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I am implementing zero order approximation method to calculate Legendre moment of an image. One simple way to calculate moments is directly using loops and calculate the double summation as given in the image-1. Then I read everywhere that kernels are separable, so using this we can do a faster implementation. Someone please elaborate how separability of kernels reduces time complexity in this case.

image-1 direct formula

Later in the paper - the author says the same things (a little difference is that - he has an exact calculation, but still kernel separability is valid in both cases) about a faster implementation as given in the image - 2.

image-2 the trick part

I will be very grateful, if someone can explain this?

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  • $\begingroup$ Could you explain what is the kernel? $\endgroup$ – Royi May 26 '18 at 6:22
  • $\begingroup$ @Royi To be true I am not able to understand exactly what are kernels in this case, thats what is causing doubts. Can this summation equation be represented as a convolution? I guess kernels are the polynomial matrices Pm(x) and Pn(y) which have to be multiplied with image function. Any thoughts that may help, even if not related to this directly would be appreciated. $\endgroup$ – Sanjeev May 26 '18 at 10:28

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