Linear dimension reduction of sequences of medical images: I. Optimal inner products

A general theory is presented for minimizing noise in linear dimension reduction of sequences of medical images when the factors and the covariance matrix and mean of the noise are given. A dimension reduction is optimal when all diagonal elements in the covariance matrix of the noise in the signal (factor) space are minimized. This occurs when the noise in the signal space is uncorrelated with the residual noise. Expressions are given for the resulting covariance matrix of the noise in the signal space. Many optimal inner products exist, which all result in the same optimal dimension reduction. Given any pair of inner products for signal space and residual space, a combined inner product exists that is also optimal. If the covariance matrices of the noise in different pixel vectors are not multiples of each other, different pixel vectors may have different optimal inner products. The averaging process in generating images from tomographic projections tends to make the covariance matrices more uniform.