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I want to add realistic noise to a simulated image but I am a little confused about the process. I want to have some Gaussian random noise representing the readout noise and to also add Poisson noise correlated with the signal in the image.

I understand how to add the Gaussian random noise; I draw samples from a Gaussian with fixed width and add the samples onto the pixel values.

I am confused about the Poisson noise though. Is this noise only correlated to the noiseless signal?

i.e I should draw samples from the Poisson distribution with $k = \mathrm{pixel value}$ (before adding the random noise) and add it to the noiseless image.

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Yes, one can superpose these two noise sources: add a Gaussian readout noise (with a spatially constant sigma) to a Poisson noise (with spatially variable number count k).

In practice, as the number counts are high for optical CCD images, one often approximates the Poisson noise by a spatially-variable Gaussian noise, using the square root of the photon number counts as sigma. If your images have some significant sky background level (e.g., hundreds of photons), this is fine even for faint sources that add only few photons. A Poisson distribution gets very close to a Gaussian already for k > 20. Note that despite this common approximation by a Gaussian, such a simulated noise contribution is still often called "Poisson noise", to stress that it originates from the spatially variable signal.

One warning: using the square root of the signal as sigma of a Gaussian to approximate Poisson noise only holds for a signal in units of photons (or almost equivalently, in units of electrons). If an image is given in ADU (analog to digital unit), the gain is required to convert the pixel values into electrons. The same holds when using a true Poisson distribution: one has to be careful that the counts have not been rescaled.

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