A Computationally Simple Procedure for Imagery Data Compression by the Karhunen-Loÿve Method

Of the several methods that have been proposed for imagery data compression, the Karhunen-Loÿve procedure minimizes the meansquare error between the original and reconstructed imagery data. In spite of its optimality property, the Karhunen-Loÿve procedure has not been widely used because of its computational complexity. The main difficulty is in the computation of the eigenvectors and the eigenvalues of the covariance matrix of the imagery data since the dimension of the covariance matrix is usually large. A computationally short procedure for calculating the eigenvalues and eigenvectors of the covariance matrix is presented. We show that the eigenvalues and eigenvectors of the N × N bisymmetric covariance matrix can be obtained from the eigenvalues and eigenvectors of two N/2 × N/2 submatrices. Since the eigenvector calculations are proportional to the third power of the matrix dimension, the proposed procedure reduces the computations by a factor of four.