Dynamic Link Library and Signal Conditioning System for Electrical Impedance Tomography Virtual Instrumentation

José Antonio Gutiérrez Gnecchi Departamento de Ingeniería Electrónica

Instituto Tecnológico de Morelia Morelia, Michoacán, México

biodsprocessing@aol.com

Abstract— This paper presents the implementation of a Dynamic Link Library (DLL) and Signal Conditioning System for on-line qualitative reconstruction and analysis of Electrical Impedance Tomography (EIT) images. Three different finite element circular meshes were developed for use with the DLL: 104, 464 and 1016 triangular elements. The library allows the user to choose between three different post-reconstruction options for data visualization: raw data, bilinear interpolation and radial basis function (RBF) interpolation. The DLL can thus be imported into a Virtual Instrumentation software, such as National Instruments LabWindows© and Agilent Technologies HPVEE© to ease the implementation of virtual instrumentation for EIT measurements. A case study is presented: in-phantom object tracking in laboratory test trials. The results suggest that the EIT DLL can be used as part of a virtual instrumentation program, reducing development time for on-line real-time analysis in multiple applications.

Keywords-Electrical Impedance Tomography, Dinamic Link Library, Virtual Instrumentation. data acquisition, digital signal processing

I. INTRODUCTION The success of radiation-based tomography for medical

use encouraged many biomedical and process engineers to develop similar techniques for medical and industrial processes. Although the specifications and requirements may differ between medical and industrial process applications a large number of tomographic imaging techniques have been developed for both. Consequently, since the early 1980's, three electromagnetic tomography sensing techniques have received considerable attention worldwide: electrical capacitance tomography (ECT), electromagnetic inductance tomography (EMT) and electrical impedance tomography (EIT).

Electrical impedance tomography is a soft-field sensing technique where the sensing field is altered by the material distribution and the physical properties (conductive or isolating) of the media being imaged [1].

The two-dimensional (2-D) EIT sensing principle is based on measurement of the electrical properties of the medium, (impedance). The technique consists of installing a number of electrodes (normally 8 or 16 electrodes) spaced equidistantly, in a single plane, around the periphery of the boundary/container, in contact with the object or media to be studied (Fig. 1).

Figure 1. Cross-section of a phantom with 8 electrodes for EIT.

The plane viewed by the sensors is then studied by applying a fixed-amplitude alternating current between a pair of adjacent electrodes (adjacent electrode measurement technique). The resulting voltages across pairs of the remaining electrodes are measured in all possible combinations except for redundant measurements yielding 20 and 104 measurements for 8 and 16 electrodes respectively. The data acquisition system shifts one electrode around the object and the measurement procedure is repeated until all possible combinations have been tried.

The image of the cross-section is then reconstructed using qualitative (i.e sensitivity backprojection) or quantitative (i.e. Newton-Raphson) methods [2].

II. BASIC PRINCIPLES OF 2-D EIT

A. Simplifying assumptions The basic governing equation of EIT can be derived

using Maxwell’s equations and three basic simplifying assumptions:

1) Quasi-static conditions hold.

2010 Electronics, Robotics and Automotive Mechanics Conference

978-0-7695-4204-1/10 $26.00 © 2010 IEEE

DOI 10.1109/CERMA.2010.128

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Alternating current sources are used in EIT. The frequency range of these current sources is typically 20-100 kHz. The quasi-static conditions can be justified provided that the wavelength of the potential distribution within the bounded volume is large compared with the maximum dimension of the volume so that the current varies synchronically throughout the volume. Therefore, Ohm’s law may be applied:

j Eσ→ →

= (1) where j is the current density (A/m2), σ is the

conductivity (S/m) and E is the electric field (V/m). 2) There are no electrical current sources or sinks

within the bounded volume. This statement can be said to be true in practice provided

there are no current sources within the region at the applied frequency. Thus:

0j→

∇ ⋅ = (2) where ∇ is the divergence operator.

