# Why does White Noise in images imply noise in adjacent pixels are independent?

My professor said in the class that for "Additive White noise, the noise in pixels adjacent to each other are independent". How?

This is what I have got so far: White noise implies that PSD (Power Spectral Density) is flat which implies Covariance matrix is an impulse function i.e.

$$\begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 0\end{bmatrix}$$

How does this imply that noise at adjacent pixels is independent? At max, this says that noise is uncorrelated.

This question asks the same thing, though that part of the question is unanswered.

• a covariance matrix will have a bit of a hard time being an impulse function. The covariance matrix should be a diagonal matrix in this case, so the matrix you've shown us is a very bad example. – Marcus Müller Feb 5 '19 at 16:34
• Yeah, I realize that if the noise is independent, then covariance matrix should be diagonal. But when I took inverse 2D DFT of a matrix of ones (Flat PSD) in MatLab, this is what I got. It confused me more. Please explain if I have understood something wrong. Thanks! – Nagabhushan S N Feb 5 '19 at 16:58
• Your 2D-DFT of a single constant 2D image has nothing to do with the covariance matrix of a set of variables! – Marcus Müller Feb 5 '19 at 17:07
• Sorry. I meant 2D IDFT. Taking Inverse DFT of PSD gives Autocorrelation right? – Nagabhushan S N Feb 5 '19 at 17:14
• yes, but your constant-entry matrix is not a PSD. – Marcus Müller Feb 5 '19 at 17:21