# gaussian noise in an image

according to Estimating acquisition noise P. 2

Estimating the mean and the standard deviation for each pixel is calculated as below:

$$\begin{equation*} i, j = 0,\dots,N-1 \\ \bar{I}(i, j) = \frac{1}{N} \sum_{k=0}^{N-1} I_k(i,j)\\ \sigma(i, j) = \left [ \frac{1}{N-1} \sum_{k=0}^{N-1}(I_k(i, j) - \bar{I}(i,j)^2) \right]^{1/2} \end{equation*}$$

My Question

1. What is $I_k$ ?

2. What should mean of each pixel be? Is it the mean of whole Image?

Thanks

• Your question has beeen answered. Do not hesitate to vote for the useful ones and accept the most suitable – Laurent Duval Feb 9 '17 at 17:26

To estimate quantities about a pixel at location $(i,j)$, you need a set of pixels in a neighborhood, close enough to $(i,j)$ in behavior, to compute statistics on.

Assuming that neither the camera nor the object at pixel $(i,j)$ move, the index $k$ may denote a sequence of images in time, as answered by @MarcusMüller.

From the other slides at CMPE 264: Image Analysis and Computer Vision, it is not evident that the presenter considers image sequences. A possibility is that the index $k$ in $[0,N-1]$ denotes a subset of $N$ pixels somehow around $(i,j)$, maybe both in time or space.

In space, $I_k$ could denote the $k$th pixel from a subblock of the whole image with $N$ pixels total, centered around the pixel with coordinates $(i,j)$. Such a lousy notation could avoid cumbersome notations for the subblock that moves with position $(i,j)$.

Pixels around the pixel with coordinates $(i,j)$ are considered as realizations of a random process modeling the center pixel, and serves to provide an estimate of the average of this pixel.

Inferring from the presentation you link to,

1. $I_k$ is the $k$th image, $I_k(i,j)$ is the pixel from that image at position $(i,j)$. This implies that there's a sequence (indexed by $k$) of images, and that also explains the existence of a "mean" image $\bar I$, which, if you look at that sum, really is just the average over all images in that sequence, for every single pixel (not over the whole image).