imnoise('gaussian') underlying equations clarification

Question

In MATLAB’s imnoise() function, when the type of noise is 'speckle', the documentation clearly states that it is multiplicative noise and states the underlying equation.

J = imnoise(I,'speckle',v) adds multiplicative noise to the image I, using the equation J = I+n*I, where n is uniformly distributed random noise with mean 0 and variance v. The default for v is 0.04.

However, no such equation is provided for the gaussian option. And there is a separate type called 'localvar'. So the equation when using imnoise(I, gaussian, mean_noise, variance_noise) should be

J(x,y) = I(x,y) + g(mean_noise, sqrt((variance_noise))

Further, my assumptions:

1. This noise g is not correlated to the spatial coordinates of the image
2. This noise g is not correlated to the intensities at those spatial coordinates
3. g is a gaussian random number generated from a gaussian distribution of mean mean_noise and standard deviation sigma = sqrt(variance_noise)

Am I right?

MAJOR UPDATE
I am unaccepting the previous answer to clear some confusion.So i checked the code for 'imnoise' in matlab and what it does is:

b = a + sqrt(p4)*randn(sizeA) + p3; where
b - image with noise added
a - original image
p4 - variance
p3 - mean

What is the range of randn()? I checked randomly and this produces values higher than 1 like 1.85. And the documentation for randn() fails to mention anything about the range. This is quite strange.

Answer 1

Yes, you are right. The noise is spatially uncorrelated (i.i.d) and also uncorrelated to the signal. Furthermore, the noise is additive and sampled from a zero-mean unit standard deviation Gaussian which is then scaled for user-provided standard deviation and offset by user-provided mean. If no variance and mean values are specified, imnoise chooses zero mean and 0.01 variance.

You can actually see the entire code for by doing >>edit imnoise in MATLAB. You should have the Image Processing Toolbox.

Regarding randn() - it produces i.i.d samples from a zero-mean unit standard deviation Gaussian. The range of a Gaussian is (-Inf Inf) and hence you see values outside the range (-1 1). MATLAB function rand() gives values in the range (-1 1) that are uniformly distributed.

Edited: Updated answer to include exact default mean and variance values.

Answer 2

the function imnoise(I, 'gaussian', mean, variance) need the variance normalized between [0 1]. So if your image is of type 'uint8', you should divide the parameter variance by 255².

Note also that the variance is different of the standard deviation sigma. If you would use sigma, you should put (sigma²/255²) as the variance parameter (because variance = sigma²).