Compression of Breast Cancer Images By Principal Component Analysis

The principle of dimensionality reduction with PCA is the representation of the dataset ‘X’in terms of eigenvectors ei ∈ RN  of its covariance matrix. The eigenvectors oriented in the direction with the maximum variance of X in RN carry the most      relevant information of X. These eigenvectors are called principal components [8]. Assume that n images in a set are originally represented in matrix form as Ui∈ Rr ×c,  i = 1,......,n, where r and c are, repetitively, the number of rows and columns of the matrix. In vectorized representation (matrix-to-vector alignment) each Ui is a N = r × c- dimensional vector ai computed by sequentially concatenating all of the lines of the matrix Ui. To compute the Principal Components the covariance matrix of U is formed and Eigen values, with the corresponding eigenvectors, are evaluated. The Eigen vectors forms a set of linearly independent vectors, i.e., the base {φ} n i=1 which consist of a new axis system [10]