I have been studying the concept of PCA and its implementation for dimensionality reduction for more than 1 month. My goal is to classify a hyperspectral image using sparse representation by the linear combination concept which is as follow:

$$y = Dx$$

So consider $D$ as a dictionary with $d\times B$ dimension where $d=3000$ is the number of samples and $B=200$ is the number of band/channel.

Now I am trying to construct the $D$ by this mean that the classes (sub-dictionaries) are well separated. Therefore I want to apply PCA to individual sub-dictionary in order to form the main dictionary.

However, my goal is to apply PCA on hyperspectral satellite imagery like this.

I have implemented the PCA in Octave and project my data on that particular low dimension. But my question is should I reduce the number of training pixels(observation=d$$) or reduce the variable dimension ($B$)?

Since I use sparse representation and dictionary concept then reducing the dimension of $d$ (observation pixels for individual classes) is more make sense rather than reducing the number of features ($B$). But I am not sure if I am right or not.

After constructing the $D$ should I transform $y$ to PCA dimension before computing its spars coefficients or not?

  • $\begingroup$ typically, PCA selects the principal components from your multi-dimensional signal, i.e. in your case, it reduces $B$ without reducing the number of observation you've got. $\endgroup$ – Marcus Müller Jan 11 at 16:03
  • $\begingroup$ But I manipulate the covariance matrix to have a pairwise pixels evaluation by means of covariance quantity, instead of computing covariance between variables. therefore I am able to compute the eigenvectors and their corresponding eigenvalues for samples (we have lots of reduction vector [pixel]) $\endgroup$ – morteza Jan 14 at 1:20
  • $\begingroup$ that can be presented by A scalar multiplication of another. So lets only use certainty, not uncertainty (in terms of variance of the data). $\endgroup$ – morteza Jan 14 at 1:29
  • $\begingroup$ The linear algebra five us two amazing concept cal Linearly-Dependent and Linearly-Independent. That is the way we can cover the main info with only the main numbers. $\endgroup$ – morteza Jan 14 at 1:30

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