Principal and Minor Eigenvectors
Chanchal Chatterjee · Apress eBooks · 2022
In Chapter 4, I discussed adaptive algorithms for the computation of the principal eigenvector of the online correlation matrix A k ∈ℜ n X n . However, in some applications, it is not enough to just compute the principal eigenvector; we also need to compute the minor eigenvectors of A k . One such application is multi-dimensional data compression or data dimensionality reduction in multimedia video transmission [Le Gall 91]. For example, in still video compression by the JPEG technique, the image is divided into 8X8 blocks. This high-dimensional video data can be reduced to lower dimensions by projecting it onto the principal eigenvector subspace of its online correlation matrix. The process of data projection onto the eigenvector subspace by a linear transform is known as principal component analysis (PCA) and is closely related to the Karhunen-Loeve Transform (KLT) [Fukunaga 90]. We approximate the eigenvectors by fixed transform vectors given by the discrete cosine transform (DCT), which is the central compression method of the MPEG standard [Le Gall 91]. It can be shown that DCT is asymptotically equivalent to PCA for signals coming from a first-order Markov model, which is a reasonable model for digital images.