe sensor at ISO 100.
(Spoiler alert: the answer turns out to be only about 100,000 available, 10,000 captured.)
The method is pretty simple too. Imagine a 1 meter x 1.5 meter matte white target, in full sun, filling the fullframe sensor as seen through a lens set on f/16, and making an exposure of 1/100 second ("sunny 16 rule" for ISO 100). From the geometry, calculate the fraction of photons that are reflected from the target and delivered to the sensor. Multiply together the total number of photons falling on the target, by the fraction delivered to the sensor, and divide by the number of pixels. The result is the number of photons per pixel.
Then the calculation goes like this:
target width: 1.5 meters
sensor width: 36 mm
focal length (arbitrary, doesn't matter in the end): 200 mm
distance from lens to target: 200 mm * 1.5 meters / 36 mm = 8.33 meters
area of hemisphere surrounding target at that distance: 2 * pi * (8.33^2) = 436 square meters
fnumber: 16
diameter of aperture: 200 mm / 16 = 12.5 mm
area of aperture = pi/4 * (12.5^2) = 123 square mm = 123e6 square meters
fraction of photons delivered to sensor: area of aperture / area of hemisphere = 123e6 / 436 = 2.82e7
photons per square meter per second: visible light power density 400 W/m^2 = 400 J/sec/m^2, divided by green photon energy 3.8e19 J = 1.05e21
(Animal Eyes, pg 27, says "10^20 photons per second per square metre". I'll go with the larger number.)
exposure time: 0.01 second
total area of target: 1.5 meters squared
photons delivered to sensor: 1.5 * 1.05e21 * 0.01 * 2.82e7 = 4.44e12
pixels per sensor: 40e6 (by assumption)
photons available per pixel: 4.44e12 / 40e6 = 111,000
The number of photons actually captured by the sensor will be about 10X less, due to the Bayer filter (1/3 per channel) and the roughly 35% quantum efficiency of CMOS sensors.
If somebody would care to check this calculation, I'd be delighted.
Rik 

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Lou Jost
Joined: 04 Sep 2015 Posts: 4264 Location: Ecuador

Posted: Wed Apr 01, 2020 1:35 pm Post subject: 


If that's right, then a 12stop dynamic range is about the most you could expect from that sensor, because at the dark end of the range, you'd be registering a single photon per exposure. _________________ Lou Jost
www.ecomingafoundation.wordpress.com
www.loujost.com 

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rjlittlefield Site Admin
Joined: 01 Aug 2006 Posts: 20649 Location: Richland, Washington State, USA

Posted: Wed Apr 01, 2020 1:59 pm Post subject: 


Lou Jost wrote:  If that's right, then a 12stop dynamic range is about the most you could expect from that sensor, because at the dark end of the range, you'd be registering a single photon per exposure. 
Yep, give or take a couple. You need 14 bit samples to count beyond 8192, so maybe you could argue that it's 13 or 14 stops.
Anyway, in all cases because of shot noise (Poisson distribution of detections) the standard deviation for individual samples is equal to the square root of the sample value. So if you fill up 14 bits, the bottom 7 or so are just noise. Recognizing that fact, a better way to allocate the bits is to take the square root of the sample value and encode that as a binary integer, which reduces the noise level to a constant ~1 in the LSB. This is the largely unrecognized reason why 8bit color with gamma ~2 actually works very well, up to the point that you start levelsadjusting the values and introducing gaps in the histogram.
Rik 

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pierre
Joined: 04 Jan 2010 Posts: 258 Location: France, Var, Toulon

Posted: Wed Apr 01, 2020 10:54 pm Post subject: How many photons are available to make a white pixel? 


Thanks for your time Rik ! _________________ Regards
Pierre 

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mawyatt
Joined: 22 Aug 2013 Posts: 2452 Location: Clearwater


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rjlittlefield Site Admin
Joined: 01 Aug 2006 Posts: 20649 Location: Richland, Washington State, USA

Posted: Fri Apr 03, 2020 1:03 pm Post subject: 


Thanks for the comments and links.
Another helpful reference is to the blog post by Jack/Almaphoto at https://www.strollswithmydog.com/photonsonapixel/ . (Thanks for that link, Pierre!)
That one summarizes his example by saying that for his D610 (24 megapixels fullframe), at base ISO (100), the green channel clips first, at an exposure that he calculates as 411,600 photons.
I always like to figure out why different methods give different results. In this case one big factor is the pixel size, his 24 MP FF versus my assumed 40 MP FF. Making that correction means that his method on my sensor would predict 246,960 photons per pixel at 40 MP.
A second big factor is that I incorrectly modeled reflectivity. My calculation overlooked that a matte reflector actually has a Lambertian reflectance profile, which is centerweighted by 2X compared to the type of uniform reflectance that my model used. Making that correction takes my number from 111,000 to 222,000 photons per pixel. That's close enough to Jack's number for my pixel size that I'm no longer looking for anything wrong in the calculation.
https://www.strollswithmydog.com/photonsonapixel/ also measures/calculates what he calls an "Effective Quantum Efficiency" (eQE) that bundles together all of the factors that can cause only some photons to produce electrons. The number that he gets for his D610 is 18.1%, versus the roughly 12% that I would have gotten from 1/3 of 35%.
So, at this point it seems that my initial numbers were too low, by something around 2 fstops.
That's significant, but not enough to change the main point that shot noise from the fundamental issue of sampling available photons is a big deal, reducing even the best pixel values to something like 8 bits of significance  16 bits total count, with the bottom half of those being noise.
Rik
Last edited by rjlittlefield on Fri Apr 03, 2020 1:14 pm; edited 1 time in total 

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Lou Jost
Joined: 04 Sep 2015 Posts: 4264 Location: Ecuador

Posted: Fri Apr 03, 2020 1:11 pm Post subject: 


Rik, weren't you looking at the MINIMUM number of photons to get a white pixel, while StrollsWithMyDog was looking at the clipping value? _________________ Lou Jost
www.ecomingafoundation.wordpress.com
www.loujost.com 

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rjlittlefield Site Admin
Joined: 01 Aug 2006 Posts: 20649 Location: Richland, Washington State, USA

Posted: Fri Apr 03, 2020 1:31 pm Post subject: 


Lou Jost wrote:  Rik, weren't you looking at the MINIMUM number of photons to get a white pixel, while StrollsWithMyDog was looking at the clipping value? 
What I'm looking at is the number of photons for a white subject, when "properly" exposed at base ISO.
It seems to me that's more like maximum, not minimum. Much lower numbers of photons could be turned into white by using higher ISO or stretching the histogram.
The difference between StrollsWithMyDog's model and mine is more like headroom  the difference between nominal white on the histogram and the somewhat larger value where the sensor completely saturates.
Rik 

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