The constant (i.e., the band mean) was in the pca primer that someone sent me a link to in this discussion. Using the Eigenvectors resulting from i.pca, I can achieve the results of i.pca using my formula below. This is essentially the same as your formula minus the constant--which doesn't really make much (of any) difference in the final result.
However, my question is about performing an *inverse pca*--getting back to the original values from the principal components maps. 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. 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. Michael On Jun 17, 2012, at 10:48 AM, Duccio Rocchini wrote: > Dear all, > first, sorry for the delay... > Here are my 2 cents to be added to the discussion. I re-took in my > hands the John Jensen book. > Accordingly > > new brightness values1,1,1 = a1,1*BV1,1,1 +a2,1*BV1,1,2..... + an,1*BV1,1,m > > where a=eigenvector and BV=original brightness value. > > I found no evidence for the mean term: "- ((b1+b2+b3)/3)" > > Michael: do you have a proof/reference for that? > > P.S. thanks for involving me in this discussion which is really stimulating! > > Duccio > > 2012/6/7 Michael Barton <[email protected]>: >> >> I think I've figured it out. >> >> If (ev1-1, ev1-2, ev1-3) are the eigenvectors of the first principal >> component for 3 imagery bands (b1, b2, b3), the corresponding factor scores >> of the PC1, PC2, and PC3 maps (fs1, fs2, fs3) are calculated as: >> >> fs1 = (ev1-1*b1) + (ev1-2*b2) + (ev1-3*b3) - ((b1+b2+b3)/3) >> fs2 = (ev2-1*b1) + (ev2-2*b2) + (ev2-3*b3) - ((b1+b2+b3)/3) >> fs3 = (ev3-1*b1) + (ev3-2*b2) + (ev3-3*b3) - ((b1+b2+b3)/3) >> >> So to do an inverse PCA on the same data you need to do the following: >> >> b1' = (fs1/ev1-1) + (fs2/ev2-1) + (fs3/ev3-1) >> b2' = (fs1/ev1-2) + (fs2/ev2-2) + (fs3/ev3-2) >> b3' = (fs1/ev1-3) + (fs2/ev2-3) + (fs3/ev3-3) >> >> Adding the constant back on doesn't really seem to matter because you need >> to rescale b1' to b1, b2' to b2, and b3' to b3 anyway. >> >> Michael >> >> On Jun 7, 2012, at 1:55 AM, Markus Neteler wrote: >> >>> Hi Duccio, >>> >>> On Wed, Jun 6, 2012 at 11:39 PM, Michael Barton <[email protected]> >>> wrote: >>>> On Jun 6, 2012, at 2:20 PM, Markus Neteler wrote: >>>>> On Wed, Jun 6, 2012 at 5:09 PM, Michael Barton <[email protected]> >>>>> wrote: >>> ... >>>>>> I'm working on a pan sharpening script that will combine your >>>>>> i.fusion.brovey with options to do other pan sharpening methods. An IHS >>>>>> transformation is easy. I think that a PCA sharpening is doable too if I >>>>>> can find an equation to rotate the PCA channels back into unrotated >>>>>> space--since i.pca does provide the eigenvectors. >>>>> >>>>> Maybe there is material in (see m.eigenvector) >>>>> http://grass.osgeo.org/wiki/Principal_Components_Analysis >>>> >>>> This has a lot of good information and ALMOST has what I need. Everything >>>> I read suggests that there is a straightforward way to get the original >>>> values from the factor scores if you have the eigenvectors. But no one >>>> I've yet found provides the equation or algorithm to do it. >>> >>> @Duccio: any idea about this by chance? >>> >>> thanks >>> Markus >> >> _____________________ >> C. Michael Barton >> Visiting Scientist, Integrated Science Program >> National Center for Atmospheric Research & >> University Corporation for Atmospheric Research >> 303-497-2889 (voice) >> >> Director, Center for Social Dynamics & Complexity >> Professor of Anthropology, School of Human Evolution & Social Change >> Arizona State University >> www: http://www.public.asu.edu/~cmbarton, http://csdc.asu.edu >> > > > > -- > Duccio Rocchini, PhD > > http://gis.cri.fmach.it/rocchini/ > > Fondazione Edmund Mach > Research and Innovation Centre > Department of Biodiversity and Molecular Ecology > GIS and Remote Sensing Unit > Via Mach 1, 38010 San Michele all'Adige (TN) - Italy > Phone +39 0461 615 570 > [email protected] > [email protected] > skype: duccio.rocchini _____________________ C. Michael Barton Visiting Scientist, Integrated Science Program National Center for Atmospheric Research & University Consortium for Atmospheric Research 303-497-2889 (voice) Director, Center for Social Dynamics & Complexity Professor of Anthropology, School of Human Evolution & Social Change Arizona State University www: http://www.public.asu.edu/~cmbarton, http://csdc.asu.edu _______________________________________________ grass-dev mailing list [email protected] http://lists.osgeo.org/mailman/listinfo/grass-dev
