FAQ8 OBTAINING A PERMITTIVITY IMAGE FROM THE CAPACITANCE MEASUREMENTS (Basic Image Reconstruction)

The number of pixels in the image usually exceeds the number of capacitance measurements by several orders of magnitude. Unfortunately, it is not possible to obtain a unique solution for each image pixel when the number of pixels in the image exceeds the number of capacitance measurements. Furthermore, image distortion can occur because ECT is an inherently soft-field imaging method (the electric field is distorted by the material distribution inside the sensor). Back to top

in many cases, the contrast between the permittivities of the materials inside the sensor is small, resulting in only limited image distortion. This allows approximate linear algorithms to be used to relate the capacitance measurements to the pixel values in the image and vice-versa. However, if the field distortion is severe, more accurate non-linear algorithms must be used. Back to top

The method which has been used with greatest success to-date is known as Linear Back Projection (LBP), and is based on the solution of a set of forward and reverse (or inverse) transforms. . Back to top

The forward transform is a matrix equation which relates the set of inter-electrode capacitance measurements C to the set of pixel permittivity values K. This transform assumes that the measured inter-electrode capacitances resulting from any arbitrary permittivity distribution K inside the sensor will be identical to those obtained by summing the component capacitance increases which occur when each pixel has its defined permittivity, with all other pixels values set to zero. This forward transform is defined in the equation below, where bold characters represent matrices :

C = S.K

K is an (N x 1) dimensioned matrix (where N is 1024 for a 32 x 32 grid) containing the set of N pixel permittivity values which describe the permittivity distribution inside the sensor (the permittivity image).

S is the forward transform, usually known as the sensor Sensitivity Matrix. S has dimensions (M x N) and consists of M sets (or maps) of N (typically 1024) coefficients, (1 map for for each of the M capacitance-pairs), where the coefficients represent the relative change in capacitance of each capacitance pair when an identical change is made to the permittivity of each of the N (1024) pixels in turn. Back to top

In principle, once the set of inter-electrode capacitances C have been measured, the permittivity distribution K can be obtained from these measurements using an inverse transform Q as follows:

K = Q.C

Q is a matrix with dimensions (N x M) and, in principle, is simply the inverse of the matrix S.

It is only possible to find the true inverse of a square matrix (where M = N). In physical terms, this is confirmation that it is not possible to obtain the individual values of a large number of pixels (eg 1024) from a smaller number of capacitance measurements (eg 66). As an exact inverse matrix does not exist, an approximate matrix must be used. The LBP algorithm uses the transpose of the sensitivity matrix, S' which has the required dimensions (N x M). Back to top.

Although we have no means of knowing which pixels have contributed to the capacitance measured between any specific electrode-pair, we know from the sensitivity matrix S that certain pixels have more effect than others on this capacitance. Consequently, we allocate component values to each pixel proportional to the product of the electrode-pair capacitance and the pixel sensitivity coefficient for this electrode-pair. This process is repeated for each electrode-pair capacitance in turn and the component values obtained for each pixel are summed for the complete range of electrode-pairs. Back to top

The LBP algorithm produces approximate, but very blurred permittivity images. For example, a typical image, for a dielectric tube inside a 12-electrode sensor, is shown in the first figure below. The LBP algorithm acts as a spatial filter with a lower cut-off frequency than that of the fundamental filter (as shown in the second figure below) and consequently produces sub-optimal images from a given set of input data. Back to top

ECT image of a dielectric tube (note the blurring)

LBP algorithm limits achievable image resolution

The forward transform (sensitivity matrix) must be calculated (or measured) for each individual sensor as a separate exercise prior to using the sensor with an ECT system. One method for calculating the sensitivity coefficient S of a pixel for an electrode-pair (i-j) is based on the use of the equation below.

S = Ei . Ej . dA

where Ei is the electric field inside the sensor when one electrode of the pair i is excited as a source electrode, Ej is the electric field when electrode j is excited as a source electrode and the dot product of the two electric field vectors Ei and Ej is integrated over the area A of the pixel.

For a sensor with internal electrodes, the components of capacitance due to the electric field inside the sensor will always increase in proportion to the material permittivity when the sensor is filled uniformly with higher permittivity material. However for sensors with external electrodes, the permittivity of the wall causes non-linear changes in capacitance, which may increase or decrease depending on the wall thickness and the permittivities of the sensor wall and contents. Back to top

The set of sensitivity coefficients for each electrode-pair is known as the sensitivity map for that pair. Back to top

For circular sensors with either internal or external electrodes, it is possible to derive an analytical expression for the electric fields and in this case, the sensitivity coefficients (and also the electrode capacitances) can be calculated accurately. For more complex geometries, numerical methods can be used to calculate the sensitivity coefficients. It is normally only necessary to calculate a few primary sensitivity maps for the unique geometrical electrode pairings, as all of the other maps can be derived from these by reflection or rotation. A set of primary maps for an 8-electrode sensor operating under protocol 1 is shown below. Back to top

Primary sensitivity maps for an 8-electrode sensor

Last updated 17-05-2002

Copyright 2001 PTL