Distorted Born Iterative Method. The Born approximation assumes a linear integral equation, while the integral equation is non-linear in the object function being sought. To extend the Born approach to non-linear problems, iterative methods must be used to solve the non-linear equation. The most popular approach to this problem is the use of the DBIM [105]. To demonstrate the implementation of the DBIM, we start by the Born integral equation b E (r ) = Einc (r ) + ∫V d r ′G(r , r ′, ϵ b ) · .κ (r ) − κ Einc (r ). (2.30) ¯ 2 ′ 2 . ′ In this formulation, the measurement data are available only outside the scatterer in the inverse scattering problem. Therefore, the only item available is Esc a(r ) = E (r ) − Einc (r ) r ∈ S, (2.31) where S is some surface outside V (see Fig. 2.9 from [103]). It is only the scattered field Esc a that bears information on the scatterer. Hence, it can be written as (2.30) as b Esc a(r ) = ∫V d r ′G(r , r ′, ϵ b ) · .κ (r ) − κ Einc (r ) (2.32) ¯ 2 ′ 2 . ′ b b Now, the preceding equation is an integral equation linear in κ2(r ) − κ2. Moreover, the error in the above equation can be easily shown to be of the order (κ2(r ) − κ2)2. In addition, for the special case where Einc (r ′) is generated by a point source a located at r ′′ ∈ S , it can be written as Einc (r ′) = G¯ (r ′, r ′′, ϵ b ) · a (2.33) Then this generates b Esc a(r ) = ∫V d r ′G(r , r ′, ϵ b ) · G(r ′, r ′′, ϵ b ) · a .κ (r ) − κ r , r ∈ S (2.34) ¯ ¯ 2 ′ 2 . ′′ Because κ2(r ) is a 3-D function with support on V , a single measurement of Esc a(r , r ′) for r on the surface S for a fixed r ′′ is not sufficient to generate enough data to solve b b κ2(r ). In other words, it is not expected that information will be accurately retrieved on a 3-D function from a 2-D function. Therefore, data for a range of r and r ′′ are needed to reconstruct κ2(r ) accurately. The problem may still be ill-posed. Therefore, the iterative method is applied to find κ2(r ) approximately. However, the new κ2(r ) can be used as the estimate κ2. Then, a new G¯ (r , r ′, ϵ b ) that corresponds to this new κ2 in (2.34) has to be found. This new Green’s function can then be found. This iterative procedure is similar to Xxxxxx’x method for solving a non-linear integral equation, except that the solution is regularised at every iteration. Specifically, the inhomogeneous Green’s function G¯ (r , r ′, ϵ b ) has a simple relation- ship with the internal electric field. From the differential equations governing the Green’s function and...
