Electrical impedance tomography (Part III) - The nonlinear model

The nonlinear model of EIT will be discussed. To obtain a tomographic image of the object a matrix of electrodes is superimposed on it's surface and series of measurements are produced. 

During the one measurement through the current electrodes the current is passed, while from measurement electrodes the potentials are measured relative to a reference electrode. This procedure is repeated with new current electrodes more and more to obtain dataset of observations.

 During the measurement procedure the observation vector

is formed, were:

- is the  number of measurements,

 - is the number of measurement electrodes, where:

 - is total number of electrodes,

- is number of current electrodes.

In the observations the information of the distribution of conductivity inside the object  is encapsulated. This information have to be retrieved.

To find the electrical conductivity we have to solve the EIT equation with respect to unknown conductivity distribution

but the voltage measured only on the surface of the object, whereas the potential distribution inside we do not know.

Consider the problem of electrical impedance tomography as a non-linear model relating the unknown discrete distribution of electrical conductivity and the vector of discrete observations :

where:

- is the vector of observations,

- is the vector containing the observations without noise,

- is the vector containing the noise,

 - is the operator of the forward problem.

The unknown discrete conductivity distribution have the form:

To solve this equation with respect conductivity the method of least squares estimation is used where is necessary to minimize the functional of the form [1]:

After expantion it in a Taylor series the formula of Newton-Raphson iteration for finding conductivity has the following form [1]:

where:

- Jacobian.


The step size is used to control the convergence of the algorithm:

In many practical applications, the computation of step size can be quite resource intensive and slow.

Gauss-Newton method is obtained by removing it:

Sources:
[1] Kolehmainen, Ville. Novel approaches to image reconstruction in di usion tomography: Ph.D. thesis — Kuopio University Publications C. Natural and Environmental Sciences, 2001.—164 p