Photon Noise
If you’ve ever taken a dim/underexposed photo you may have noticed you can brighten it in photoshop but the quality never matches that of a photo that was properly exposed. This can be seen below, the images are both bright but the one on the right was exposed to 50 times more light.
So what’s the difference between the properly exposed image and the underexposed image? Noise! There are several types of noise that affect images but this post is about one particular source of noise called photon noise (or shot noise in non-imaging contexts). Understanding the constraints imposed by noise is crucial when you’re trying to push the limits of an imaging system.
Counting Hail Stones Again
In The Units of Spectral Radiance, I talked about setting a bucket outside in a hail storm to quantify the rate of hail.
We said that for a one minute measurement we collected 50 hailstones in our bucket. The interesting thing is that even if the real rate of hail remained perfectly constant a repeat of this experiment would probably not yield the exact same count of hail stones. Sometimes you might get 46, sometimes 58; it would all be in the ballpark but it would be different by small random amounts.
This random fluctuation in the measurement is called shot noise and it’s present any time you are counting independent events. To be clear this randomness is not describing random atmospheric conditions related to specifically to hail, the random fluctuations appears anytime you count discrete independent events. This random fluctuation is similar to how a series of 10 coin flips are not always 50/50 heads and tails.
Idealized hail stones are independent events because a stone’s arrival time isn’t affected by the stones around it. Some other things we could count that would show this phenomenon:
- Number of cars passing through an intersection per minute
- Number of phone calls being placed to a call center per minute
In each of these cases if you tally the number of events in each minute for 10 minutes, each tally would be a deviation around the true mean due to shot noise.
Here’s a histogram showing 100 measurements. Each measurement is the number of cars counted passing through an intersection during a 1 minute interval. The true mean-cars-per-minute of this constant ideal street is 20. Notice some of the results deviate from the true mean by 70%!
Poisson Distributions
I was able to simulate the car counting above because luckily for us, shot noise deviation for a given mean follows a Poisson Distribution. Why, you ask, do things in the universe have a Poison Distribution of noise when counted? The answer is I have no idea but Khan Academy has you covered.
Here’s our car counting simulation data along side the Poisson Distribution for a mean of 20.
Now let’s repeat this same simulation except instead of 20 cars per minute let’s pretend we’re on a massive highway with a true mean of 1000 cars per minute and we again tally cars per minute 100 times.
previously, shot noise caused our to be measurements off by as much as 70%. Now the largest deviation, 925, is only 8% from the mean of 1000. This shows that our ability to accurately count independent events improves as the number of events increases.
It turns out that as the number of counted events (signal) increases, the standard deviation of the Poisson Distribution (noise) increases more slowly, as the square root of the signal. Another way to state that is that the signal to noise ratio (SNR) favors signal as signal grows.
Cat Pics and Photons
A camera is a glorified photon counter and each pixel of an image is an experiment to count how many photons were streaming out of that point for a given period of time. Just like cars, phone calls, and hail, photons exhibit shot noise. In this context it’s also called photon noise.
If we take a very dim photo our camera pixels may only capture a few hundred photons each, which we now know would result in a poor estimate of the true mean number of photons. This is what we saw in the first image in this post. As the mean number of photons being captured by a camera pixel increases the measurement of how many photons there were becomes more accurate.
Above is a simulation of signal to noise ratio improvement as the mean number of photons increases by 10x. Each pixel is its own counting experiment that suffers from noise. The bright parts of the image already have more photons than the dim parts so the signal to noise ratio varies throughout the image. The darkest parts of the image have such a poor signal to noise ratio that you often don’t measure a single photon.
In a real camera, you can capture more photons and improve your signal to noise ratio by:
- increasing the exposure duration
- using a wider aperture
- using a flash (BYOP bring your own photons)
No matter how much you brighten an underexposed image, you can’t escape the fact that the pixels suffer from irreparable photon noise (and likely a couple other sources of electronic noise we didn’t talk about). Your only hope is to find more photons to count.
