One of the goals of current research in medical ultrasound is to develop techniques to quantitatively differentiate between tissue types and tissue states on the basis of the changes in the backscattered signal caused by differences in the elastic properties of the tissues under study. In this dissertation, we proposed to classify white matter and gray matter of adult brain tissues. The first step in this investigation was to develop a linear dichotomizer for the classification of white matter and gray matter using a power spectrum eigenvector approach with Linear Discriminant Analysis (LDA). The LDA classifier is described in Chapter 4. The successful classification rates for pure white matter and pure gray matter tissues were 99% using the LDA. When the LDA was used in transition regions the successful classification rates for both type of tissues dropped to 56%. We concluded that the weakness with this classification scheme was that it did not exploit the information at different scales.