Re: [WM]: Normalized Correlation

[Joseph O'Ruanaidh <joruanaidh@genuone.com>]

I wish that I had thought of this all by myself.
In fact this is just a part of the theorem that proves
that m-sequences are almost orthogonal. (ref. Pickholtz)

Cross correlation of m-sequences is like adding
using exclusive-or every integer from 1 to (2^N)-1 inclusive. There is
always only one more odd than even - which means that it gets close to
zero (giving a slightly negative cross correlation)

To make m-sequences orthogonal the missing all zero 
state needs to be included. This is not hard to do in
software but in hardware a shift register would
lock in an all zero state.

Joe

Cagatay Candan wrote:
> 
> At 10:50 AM 8/14/02 -0400, Joseph O'Ruanaidh wrote:
> >However, given that these are BITMAP images we might want to try
> HAMMING
> >distance. This would be the number of bits that are different. The 
> >maximum number of bits they can have in common is N,  the length of 
> >the
> 
> >vector, so let's divide by N to normalize. Okay that works.
> >
> >What is the relationship between the two measures?
> >The first is based on Euclidean distance (sum of squares)
> >
> >Look at these tables:
> >
> >XOR     1     0
> >----------------
> >1       0     1
> >0       1     0
> >
> >
> >x       -1    1
> >----------------
> >-1       1   -1
> >1       -1    1
> >I hope you can see that they are isomorphic. If you
> >changed all your binary zeros to ones and binary ones to minus ones 
> >the
> 
> >two distances would be equivalent (different numbers saying exactly 
> >the
> 
> >same thing).
> 
> This is a great observation.  To see the isomorphism more clearly, 
> please have a look at this table:
> 
> set_1= { even, odd }  op_1={addition}
> set_2= { negative, positive}  op_2={multiplication}
> 
> set_1& op_1             %               set_2 & op_2
> %%%%%%%%%%%%%%%%%%%%%%%%%%
> even+even=  even      %    positive*positive=positive
> even+odd =  odd       %     positive*negative=negative
> odd+even =  odd       %     negative*positive=negative
> odd+odd  =  even      %     negative*negative=positive
> %%%%%%%%%%%%%%%%%%%%%%%%%%
> 
> The essence of the whole thing is: Both operations map the set 
> elements identically. Therefore I may call this operation as binary 
> addition if that is my wish.  And my marsian friend may call it 
> something else if that is
> 
> his wish. But his computer and my computer are doing **exact** same 
> thing when we execute this routine. Just we have different labels on 
> it.
> 
> best wishes,
> cagatay


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