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We have developed, using both partial differential equation approaches and integral equation formulations, a precise method to invert acoustic or electromagnetic scattering data from macroscopic concealed objects. Our approach makes use of the ideas associated with our exact solution of partial differential equations as described in our paper where we were able to collapse the number of equations by elimination of transcendentals therefore preserving the absolute mathematical precision inherent in the partial differential equation formulation. Our mathematical method, as a consequence, has not encountered the traditional loss of precision when inverting the scattered data. The unrestricted wavelength range allows us to penetrate any material which may surround the object and differentiate between the object and the media. For this reason we have applied our inversion scheme to landmine detection as we can penetrate and differentiate under both wet and dry conditions. Also, we are able to account, under certain conditions, for dielectric nonlinearities of material in the concealed object. Therefore, we are able to build in density dependent false colors a 3D grid representative of both the media and of the embedded object including the internal structure of the object. We have surveyed the literature on the subject of recovery of physical location of concealed objects and we have found that most of the present applications such as land mine detection, and we have found that most of the present applications have shortcomings due to the physical changes that are present in the surrounding media or the discontinuities of physical properties of the media. For all the above reasons we believe that we may have the most versatile and mathematically precise approach to the solution of this problem.
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