Additive poisson noise to an image

Question

I have written a function to add poisson noise to an image using numpy with np.random.poisson(..). The image is already in numpy array form, using grayscale (0-255). I am wandering if it makes more physical sense to provide the numpy function with the pixel values as the rates for the distribution, or use a set value over all the image.

In the first case, the function will be expressed as:

import numpy as np
def poisson_noise(X):
    noise = np.random.poisson(X, X.shape)
    return noise + X

In the second:

import numpy as np
def poisson_noise(X):
    noise = np.random.poisson(CONSTANT_RATE, X.shape)
    return noise + X

In the first case, the pixels with higher grayscale value (lighter) will be more influenced by the noise, would that have any physical interpretation ?

Thank you!