[GRASS-dev] PCA question

[Nikos Alexandris]

Thank you for the details Michael!

(cc-ing to Dr. Koutsias, this discussion might be of his interest)

Michael Barton:

> > [...]

> >> However, my question is about performing an *inverse pca*--getting back
> >> to the original values from the principal components maps.

Nikos:

> > Michael, getting back to the original values can _only_ be done if one
> > does not "touch" the data in any of the intermediate steps, i.e. Input >
> > EVD (or SVD) > Inverse PCA > Original values.

> > If one alters the data at any step prior to the Eigenanalysis or SVD, I
> > don't think it is possible to land back on "level 1". From the moment
> > that global stats of a multivariarte dataset, subject to PCA, are
> > changed, one will probably jump into a "new" reality.  This means that it
> > takes an extra effort to interpret the "new" stuff.

Michael:
 
> Right. That is why I'm not doing any alteration of the data after
> transforming to PC's.
> >> The idea of pca sharpening is that you perform a pca, then do an inverse
> >> pca substituting the pan band for pc1 to enhance the resolution.

Nikos:

> > I haven't tried PCA sharpening.  So, apologies for my simplistic
> > question(s), I just want to understand the trick here.

> > Which resolution is to be enhanced?  The geometric?  Is it meant to keep
> > PC1 and mix it with the rest, or keep the Pan and throw away PC1?

> > Principal Component 1 will contain the highest variance of your input data
> > -- which, in fact, is a composition of different amount of information
> > originated from all input bands. If you throw that away you are left with
> > a dataset which is likely to be useless!

Michael:
 
> The way this works is to:
> 
> 1) transform 3 lower resolution bands to 3 principal components

> 2) replace PC1 with the higher resolution panchromatic band (under the
> reasonable assumption that the pan band will include more of the total
> spectral variability than will any more spectrally limited band). Histogram
> matching the pan band to PC1 is recommended here.

This assumption is, indeed, necessary and sounds pretty rational. Interesting 
stuff.

> 3) do an inverse PCA to get back to the original bands with a similar range
> of spectral response but with higher spatial resolution.

> There have been--and continue to be--studies of the performance of different
> pan sharpening algorithms from various perspectives. For myself, pan
> sharpening can help with visually resolving more features in greater
> detail. But this is at the cost of making it considerably more difficult to
> understand what the pixel values of the enhanced bands mean.

> >> The equations I show below seem to work, but what I've read indicates
> >> that it is not possible to *exactly* get the original values back; you
> >> can only approximate them.

Nikos:

> > As Markus' demonstration showed in another post, the results can be close
> > enough so the differences can be disregarded. As far as I have understood
> > PCA, it depends on how many decimals are taken into account, while doing
> > all the math and _not_ effectively altering the data at any of the
> > intermediate steps.

Michael:

> Yes. Markus' demo made me more comfortable with the algorithm overall. When
> you replace PC1 with the pan band, of course, you don't get back to the
> original values. But the ranges look pretty good now.
 
> I'll attach the script here in case you want to try it. Ver. 2 and 3
> represent different kinds of optimizing for calculation speed. v3 only
> works with a new GRASS python function that Hamish committed to trunk
> yesterday. V2 should work with all current versions of GRASS.

Thanks a lot. Noted on my ToDo list: "Check MichaelB's pan-sharpening scripts 
(after next week)".

> Here are some helpful references:

> Amarsaikhan, D., & Douglas, T. (2004). Data fusion and multisource image
> classification. International Journal of Remote Sensing, 25(17), 3529–3539.

> Behnia, P. (2005). Comparison between four methods for data fusion of ETM+
> multispectral and pan images. Geo-spatial Information Science, 8(2),
> 98–103.  doi:  10.1007/BF02826847

> Du, Q., Younan, N. H., King, R., & Shah, V. P. (2007). On the Performance
> Evaluation of Pan-Sharpening Techniques. Geoscience and Remote Sensing
> Letters, IEEE, 4(4), 518 –522.  doi:  10.1109/LGRS.2007.896328

> Karathanassi, V., Kolokousis, P., & Ioannidou, S. (2007). A comparison study
> on fusion methods using evaluation indicators. International Journal of
> Remote Sensing, 28(10), 2309–2341.  doi:  10.1080/01431160600606890