FAQ9 ADVANCED IMAGE RECONSTRUCTION METHODS

It is possible to improve the image resolution and accuracy to values much closer to the theoretical limit by the use of iterative techniques. The idea is to use the forward and inverse transforms alternately to progressively correct the pixel values, and is based on the assumption that the forward transform is reasonably accurate if the field distortion is low but that the inverse transform may be very inaccurate. This technique is conceptually similar to the practice of correcting the distortion of an imperfect amplifier by the use of negative feedback. Back to top

The iterative method operates as follows:

The set of capacitances C1 for one image frame are measured and a set of initial pixel values K1 are calculated using (the inaccurate) forward transform. These approximate permittivity values K1 are then used to back-calculate a set of capacitances C2 using the (relatively accurate) inverse transform. A set of error capacitances dC = (C2 - C1) are calculated and used to generate a set of error permittivities dK using equation the inverse transform. These error permittivities are then used to correct the previous set of permittivities to generate a new set of pixel values K2, where K2 = (K1 - dK ) . These new permittivities K2 are then used to calculate a new set of capacitances C2 and the sequence is iterated until the permittivity values converge to the correct solution.

A number of additional steps are possible, including truncating the image pixels to lie within the known calibration range at each iteration and applying gain and truncation factors to the error capacitances. However, it is important to check that the permittivity values converge in order to ensure a valid solution. Experience shows that this method can produce images of good resolution, close to the theoretical maximum achievable with a given measurement protocol and number of electrodes. An example of an image obtained using this technique is shown below. This image was constructed using the same data used to reconstruct the blurred image shown in the rsponse to FAQ8 Q8 Back to top

Improved image of dielectric tube obtained using iterative methods.

Although the iterative method produces good images, it cannot be used on-line because of the time taken to carry out the relatively large number of iterations required to produce the image. It is possible to develop better inverse transforms for Q by using more advanced mathematical concepts for deriving approximate inverses of matrices. Two examples are methods developed by Landweber and Tikhonov. Back to top

In Landweber's method, the inverse transform Q is given by the equation:

QL = V . F ( W, t, R ) . U'

where: V, W and U are the matrices obtained by applying the Single Value Decomposition (SVD) process to the sensitivity matrix S , F is the SVD filter function matrix, U' is the transpose of U and

f = ( (1 - ( 1 – w )R ) / w

where: f is one element of the filter matrix F, w is one element of the diagonal matrix W, L is the Landweber transform parameter and R is the number of iterations.

In Tikhonov's method, the inverse transform Q is given by the equation:

QT = S' . ( S . S' + t . I )-1

where: S is the sensitivity matrix, S' is the transpose sensitivity matrix, t is the Tikhonov transform factor, and I is the identity matrix.

Some insight into the mechanism of operation can be seen from the figure below which shows the Landweber transform plotted as an equivalent set of primary sensitivity maps. By comparing these transforms with the figure shown in response to FAQ8 Q11 it is clear that the Landweber transforms have far more structure. Consequently, they produce more detailed images from the same capacitance data, as shown in the second figure below. However, they also produce some spurious artefacts in the image. These can be reduced, while further improving the image, by carrying out a small number of iterations using the appropriate inverse transform in place of the transpose sensitivity matrix, as the third figure below, obtained after only 5 iterations, illustrates. The attraction of these techniques is that they are fast enough for use on-line. Back to top.

Improved image of dielectric tube obtained using iterative methods.

Improved image obtained in a single step using Landweber's method.

Further improvement obtained by iterating Landweber's method and truncating pixel values.

Last updated 17-05-2002

Copyright 2001 PTL