There is a fundamental difference between adding Gaussian noise and applying Poisson noise. In practice, people often talk about adding Poisson noise anyway, but this is not accurate. I will be looking at this from the image processing perspective in this article, and I’ll show purely visual examples. An application of this could be a simulation where you want to add noise to an image, and you known that this noise is Poisson distributed in the system that you are simulating.
Gaussian noise is typically generated separately and independently from the original image and then added to it (hence, additive noise). When you apply Poisson noise, on the other hand, you take the original image and ask the question “what would these individual pixels intensities be if they were produced by a Poisson process?”. This means that Poisson noise is correlated with the intensity of each pixel. Gaussian noise is independent of the original intensities in the image.
Why is this Difference Important?
There is the risk is that you use the common knowledge that Poisson noise approaches Gaussian noise for large numbers, and then simply add Gaussian noise with a fixed variance to the original image. This adds noise that is too strong in the darker parts of the image. This is demonstrated in the image below.
The first row of the image shows squares with an increasing photon count, from 0 to 5 increasing in steps of 1 (admittedly very low signal levels, but an example with higher values follows). All the pixels in a square have the same value. In the second row, Poisson noise has been applied. In the third row, Gaussian noise has been added, with the variance adapted to the pixel values, as an approximation of Poisson noise. The variance of the Gaussian noise can be adapted to the pixel values using the relation
Adapting the variance makes the Gaussian noise very close to the Poisson noise, except for the darkest squares. The fourth row shows the mistake that must be avoided, which is using Gaussian noise with a fixed variance. Negative values have been set to zero (also in the image that follows) for the rows with Gaussian noise, emulating the effect that a typical detector does not produce negative values.
The illustration below shows the same thing for 0 to 50 detected photons, increasing in steps of 10.
The second and third rows are now very close together, so it seems that you can get away with this for even these still very small signal levels. The fourth row still looks different, since the effect of the variance being too high remains for the darker parts of the image. This effect stays visible for even higher photon counts.
So the conclusion must be that, if you need Poisson distributed noise, you cannot not simply replace it by Gaussian noise with a fixed variance.