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Problem and Question

I am doing a research project and I haven't taken any statistical mechanics before so I am confused about some core aspects of my research.

Currently, I have an image time series of pixel intensity values and I am trying to perform an animation of the Generalized Spatiotemporal Correlation Function (STICS) defined as $$ r(\zeta, \eta, \tau) = \frac{\langle\langle \delta i(x,y,t) \delta i (x + \zeta, y + \eta, t + \tau\rangle_{xy} \rangle_t}{\langle i\rangle_t \langle i \rangle_{t + \tau}}, \tag{1} $$ where $\zeta$ and $\eta$ are spatial lag variables that represent shifts of the image in $x–y$ space for calculation of spatial correlation $\langle \cdot \rangle_{xy}$, whereas $t$ is a temporal lag variable representing shifts in time in the $x–y–t$ image series for calculation of the temporal correlation $\langle \cdot \rangle_t$. The angular brackets in the denominator simply represent calculation of a mean intensity with $$ \delta i(x,y,t) = i (x,y,t) - \langle i \rangle. \tag{2} $$

Now, say I have $n$ images each with a pixel grid with intensity values, which I obtain from a Python simulation. Such as

enter image description here

What would be a procedure to generate an animation of Equation $(1)$ , a correlation function temporal evolution, in order to get something like

enter image description here

What I have done

I was able to generate the images simulations and to store the values intensity fluctuation pixel values of $(2)$ conveniently, but I am stuck on understanding how and what to do with $(1)$. Particularly, I can't comprehend the numerator of $(1)$. From online search, the brackets seem to represent expected values? One online reference defines this angular bracket as $$ \langle A \rangle = \frac{1}{N} \sum_{i = 1}^{N} A_i, $$ but I don't see how this applies in my case, given that I have 3 dimensions. Can someone explain, by giving a short example, on how to get one frame of the correlation function temporal evolution (as in the picture above), using computations from $(1)$ in conjunction with my time series of $n$ images? Or any other tips for that matter?

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  • $\begingroup$ Are you using numpy arrays? It sounds like using the np.mean function may be what you are looking for, as it is able to work on multidimensional arrays. $\endgroup$ Commented Oct 31, 2022 at 9:04

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