3) The conductivity distribution, σ, is isotropic. This assumption is made by most workers since the early

80´s due to the added complications resulting from allowing anisotropy. Now, the relationship between the electric field intensity E and potential φ is given by:

E ϕ→

= ∇ (3) Combining equations (1), (2) and (3) yields Poisson’s

equation: ( ) 0σ ϕ∇⋅ ∇ = (4)

Equation (4) describes the potential distribution everywhere within an inhomogeneous isotropic region. For the special case of a homogeneous conductivity distribution, σ is a constant and the relationship further reduces to Laplace’s equation:

2 0ϕ∇ = (5) Poisson’s equation is a second order partial differential in

φ and a unique solution exists if sufficient boundary conditions are specified: Dirichlet conditions (known voltages on the boundary), Neumann conditions (known currents on the boundary) or Robin conditions (combination of both). In EIT, the applied alternating current is known and the resulting voltages developed across pairs of electrodes are measured along the periphery of a given volume or vessel. Therefore the boundary conditions in 2-D EIT are:

0

0

at earth

between electrodes

j for each electrode

ϕϕση

ϕση

=

⎛ ⎞∂ =⎜ ⎟∂⎝ ⎠⎛ ⎞∂ =⎜ ⎟∂⎝ ⎠

(6)

The governing equation (5) is a very complicated partial differential equation and there is no analytical solution for an arbitrary conductivity distribution. Therefore it is necessary to employ numerical methods to solve the forward and inverse problems.

B. Forward problem

The initial step in resistivity/conductivity image reconstruction requires solving the forward problem: given the resistivity/conductivity distribution within the boundary, σ, into which current is injected and corresponding voltage measured, find the internal voltage and current density functions. Amongst the most common numerical techniques are the Finite Element Method (FEM) and the Finite Difference Method (FDM). The FEM gives a piece-wise approximation to the Poisson’s equation whereas the FDM gives a point-wise approximation. For this reason it is common to find in the literature that FEM is usually employed. The use of the FEM largely simplifies the solution of equation (4) into an algebraic problem (linear set of equations), which can be solved using numerical techniques [3]. The initial step consists on discretizing the volume under consideration into small elements with constant conductivity (Fig. 2A). Triangular elements are normally used (Fig. 2B) because they are the simplest polygonal elements into which any two-dimensional region can be subdivided. The finite element method seeks an approximation function Φ(x,y), to the exact solution φ(x,y) in a piece-wise manner. It can be shown that the conductance matrix can be obtained from (7):

[ ]Y C→ →

⋅Φ = (7) The Y conductance matrix is an n x n sized matrix where

n is the number of nodes. This matrix is symmetrical, positive definite, double centred and sparse (Fig. 3). Φ is an n-sized vector which represents the voltage at n points inside and on the boundary of the vessel. C is an n-sized vector which contains the applied current on the boundary.

Figure 2. A) FEM Model of a circular vessel and B) two-dimensional

triangular element.

Figure 3. Lower non-zero values and diagonal positions (1016 element mesh and 561 nodes).

649

11

22

3211 15

3322 23

4432 33 35

5144 45

51 53 54 5553

54

55

0 0 00 0 00 00 0 0

00

Tkkk

k kk

k kk

k k kk

k kk k k k

kkk

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥→⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥

⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

The computational time required for solving the linear set of equations can be reduced using sparse matrix and vector techniques. The C++ code used in this work uses Cholesky factorization and skyline storage scheme (8):

(8) The skyline storage scheme reduces memory space usage

and computational time by storing only the significant values located in the lnz (lower non-zero) diagonal region. This results in a considerable reduction of memory usage. For instance, a 104 triangular elements and 69 nodes matrix occupy originally 2415 lnz locations; for 464 and 1016 elements the original number of locations are 35245 and 146611 respectively. Using the skyline storage scheme reduces the memory allocation usage to 449, 2865 and 8691 for 104, 464 and 1016 triangular elements FEM respectively, and speeds up the computational process considerably.

C. Inverse problem The resistivity values are obtained by solving the inverse

problem: given the internal voltage and current density distribution, find the internal resistivity ρ.

