NetNews Usenet Archive 1992 #27/NN_1992_27.iso/spool/comp/ai/neuraln/4323

Internet Message Format | 1992-11-19 | 2.3 KB

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From: [e-mail]. (Konstantinos Diamantaras)
Newsgroups: comp.ai.neural-nets
Subject: Re: principal component analysis and "wavelets"
Message-ID: <[e-mail]>
Date: 18 Nov 92 16:11:33 GMT
References: <[e-mail]>
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Organization: Siemens Corp. Res., Inc.
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In article <[e-mail]> [e-mail] (Daniel Brown) writes:
>I posted about a week ago with a question about preprocessing
>of image input.  I only received a few responses, and I need
>more information on them:
>
>    i) What is principal component analysis (PCA)?

I am not much of a wavelet person, but I can tell you what PCA is.
It is related to the Karhunen-Loeve transform, and eigenvalue decomposition
of the autocorrelation matrix of your input. Simply put, if you
have an n-dim stochastic signal x with autocorrelation matrix R=E{xx'}
and you look to map it LINEARLY down to m dimensions (where m<n)
via a transformation of the form
y=Wx
(W is an mxn matrix) so that when you optimally reconstruct x from y
you get the minimum mean-square error possible, then the best matrix
to use is W=[e_1 e_2 e_3 ... e_m]' where e_i are the principal eigenvectors
of R (i.e. those eigenvectors associated with the maximum eigenvalues).
The minimal error you get is just the sum of the smallest eigenvalues of R.
That is a classic result originally shown by Hotelling (1933) and
was taken over by Karhunen and Loeve (in the 50's I think) for the analysis
of continuous-time stochastic processes.

You can find discussions regarding the Karhunen-Loeve transform
(which is really the same as PCA) in any good book on image processing
theory, such as Anil K. Jain, "Fundamentals of Digital Image Processing",
Prentice Hall, 1989, or Arun N. Netravali and Barry G. Haskel,
"Digital Pictures: Representation and Compression", Plenum Press, 1988,
among many others.

>    ii) Has anyone heard of "wavelets" used in
>       preprocessing of image data?
>
>Thanks in advance,
>
>Dan Brown
>[e-mail]
>
>I will post a summary of replies ...

Hope this helps.

Best regards

        Kostas Diamantaras