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Tuesday, March 11, 2014

Mean, median, myriad, and meridian computation

While the mean calculation is a trivial task, the median calculation is not trivial. It can be determined directly by sorting the samples, but the sorting process is computationally expensive *. Therefore, many researches have been done to develop fast algorithms to compute the median **. The myriad is more difficult to calculate than the median. The myriad can be obtained by minimizing the associated cost function, or by finding polynomial roots of the derivative polynomial of the cost function. The derivative polynomial is (2N-1)-th order polynomial, and the only real value satisfies the condition to be the myriad. The meridian is much more complicated than the others. Because the cost function has a finite number of local minima (input samples), multiple roots can be found through the derivative polynomial of the cost function ***. On the other hand, the meridian is one of the local minima of the cost function, so the meridian is one of the input samples. By evaluating the cost of every input sample, the meridian can be obtained.

* Kalluri, Sudhakar, and Gonzalo R. Arce. "Fast algorithms for weighted myriad computation by fixed-point search." Signal Processing, IEEE Transactions on 48.1 (2000): 159-171.
** ANSI C implementation of median search algorithms:
*** Aysal, Tuncer C., and Kenneth E. Barner. "Meridian filtering for robust signal processing." Signal Processing, IEEE Transactions on 55.8 (2007): 3949-3962.

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