The inverse conductivity problem consists of determining an unknown conductivity distribution inside a physical body from electric measurements on the surface of the body. This problem was first introduced in mathematical form by Calderón in 1980. Most important application of Calderón's problem is the medical imaging technique known as electrical impedance tomography (EIT). For example, one may examine a cross-section of a patient's chest. The tissues and organs in the body have different conductivities, a fact which enables one to form an image from the conductivity distribution. By applying a basis of current patterns on electrodes attached around the patient's chest and measuring the resulting voltages on the electrodes, the solution to the inverse conductivity problem yields a 2-D image of a cross-section of the chest. In this geometry, several of the clinical applications include monitoring heart and lung function, diagnosis of pulmonary embolis (a blood clot in the lung), diagnosis of pulmonary edema, monitoring for internal bleeding, and the early detection of breast cancer.
Other applications of EIT include subsurface flow monitoring and remediation, underground contaminant detection, nondestructive evaluation, and industrial process monitoring. For more information on EIT, see the articles Cheney, Isaacson & Newell [SIAM Review 41 (1999)] and Mueller & S [SIAM J. Sci. Comp. 24 (2003) PDF].
The main direction of my EIT research is designing a practical reconstruction algorithm based on theoretical uniqueness proofs. The starting point is the joint work with Jennifer Mueller and David Isaacson [Inverse Problems 16 (2000) PDF], where a practical algorithm is presented based on the two-dimensional uniqueness proof by Adrian Nachman [Annals of Mathematics 143 (1996)]. A regularization approach for the method is introduced in the article Mueller & S [SIAM J. Sci. Comp. 24 (2003) PDF]. Our algorithm involves numerical solution of a d-bar equation originating from inverse scattering theory. A fast algorithm for the solution of d-bar equations is presented in Knudsen, Mueller & S [J. Comp. Phys. 198 (2004) PDF].
The above numerical work is tested on simulated data only. We examine the properties of our method when applied to measured data in Isaacson, Mueller, Newell & S [IEEE Tr. Medical Imaging 23 (2004) PDF], and find that our method produces useful reconstructions and recover the conductivity values more reliably than linearized algorithms. See the figure below for a reconstruction of a chest phantom from data measured in the Rensselaer Polytechnic Institute EIT laboratory.
More recently, the regularized d-bar method was applied to in vivo human data. The reconstructed video sequence enables monitoring cardiac activity, as reported by Isaacson, Mueller, Newell and S in [Physiological Measurement 27 (2006) PDF].
In addition to the above reconstruction method I am interested in algorithms recovering partial information about the conductivity from boundary measurements. One important problem is recovery of conductivity and its normal derivative at the boundary, since this is crucial input information for algorithms for reconstruction of the conductivity in the interior. A boundary reconstruction method is introduced in Nakamura, S, Tanuma & Wang [ Computing (2005) PDF]. Moreover, together with Masaru Ikehata I have published two works on finding inclusions in known background conductivity, see the articles [Inverse Problems 16 (2000) PDF] and [Inverse Problems 20 (2004) PDF]. See the figure below for the recovery of inclusions in homogeneous background from simulated noisy data using Mittag-Leffler's function.
In a joint effort with Takanori Ide, Hiroshi Isozaki, Susumu Nakata and Gunther Uhlmann, hyperbolic geometry was used to derive an algorithm for detecting inclusions in known background from localized EIT measurements. See the submitted manuscript here; the article will appear in Communications on Pure and Applied Mathematics. The figure below shows an example conductivity, the domain that can be recovered by probing the inclusion using spheres, and reconstruction from simulated data. Relative error of reconstruction is 16%.
