Linear dimension reduction of sequences of medical images: II. Direct sum decomposition

Using unitary transformations together with a previously described statistical theory for optimal linear dimension reduction it is shown how pixels in a sequence of images can be decomposed into a sum of variates, covariates, and residual vectors, with all covariances equal to zero. It is demonstrated that this decomposition is optimal with respect to noise. In addition, it results in simplified and well conditioned equations for dimension reduction and elimination of covariates. The factor images are not degraded by subdivision of the time intervals. In contrast to traditional factor analysis, the factors can be measured directly or calculated based on physiological models. This procedure not only solves the rotation problem associated with factor analysis, but also eliminates the need for calculation of the principal components altogether. Examples are given of factor images of the heart, generated from a dynamic study using oxygen-15-labelled water and positron emission tomography. As a special application of the method, it is shown that the factor images may reveal any contamination of the blood curve derived from the original dynamic images with myocardial activity.