Re: Principal Components (Hotelling of KL Transform) of 2 Band (Red and Green) Real Color Image

Jonathan Campbell wrote:
PeterOut wrote:
On Jan 6, 5:37 am, Jonathan Campbell <jg.campbell...@xxxxxxxxx> wrote:
PeterOut wrote:
Say you have a 2 channel image (red and green channels for example)
and you want to get the principal components in order to do a
Hotelling (Karhunen-Loève) transform on the image so as to minimize
[...]
Thank you very much for your reply. You are right. I checked and it
would appear that it is indeed the covariance matrix that I should
use. However, it would appear that I would have the same problem
since covariance is reflective and, consequently, the covariance
matrix between the two data sets would be

M=1 a
a 1


Not unless the variance of both channels is 1? As I say, I have never applied PCA/K-L to 2-d. data, so I'm a little hesitant.

Variance-covariance matrix = Cov = E{ (x - mu) (x - mu)^T } ? 2 x 2 in your case. mu = mean.


Just looking at equations that I use for determining major and minor axes for 2-d. shapes (Masters, 1994, p. 316); same problem and I know these 'work'.

Cov = [a b; b c]

Let r = sqrt((a-c)^2 + 4 b^2)

Eigenvalues: lmax = (a+c+r)/2, lmin = (a+c-r)/2

Eigenvectors: vmax = (a-c+r, 2b); vmin = (a-c-r, 2b); checks with your eigenvectors I think, but that gets us no further.

Masters warns that if the eigenvalues are nearly equal, then the eigenvectors are ill-defined; doesn't that mean a circularly-symmetrical data cloud?

@Book{masters,
author = "T. Masters",
title = "Signal and Image Processing with Neural Networks: C++
sourcebook",
publisher = "John Wiley \& Sons",
address = "New York",
year = "1994"
}

Best regards,

Jon C.
.

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