gaussian noise in an image
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.