INTRODUCTION TO ELECTRICAL IMPEDANCE IMAGING

YS FIELD IMAGING GROUP

Electrical impedance imaging system was designed to reconstruct resistivity (or conductivity) image of a subject using injected currents and voltage measurements. In electrical impedance imaging, a chain of electrodes is attached to the surface of a conducting body where different current patterns are applied and their corresponding voltage potentials are measured. Imaging distribution of electrical conductivity in an object by this method has been received considerable attention because of its low cost and possible applications such as medical imaging, corrosion detection, crack detection, etc. However, static EIT(electrical impedance tomography)imaging is still far from clinical applications due to the severe ill-posedness which is caused by the inherent low sensitivity of boundary measurements to any changes of internal tissue conductivity values. Moreover, EIT has highly nonlinear relation between the target (conductivity distribution) and the current-voltage data. So, EIT has quite different nature from CT(computerized Thomography) and MRI(Magnetic Resonance Imaging) which have linear relation between the target image value and the observed data obtained by X-ray or NMR signal.

The electrical impedance imaging system can be modeled by a elliptic partial differential equation. Suppose that an electrically conducting body, whose conductivity distribution is to be reconstruct, is occupied in the region Ω ⊂ ℝ^{n}(n = 2, 3). The conductivity, denoted by σ, is a symmetric, positive definite matrix which has jump across an interface between different regions. Across the interface, the normal component of the electric field and the tangential component of the current density are discontinuous. Our goal is to identify this interface using this jump relation carefully. For simplicity, let us assume σ is a scalar valued function. When we inject a current, denoted by a function g on ∂Ω, to the subject, electric current density J will be set up within the object with J · ν|_{∂Ω} = g where ν(x) denotes the outward unit normal vector to the boundary. Since J = −σ∇u where u is the voltage potential, after normalization u is governed by the following Neumann problem

\{ \begin{matrix}∇ · (σ(x)∇u(x)) = 0 in Ω \\ σ(x)∇u = g on ∂Ω, ∫_{∂Ω} u = 0. \end{matrix}

It is well known that, for a given g ∈ H^{−1/2}(∂Ω), the above Neumann problem has unique solution up to modulo constant within H^{1}(Ω). From now on, we denote the solution u of the Neumann by u[σ, g] because it is determined uniquely by the conductivity distribution σ and the Neumann data g.

To get better information on the target image σ, we may apply several currents g_{j}, j = 1, ·, N and measure the corresponding boundary voltages f_{j} := u[σ, g_{j}]|_{∂Ω}. In practice, N, the number of applied currents, is limited by the number of electrodes which are attached on the body.

The inverse problem of reconstructing σ is to invert the highly nonlinear map T : C → [L_{0}^{2}(∂Ω)]^{N} defined by

T (σ) = f := (f_{1}, f_{2}, ⋯, f_{N})

where C is a set of all possible conductivity σ. To search T^{−1}(f), it is necessary to interpret how the change in conductivity affect the current-to-voltage relation. Also, it should be taken account into inevitable noise in measurements and modeling errors. Indeed, since the data f always is distorted by some measurement error, we seek an approximate solution which is selected out of the set

C_{ϵ}[f] := {σ ∈ C : ∥T (σ) − f∥_{L^{2}(∂Ω)} ≤ ϵ := a given noise level}.

In order that an approximate solution is close to a true solution, the diameter of the set C_{ϵ}[f] should be reasonably small in some appropriately chosen metric. The diameter C_{ϵ}[f] is large when small perturbations of data cause large distortions to the solution in a given metric. What could be an appropriate admissible class C and reasonable metric? Considering all possible conductivity distribution σ of the human body, it is reasonable to assume that the admissible class C is contained in the set of all positive piecewise C^{1}−functions. As a metric for C, a metric involving total variation and the standard H^{1}(Ω)-norm could be effective to distinguish discontinuous images. Once a metric is fixed, the diameter C_{ϵ}[f] depend on the choice of C and applied currents pattern. To reduce the diameter of C_{ϵ}[f], unnecessary functions should be excluded from the admissible class C using a priori information about the physical structure of the body so that in a small class C the size of C_{ϵ}[f] could be shrunk significantly. Choice of applied currents also plays an important role on the size of C_{ϵ}[f]. For example, high oscillatory current-voltage data usually have poor information for presence of anomalies lying in deep inside body because most of currents flow near the boundary.

Let us focus on a simple model such as the case of piece-wise constant conductivity. Suppose that C is the set of σ which is of the form

σ = σ_{0}χ_{Ω\∪_{j} D̄_{j}} + ∑_{j} σ_{j}χ_{D_{j}}

where σ_{0}, σ_{j} are positive constants and D_{j} are smooth domains contained in Ω. We assume that the diameter of C_{ϵ}[f] is reasonably small for appropriate chosen currents, although we do not have any mathematical evidence. Then, our question is how to find one out of C_{ϵ}[f] numerically with economical value. Most of reconstruction processes use iteration schemes which require to solve the forward problem for each change of the conductivity until the calculated boundary data T (σ) is matched to the measured data f within noisy level ϵ. In these methods, reconstruction algorithms requires tremendous computational time and its iteration scheme may fail to converge to an approximate solution without taking a good initial guess. In our knowledge at the present, it is too laborious to develop instant reconstruction algorithm for the full bulk shape of D_{j}. Instead, it could be worthwhile to extract only necessary partial information of D_{j} such as location and size. This paper deals with both the mathematical theory and the numerical algorithm for these aspects.