Stability and dimensionality of Karhunen-Loêve multispectral image expansions

The Karhunen-Loêve (K.L.) expansion is a useful tool for the representation, pre-processing and orthogonal coding of multispectral imagery: Each spatial pixel is analysed independently as the K.L. transform is taken in the spectral dimension, i.e. along the various N spectral channels. The eigenvectors are those of the covariance matrix. The (principal) eigenimages are thus “false color” images, which can be viewed without decoding as the spatial topology is unchanged, and the higher order principal images present a strong contrast enhancement.⁽¹⁾ These principal images are also uncorrelated, a very desirable feature for many applications including clustering.⁽²⁾

The source dependency of the eigenvectors, however, introduces “instability” in the form of pronounced statistical noise on some principal images. This paper gives the results of a numerical study carried out on a 7 channel Daedalus Multispectral Scene. The uncertainties of the eigenvalues and eigenvectors are evaluated from two “drawings” of the pixels of the raw data.

Both the numerical results of the study and the direct viewing of the principal images show that three out of the seven have so much noise that they do not yield any useful information. Only the first two principal images have excellent stability, and they contain most of the total contrast variance of the scene. Two other principal images of lower order are also stable, but, contribute very little to the total contrast variance. These images carry texture information rather than homogeneous zone clustering information.