TABLE OF CONTENTS A new method for finding small inhomogeneities from surface scattering data is proposed and mathematically justified in this chapter. The presentation follows [107] and is based on [105]. The method allows one to find small holes and cracks in metallic and other objects from the observation of the acoustic field scattered by the objects.
In many applications one is interested in finding small inhomogeneities in a medium from the observation of the scattered field, acoustic or electromagnetic, on the surface of the medium.
We have two typical examples of such problems in mind. The first one is in the area of material science and technology. Suppose that a piece of metal or other material is given and one wants to examine if it has small cavities (holes or cracks) inside. One irradiates the metal by acoustic waves and observes on the surface of the metal the scattered field. Prom these data one wants to determine:
are there small cavities inside the metal?
if there are cavities, then where are they located and what are their sizes?
Similar questions can be posed concerning localization not only of the cavities, but any small in comparison with the wavelength inhomogeneities. Our methods allow one to answer such questions.
As a second example, we mention the mammography problem. Currently x-ray mammography is widely used as a method of early diagnistics of breast cancer in women. However, it is believed that the probability for a woman to get a new cancer cell in...
