PCA compression: Am I supposed to center the data? Project the centered data or the original? I am trying to use PCA to compress pictures and audio. For example in the case of audio, the audio is cut into blocks and then the blocks are fed into the data array as new cases.
Question: The data matrix for PCA is generally centered, sometimes even normalized. For compression application, should this centering and normalization be done, given that I want to store the original data and not a transformed version? If the centering is applied should I project the centered data or the original data?
 A: 
For compression application, should this centering and normalization
  be done,

Yes, it should be done by default. The only problem with centering could be if your data is sparse (centering messes up sparsity), but images typically aren't stored in sparse arrays. 
As to why you should do it? This question has been asked multiple times, see for example How does centering the data get rid of the intercept in regression and PCA?


For compression application, should this centering and normalization
  be done, given that I want to store the original data and not a
  transformed version?

The point of compression is to store less data per example, so you have to store transformed data and be able to restore it.
This is not a problem with standarisation - mean subtraction and dividing by feature's standard deviation is reversible. 
What you'd want to do is given your compressed data (coefficients of principal components) retrieve data using PCA, and then add the mean and scale it according to standard deviation. This means that your decompression algorithm, in addition to storing principal components, has to store two more matrices (the one containing means and the one containing standard deviations).
