Electrical impedance tomography (Part IV) - Jacobian

In EIT problem solving it is necessary to calculate the derivatives of the measured voltages respect to conductivity. Complete matrix of partial derivatives is called the Jacobian, sometimes called the sensitivity matrix and its rows are called sensitivity maps [1].

Jacobian - the matrix of the sensitivity of the voltage across the electrodes to the change in electrical conductivity within the object in the application of different current patterns.

The Jacobian of size

is as follows:

The Jacobian can be computed quite effectively as an integral of the scalar product of gradients of current and measurement fields [1]. Measurement fields can be calculated assuming that the measurement patterns are converted into current patterns.

The potential field produced by passing a current through the current electrodes  is:

The potential field produced by passing a current through the measuring electrodes is:

- is the m-th measurement at the d-th current sheet template, then the formula for calculating the sensitivity matrix is as follows:

Properties of sensitivity matrix (Jacobian) largely determine the characteristics and behavior of the solution of the EIT.

To investigate the properties of the matrix we use the singular value decomposition (singular value decomposition, SVD). Singular value decomposition of the matrix J of size RxK is its representation in the following form:

where:
 U     is orthogonal matrix of size RxR,
 V     is orthogonal matrix of size KxK,

is matrix of size RxK.

On the main diagonal of matrix the singular numbers are contained in descent order and all non-deagonal elements are equal to zero.

From singular value decomposition we can calculate the Moore-Penrose pseudoinverse matrix:

Condition number of matrix is:

where:
  - is the largest singular value,
 

- is the smallest singular value,

- is a norm of matrix.

Singular value decomposition is an important tool for ill-posedness investigation. The plot of the eigenvalues characterizes the instability of the inverse problem (Fig. 1). 

Fig. 1

Note the properties of the sensitivity matrix:

1. Jacobian matrix is rank-deficient. This suggests that one or more rows and columns of the matrix are linear combinations of other rows and columns. The linear dependence is showed by a jump in singular values ​​of singular spectrum. As a rule, the smallest singular values ​​equivalent to or less than machine precision.
2. The Jacobian has a large condition number.
3. A relative decline of singular values ​​and Picard coefficients have a special meaning. For ill-posed problems with noisy observations a stable solution exists when a Picard condition is satisfied [2]. According to it, the coefficients

should fall to zero faster than the singular values. From (Fig.1) we can see that not all singular numbers should be used to get suitable solution.

Sources:

[1]  Polydorides, N. Image Reconstruction Algorithms for Soft-Field Tomography: Ph.D. thesis / Polydorides — Manchester, United Kingdom: N. UMIST, 2002. —250 p.

[2] Hansen P. C. Regularization Tools A Matlab Package for Analysis and Solution of Discrete Ill-Posed Problems / Hansen P. C. —Denmark: Technical University of Denmark, Department of Mathematical Modeling, 1998. — 111 p.