In 1971 Geselowitz demonstrated that the change in conductivity distribution in a given region inside a conductive body can be computed from the observed change in the mutual impedance acquired on the object’s boundary (Fig. 4). In the early 1980s [4], the forward problem of relating a conductivity distribution to resulting transfer impedance was defined by using the sensitivity theorem of Geselowitz, whose compensation theorem states that if the conductivity distribution inside the object changes from σ(x,y) to σ(x,y) + Δσ(x,y) the change in mutual impedance Δz throughout the volume is given by (9):

dv

IIz

v ψφ

σσψφσ )( Δ+∇∇Δ−=Δ ∫ (9)

+Iφ

-Iφ

+Iψ

-Iψ

AB

C

Dσ(x,y)

Figure 4. Potential distribution, ψ, when current Iψ is applied to electrode

port A-B and potential distribution, φ, when current Iφ is applied at the electrode port C-D

If the change of conductivity Δσ is very small, it can be shown that (9) can be simplified to (10)

dvII

zv ψφ

ψφσ ∇∇Δ−=Δ ∫ (10)

The potential gradients ∇ φ and ∇ ψ are now independent of Δσ and the dependence of a small change in Δz in the impedance boundary data to small changes of conductivity within the volume is governed by the integral (11):

ψφ

ψφIIv

∇∇∫ (11)

which is termed the sensitivity coefficient. For a discretized volume this integral corresponds to the sensitivity coefficient matrix S. A number of researchers have addressed the linearized EIT image reconstruction problem using quantitative and qualitative methods. This work uses the qualitative weighted sensitivity backprojection method where the reconstructed pixel grey levels are obtained by (12):

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

mref

p

ref

p

ref

p

nmnn

m

m

vv

vv

vv

sss

ssssss

g

ln

ln

ln

2

1

11

22221

11211 (12)

where υp is the perturbation voltage and υref is the reference value. In simple terms, this means that although the backprojection method does not yield the exact impedance values throughout the volume, the reconstructed images represent the changes in impedance values within the bounded volume.

D. Image post-processing The result of the image reconstruction process is a set of

impedance values corresponding to each triangular element (Fig. 5A).

Figure 5. A) Reconstructed values ,B) interpolation and C) 10-level contour tracing procedure. D) Isometric view.

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It is necessary to conduct further post-processing operations such as interpolation (Fig. 5B) and contour tracing (Fig. 5C) to extract useful information for image analysis. A common choice is bilinear interpolation (Fig. 6A). For the 2-D case, the task is to find intermediate points p(x’,y’) given by:

)1,1( )1,( ),1( ),( )','( 4321 +++++++= yxpdyxpdyxpdyxpdyxp (13)

where the bilinear interpolators di:

∑=

−= 4

1

1

jj

ii

d

dd (14)

represent the distances from the location of p(x’,y’) to the four neighbouring points. An alternative method previously used for image analysis [5][6] and on-line control purposes [7] is Radial Basis Function interpolation (RBF) (Fig. 6B). For two-dimensional interpolation the aim is to find an approximation:

( ) ( )( )2 2

2

2 2

1

( ) ( )

2

1

( , ) ,

( , )

i k j k

n

k k kk

x x y yn d

kk

F x y w x x y y

F x y w e

φ=

⎛ ⎞− + −⎜ ⎟−⎜ ⎟⎜ ⎟⎝ ⎠

=

= − −

=

∑

∑

(15)

Figure 6. A) Bilinear interpolation procedure and B) Radial Basis Functions.

E. Software for EIT image reconstruction Over the past 10 years, there has been a significant effort

to produce open-source, modifiable software that can reduce EIT development time, provide means for comparison with new developments and facilitate worldwide contribution.

For instance, the EIDORS software suite for MATLAB, has evolved from a 2-D mesh generation software to a more versatile suite that includes multiple algorithm support, generalized model formats, interface software for common EIT systems and Octave and MATLAB support [8]. Although it is possible to use Virtual Instrumentation (VI) software, such as HPVEE, to access the EIDORS software suite through the MATLAB engine, the frequent calls between the programs reduce the overall speed restricting its use for on-line EIT image reconstruction and analysis.

Here the author proposes that coding the EIT image reconstruction routines as dynamic link libraries, can also facilitate EIT development and improve the speed for on-line image analysis.

III. EIT VIRTUAL INSTRUMENTATION. Virtual Instrumentation can be considered as a collection

of specialized subunits [9] that make up to instrument as a whole. The usefulness of VI for diverse applications is now widely addressed for a diverse number of applications ranging from signal processing [10] to control [11]. One of the advantages is that a virtual instrument can be constructed using commercial software and off-the-shelf data acquisition systems [12] as well as software and hardware designed for any specific application.

