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I have particle activity as shown in the left pane of animation below. The activity is clustered and it moves slowly. Sometimes these clusters merges together. On the right side of it, its shuffled version is shown. Each frame of the activity on the left is shuffled by a fixed permutation matrix P.

<code>enter image description here</code>

The problem I have is the inverse problem. Often I record what looks like activity on the right. Shuffling is allowed. My hypothesis is that there is a permutation $P^{-1}$ (fixed for each frame) which can transform activity on the right to the left.

Not sure if this has a unique solution or the problem is even well-posed. I am not familiar with literature on this. Any tip, hints or reference would be of great help.

PS: Are there some features which remains invariant post permutation?

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