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

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
the correlation between the red and green channel.  You would get the
correlation matrix but, since correlation is reflective, the
correlation matrix M would be given by.
M=1 a
     a 1

where a is the correlation coefficient between the two bands.  It
seems that, whatever the value of a is, the eigenvectors would be
(0.7071,0.7071) and (-0.7071,0.7071).  So the transformation would be
the same regardless of what is in the input images.  This does not
seem very useful since I thought the idea of the Hotelling transform
was to rotate the axes to minimize the correlation between the bands.
Is there something I'm misunderstanding?

I use K-L / PCA a lot, but never on 2-d. data; and I always use the
covariance matrix. What happens if you use the covariance instead of
correlation?

Otherwise, sci.stat.math may be able to help.

Best regards,

Jon C.

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

This would lead to eigenvectors of (0.7071,0.7071) and
(-0.7071,0.7071) regardless of the covariance values. How does it
would for you? I do not think the dimensionality of the data set
would make a difference.

Thanks again,
Peter.
.

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