How to make series circle shift invariant

I have a series of numbers. They may be integer, floating, or binary. Length of series is constant. I need criteria which is circle shift invariant. For more information assume series S1, S2, S3, S4 and S5:

S1 = {A, B, C, D, E, F}
S2 = {F, A, B, C, D, E}
S3 = {D, E, F, A, B, C}
S4 = {A, C, D, F, E, B}
S5 = {A, B, C, D, E, G}

where A, B, C, E, F, and G are numbers. S2 and S3 is resulted by shift circle of S1. So the criteria should return same values for these three series. S4 have same number of S1, S2, and S3 but order of numbers are not the same. S5 and S1 are similar so I want to criteria of S1 and S5 be similar.

I want to use them as descriptors in image processing. In other words, I want to make a feature (like LBP, or HOG) rotation invariant. For example mean and variance can be used but I need more. Are there any suggestion?

• So what is your problem /question? You are describing the situation, and mentioning something about criteria, but it is unclear what you want to achieve and why you are not able to do this? – Irreducible Jun 6 '18 at 10:47
• I need criteria for these series for comparing them as I said. These criteria should be circle shift invariant. – Babak.Abad Jun 6 '18 at 11:18