The proposed VI scheme for EIT measurements is shown in Fig. 7. The black colored lines show the schematic diagram of the virtual instrument implemented in this work. The grey lines indicate that, being a modular instrument, any number of data acquisition systems, signal source(s) and/or software options can be chosen to build the virtual instrument.

.

Figure 7. EIT VI A) personal computer B) different user interfaces, C)

DAS. D) EIT SCS, E) Phantom/object to be studied.

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A. Hardware A personal computer is the basis for the virtual

instrument. A DT3010/32 (Data Translation©) data acquisition system (DAS) installed in the PCI port was used for this work. The EIT Signal Conditioning System (EIT SCS) exploits the capabilities of the DT3010/32 board to generate and measure data simultaneously. The analog sinewave feeds a Voltage Controlled Current Source (VCCS) to produce the current excitation signals. The voltage sinewave can be obtained from the DAS or from a signal generator triggered from the DAS. A multiplexer array is controlled through the digital output port to select the excitation and measurement electrode pairs. The resulting voltage signals are fed back to the PC through the DAS analog input port.

B. Software The EIT image reconstruction software coded as DLL,

solves the forward and inverse problems and performs the data interpolation procedures described in section 2. Fig. 8 shows and example program using HPVEE. Once the DLL is imported the EIT solver functions are available throughout the program. The EIT reconstruction software requires a number of parameters: mesh size (104, 464 and 1016 elements), interpolation choice (raw data, bilinear, RBF), interpolation grid size and RBF Gaussian width σ.

IV. 4. RESULTS AND DISCUSSION Fig. 9 shows the experimental set-up for on-line detection

of object tracking using the instrument.

Figure 8. Example program using HPVEE

Figure 9. A) PC, B) DT3010 screw terminal panel, C) EIT SCS, D) Phantom. E) optional signal generator and F) power supply.

Figure 10. A) Initialization and calibration processes, B) data acquisition and image reconstruction, C) post-processing, interpolation and image

enhancement.

A typical on-line EIT program is shown in Fig. 10. The program starts by performing initialization and calibration operations by taking a reference image prior to the insertion of perturbation objects. Conductive (1/2” copper pipe) and isolating (1/2” PVC pipe) objects were then inserted in the phantom filled with tap water (280 mS/cm). First, the pipe was moved around the phantom for evaluating the on-line object tracking capabilities of the virtual instrument. Fig. 11 shows a sequence of images obtained while the PVC pipe is moved inside the phantom from the 4 o’clock position towards the 1 o’ clock position. In a similar manner, Fig. 12 shows the sequence of images obtained using a copper pipe, moving inside the phantom.

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Figure 11. Tracking the A) PVC pipe from B) 4 o’clock C) through to D) 1 o’clock.

Figure 12. Sequence of resulting images from moving the copper pipe from

A) the 4 o’clock B) through C) to the 1 o’ clock position.

The total update time is approximately 2.6 seconds. The data frame acquisition process takes 0.152 seconds using a 10 kHz excitation signal and 15 cycles per measurement using a PC Pentium IV dual core at 3GHz. Since the data frame capture is faster than the image reconstruction, moving the object inside the phantom appears as a blurred motion artifact. Although the proposed VI, using the image interpolation procedures is not suited to image fast process (i.e. fast moving flows) it could be suitable for static measurements and/or measurement of slow processes (liquid transport through porous media). If a more modest image is required, raw data can provide up to 5 images per second that can later be processed off-line. Still the proposed scheme allows rapid prototyping and development of EIT measurement systems.

V. CONCLUSIONS Implementation of the image reconstruction software as a

DLL permits the user to build a versatile virtual instrument for multiple applications. Interpolated images are more useful for image analysis than using raw data. The price to pay is an increased computational time. Although the total update time can result slower than that of dedicated EIT systems, a VI scheme allows rapid prototyping for static and relatively slow processes with considerably useful images. When the process occurs rapidly, a lower resolution, raw data image can be used to elucidate the process dynamics at a rate of 5 images per second. The modular nature of the VI, allows the use of multiple DAS and/or particular EIT signal conditioning systems. The case study presented here uses the HPVEE program but any other Virtual instrumentation software that permits the use of DLL compiled functions can also be used. Overall the system described in this work preserves the modular nature of a VI system.

VI. ACKNOWLEDGEMENT The author acknowledges the financial support from

CONACYT under grant CB-2008-01-107083.

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