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?


  • $\begingroup$ Your question has beeen answered. Do not hesitate to vote for the useful ones and accept the most suitable $\endgroup$ – 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).
  2. answered by 1.

I don't know why the number of images in that sequence is equal to the width and is equal to the height of these images, but it's just how they defined that. Since you have the presentation, I assume you also were in that lecture or have some form of scriptum so that you're probably better prepared to interpret these slides than we are.

If you just found this presentation on the internet: Maybe it'd be easier to use a textbook, since these tend to contain all the context one needs to understand a topic.


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