Analytical Algorithms for the Inverse Source Problem in a Sphere

A homogeneous sphere is excited by a point source lying inside the sphere. Analytical inversion algorithms are established concerning the determination of the physical characteristics of the sphere as well as the location and strength of the source. The basic quantity utilized in these algorithms is the total field on the sphere which is assumed to be known. The investigation of the above described problem is motivated by various applications in medical imaging. Introduction A point source inside a homogeneous spherical conductor constitutes a simplified yet realistic model for investigating a variety of applications in brain imaging [1], [2]. Locating point sources using surface measurements is an example of an inverse source problem [3]. We consider the basic static problem consisting of Laplace’s equation in a ball Vi with boundary ∂V . The goal is to identify a point source lying in Vi from Cauchy data on ∂V . There are fields both inside and outside the sphere, with appropriate interface conditions on the sphere. The inverse problem is to determine the location and strength of the source knowing the total field on the sphere. The internal conductivity is also to be found. We obtain exact and complete results by developing analytical inversion algorithms utilizing the moments obtained by integrating the product of the total field on the spherical interface with spherical harmonic functions. All the information about the primary source and the ball’s physical characteristics is encoded in these moments. The presented method is simple, explicit and exact (given exact data). Other analytic inversion algorithms for determining static point dipoles as well as acoustic point sources inside a homogeneous sphere are presented in [4]. 1 Mathematical Formulation Consider a homogeneous spherical object of radius a, surrounded by an infinite homogeneous medium. Denote the exterior by Ve and the interior by Vi. A point source lies inside the sphere at an unknown location r1 ∈ Vi. We will determine the source, using information on the spherical interface. Denote the field outside the sphere by ue and the total field inside by ui. Then, ui = u pr + u, where u is the primary field due to the source (u is singular at r1) and u sec is the secondary (regular) field. The field ue is regular and satisfies an appropriate far-field condition. The fields ue and ui are related by transmission conditions on the sphere. For the primary field, we choose a point source, u(r; r1) = A |r− r1| , r ∈ R\{r1}, (1) where A is a real constant. Introduce spherical polar coordinates (r, θ, φ) for the point at r so that the source is at (r1, θ1, φ1) with r1 = |r1| < a. Then, the transmission conditions are ue = ui and 1 ρe ∂ue ∂r = 1 ρi ∂ui ∂r at r = a, (2) where ρe and ρi are constants. Since we deal with a static problem, both ue and u are governed by Laplace’s equation. The field ue decays to zero at infinity. In the context of electrostatics, ρe and ρi are inverse conductivities. 2 Inverse Source Problem A static point source lies at r1 and generates the field u. Near the sphere (r1 < r < a), separation of variables gives the expansion