Application to the Inverse Scattering Problem for Low Contrast Mediums

and belong to the space ^ - S

 
 

  . Then the integral equation (4.84) has a unique solution in the space , if and only if, the unique solution of (4.90) coincide with the unique solution of (4.91). The mapping from the space to



is continuous and given by

<

is defined by (4.77).

4.4. Application to the Inverse Scattering Problem for Low Contrast Mediums

The scattering of an acoustic wave by a penetrable inhomogeneous medium (a scatter) is normally discussed in either the frequency or the time domain. This means that the scatter is either illuminated with a time harmonic/periodic field, or with a time varying field. In the first case, the underlying PDE is the Helmholtz Equation, whereas in the later case, the PDE will be the Wave Equation. For the Inverse Medium Problem properties of the bounded inhomogeneous medium may be found from measuring the fields outside the inhomogeneous medium.

When considering medical applications, two different contexts appear naturally. One is where only the location of an object is of interest, and this could be when trying to examine whether there is a tumor in the body and how big it is [CCM00]. The other is when internal properties are needed - such as for instance the glucosamine production of a tumor for predicting survival rates and recommending treatment plans for patients.

The latter is a much more difficult problem, since local variations are needed, in contrast to the first, where only an overall change is of interest.

For the method to be proposed here the forward scattering problem (of illuminating an object) will be discussed in both the frequency and time domain. The incident field will be the composition of an incident spherical field and the scattered field hereof by a homogeneous medium. The mathematical formulation of these two problems are given in table 1

The relationship between the two formulations are that ^{  
*  ! 
  }. If in addition the medium is assumed to be non-absorbing the coefficients in the PDE’s will

Frequency domain equations Time domain equations

Table 1. Table of the mathematical formulation for the acoustic scattering problem in the frequency and the time domain respectively.

satisfy

denotes the characteristic function for an open set ,^{ } the speed of sound outside the inhomogeneous medium and^
 the refractive index.

The inverse time dependent problem is from knowledge of the incident and scattered fields for all times on a sphere encircling the inhomogeneity to reconstruct the refractive index.

A common way of representing the solutions of the scattering problem in the fre-quency domain is by the Lippmann-Schwinger integral equation.

    

Since ^{   } has compact support equation (4.95) is a Second Kind Fredholm Prob-lem for ^{  } with

B

the Green’s function for the Helmholtz Equation and ^ the incident field.

We construct an additional integral equation that can be used for solving the prob-lem. Let^{ 
*    } denote free-space solutions (^{ }  constant respectively). By introducing ^{     } and denoting ^{  } the convolution between^{  } it follows

from Green’s Theorem that Assuming that the contrast of the medium is low and without absorption (^{
 ; } ,



 ;   ) the second terms in (4.97) and (4.96) respectively are of second order compared to the r.h.s. In physical terms this means that the scattered field is very small, since for a low contrast medium most of the field will be transmitted through the medium.

Hence, for mediums with low contrast (4.97), (4.96) reduces to

>    

which are equations that we will be able to construct explicit solutions for.

4.4.1. Reduction to a One-dimensional Integral Equation. After having listed the forward problem in table 1, and defining the necessary integral equations, we introduce the inverse problem of finding ^
 and elaborate on how the transformation theory applies to solving (4.98). Let^
 be compactly supported in^{  *A} , a ball with center at zero and radius^ .

For the 3-dimensional Helmholtz Equation, a general free-space solution ^{ } can be written as^{
   
  4 4Z} 

of (4.99) (or (4.96)) reduces to Our inverse problem of finding^
 is therefore reduced to that of finding^


-  

If ^8
* satisfies the assumptions of theorem 4.15 then the result of theorem 4.15 applies to^
 -  .

Corollary 4.16. Assume ^
* in (4.101) satisfy the conditions in Theorem 4.15, then equation (4.101) has a unique solution and the permittivity can be found as



F 
   

        

- 
 *44Z   9  (4.102)

4.5. Conclusion

We have constructed classes of integral kernels such that the Generalized Fourier Trans-form pairs^{  }

S

 are continuous mappings and such that elements in the classes are found through an Abelian map^{ 
} .

New spaces of -transform with inverse maps have been constructed and a space of kernels that are products of a Bessel and a Hankel function have been constructed.

To the authors knowledge, no theory handling these classes of kernels exist.

The theory’s application to the Inverse Medium Problem has been established by solving a first kind problem representation explicitly. This representations of the problem in terms of time dependent data is also new to the authors knowledge.

Bibliography

[Tr67] F. Treves, Topological Vector spaces, Distributions and Kernels. Academic press, 1967.

[GR79] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products. Academic Press, 1979, ISBN 0-12-294760-6

[SSK98] A. Kilbas et. al On inversion of H-transform in^
 -space Internat. J. Math. & Math. Sci Vol 21,(1998), No 4

[CK98] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory Springer Verlag, 1998, ISBN 354062838X

[CCM00] D.Colton J. Coyle and Peter Monk, Recent Developments in Inverse Acoustic Scattering Theory Siam Review, 42 (2000), No 3, pp 369-414