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The chromatic structure of natural scenes

2
sandwichmaker
10-17-2002
12:11 PM ET (US)
Uncorrelated means that the covariance E[[X-Ex][Y-Ey]] for X,Y is zero. Independence means that their densities can be factored into pieces which only depend on X and Y respectivly.

Two variables X and Y are uncorrelated if
E[[X-EX][Y-EY]] = E[XY - XEY - YEX - EXEY]
                = E[XY] - EXEY
                = 0

i.e. EXY = EXEY

The requirement for independence is much stronger

it is

Eg_1(X)g_2(Y) = Eg_1(X)Eg_2(Y)

where g_1 and g_2 are arbitrary measurable (read functions which have a finite integrals) functions.

hence, independence implies zero correlation but not the otherway around.
Edited 10-17-2002 12:35 PM
1
Kristin BransonPerson was signed in when posted
10-16-2002
03:40 PM ET (US)
I didn't understand much of this paper. I'm looking forward to hearing about ICA, as I've never studied it but always have wanted to. I hope that Andrew can explain the difference between independent and uncorrelated and all those terms that seem like they should mean the same thing, but don't. I also don't know much about optics, so I think the main contribution of the paper is lost to me. Specifically, I couldn't really separate the contributions of this paper from the previous work mentioned. These questions are so general that I don't expect them to be addressed in the discussion board.

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