Why is Camera Read Noise Gaussian Distributed?

As a microscopist I work with very weak light signals, often just tens of photons per camera pixel. The images I record are noisy as a result¹. To a good approximation, the value of a pixel is a sum of two random variables describing two different physical processes:

1. photon shot noise, which is described by a Poisson probability mass function, and
2. camera read noise, which is described by a Gaussian probability density function.

Read noise has units of electrons, which must be discrete, positive integers. So why is it modeled as a continuous probability density function²?

The Source(s) of Read Noise

Janesick³ defines read noise as "any noise source that is not a function of signal." This means that there is not necessarily one single source of read noise. It is commonly understood that it comes from somewhere in the camera electronics, but "somewhere" need not imply that it is isolated to one location.

The signal from a camera pixel is the number of photoelectrons that were generated inside the pixel. I imagine readout of this signal as a linear path consisting of many steps. The signal might change form along this path, such as going from number of electrons to a voltage. At each step, there is a small probability that some small error is added to (or maybe also removed from?) the signal. The final result is a value that differs randomly from the original signal.

Importantly, I do not think that it matters which physical process each step actually represents; rather there just has to be many of them for this abstraction to be valid.

"But aren't there only a handful of steps?" you might ask. After all, linear models of photon transfer typically consist of a few processes such as detection, amplification, readout, and analog-to-digital conversion. I am not referring to these when I use the term "step." Rather, I am referring to processes that are much more microscopic, such as passage of a signal through a transistor or amplifier chip. At the very least Johnson noise, or random currents induced by thermal motion of the charge carriers, will be present in all of the camera's components.

Read Noise is Gaussian because of the Central Limit Theorem

The reason for my conclusion that I can ignore the details so long as there are many steps is the following:

I can model the error introduced by each step as a random variable. Let's assume that each step is independent of the others. The result of camera readout is a sum of a large number of independent random variables. And of course the Central Limit Theorem states that the distribution of the sum of random variables tends towards a normal distribution, i.e. Gaussian, as the number of random variables tends towards infinity. This happens regardless of the distributions of the underlying random variables.

So read noise can appear to be effectively Gaussian so long as there are many steps along the path of conversion from photoelectrons to pixel values and each step has a chance of introducing an error.

Sums of Discrete Random Variables

I encountered one conceptual difficulty here: the sum of discrete random variables is still discrete. If I have several variables that produce only integers, their sum is still an integer. I cannot get, say, 3.14159 as a result. Does the Gaussian approximation, which is for continuous random variables, still apply in this case?

This question is relevant because the signal in a camera is transformed between discrete a continuous representations at least twice: from electrons to voltage and from voltage to analog-to-digital units (ADUs).

Let's say that I have a discrete random variable that can assume values of 0 or 1, and the probability that the value is 1 is denoted \( p \). This is known as a Bernoulli trial. Now let's say that I have a large number \( n \) of Bernoulli trials. But the sum of \( n \) Bernoulli trials has a distribution that is binomial, and this is well-known to be approximated as a Gaussian when certain conditions are met, including large \( n \)⁴. So a sum of a large number of discrete random variables can have a probability distribution function that is approximated as a Gaussian.

This does not mean that the sum of discrete random variables can take on continuous values. Rather, the probability associated with any one output value can be estimated by a Gaussian probability density function.

But how exactly can I use a continuous distribution to approximate a discrete one? After all, if the random variable \( Y \) is a continuous, Gaussian random variable, then \(P (Y = a) = 0 \) for all values of \( a \). To get a non-zero probability from a probability density function, I need to integrate it over some interval of its domain. I can therefore integrate the Gaussian in a small interval around each possible value of the discrete random variable, and then associate this integrated area with the probability of the obtaining that discrete value. This is called a continuity correction.

Example of a Continuity Correction

As a very simple example, consider a discrete random variable \( X \) that is approximated by a Gaussian continuous random variable \( Y \). The probability of getting a discrete value 5 is \( P (X = 5) \). The Gaussian approximation is \( P ( 4.5 \lt Y \lt 5.5 ) \), i.e. I integrate the Gaussian from 4.5 to 5.5 to compute the approximate probability of getting the discrete value 5.