Abstract: The detection method and apparatus detects abnormalities having a first signal propagation property within a host medium having a second signal propagation property based upon signals which have been reflected, scattered and/or absorbed by the host medium and any abnormalities therein. The detection apparatus includes a signal source for introducing a signal into the host medium which varies with respect to a predetermined signal parameter, such as time, frequency or source location. The detection apparatus also includes a plurality of detectors disposed along some or all of the boundary of the host medium for detecting the signal following propagation through the host medium and any abnormalities within the host medium. The detection apparatus further includes a signal processor for constructing at least one partial differential equation to describe the propagation of a regularized signal through a medium having the same shape as the host medium.