Linear Combination

For the Morro Bay TM scene there are 7 spectral bands. Thus each pixel has 7 values. The pixel in row i, column j of the image is a vector:

x(i,j,1) x(i,j,2) x(i,j,3) x(i,j,4) x(i,j,5) x(i,j,6) x(i,j,7)

x(i,j,1) is the value of band 1 in row i, column j, x(i,j,2) is the value of band 2 in row i, column j, etc.

A linear combination of these values, to calculate the first Principal Component, would look like:

This multiplication and addition is carried out for each of the picture elements, pixels, in the image. The Principal Components Analysis is the calculation of the values of the set of vectors a and then the multiplication of the image data by them to get the projections of the data points onto the Principal Components.