Regarding the nature of white noise in images
A signal is just a varying quantity over an axis.

Think of a greyscale image.

Think of a single row of that image. Can you see how the intensity evolves over the distance on the horizontal axis?

In an image, you get two axes (x and y), whereas in sound, you just get one.

In (mono) sound, your instantaneous intensity is the only quantity varying. In an image, that quantity might be brightness alone, or brightness, hue and saturation, or red, blue and green intensities, or whatever channels your color model defines.

Anyway, the point about white noise is it's autocorrelation function; and that's easy to define on the two axes of an image, exactly the same as for sound: 


Rows of an image: If you shift the row(s) by any amount (other than 0), then the expected sum-of-products between the original and shifted row is 0. 
Columns of an image: If you shift the columns(s) by any amount (other than 0), then the expected sum-of-products between the original and shifted column is 0. 


If both properties are given (and in fact, the correlation in each direction, even if you "sliced" your image arbitrarily in diagonals), then your 2D noise is white.