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1 Forward Problem Solution as the Operator of Filtered and Back Projection Matrix to Reconstruct the Various Method of Collecting Data and the Object Element in Electrical Impedance Tomography K. Ain 1,3, D. Kurniadi 1, Supriyanto 1, O. Santoso 2, R.A. Wibowo 3 1 Engineering Physics Program, 2 Informatics Program, ITB, Bandung - Indonesia 3 Physics Departement - Airlangga University, Surabaya Indonesia khusnulainunair@yahoo.com Abstract. Back projection reconstruction has been implemented to get the dynamical image in electrical impedance tomography. However the implementation is still limited in method of adjacent collecting data and circular object element model. The study aims to develop the methods of back projection as reconstruction method that has the high speed, accuracy, and flexibility, which can be used for various methods of data collection and model of the object element. The proposed method uses the forward problem solution as the operator of filtered and back projection matrix. This is done through a simulation study on several methods of data collection and various models of the object element. The results indicate that the developed method is capable of producing images, fastly and accurately for reconstruction of the various methods of collecting data and models of the object element. Keywords: forward problem, back projection, operator matrix, collecting data, object element model, tomography, electrical impedance. PACS: 02 INTRODUCTION Electrical Impedance Tomography (EIT) is an imaging technique which determines the electrical conductivity distribution within a medium using current source injection and voltage measurement from a series of electrodes on its surface. Compared to CT- Scan or MRI, EIT produces relatively poor image reconstruction. Despite its limited resolution, EIT has several advantages, namely portable, real time, noninvasive, low cost and can generate functional images [1][2]. The EIT problem is categorized as an ill-posed and non-linear inverse problem, so that the solution is very complex [3]. One of the fast and efficient reconstruction methods is linearization based on what is referred as back projection methods. The back projection algorithm was first proposed by Barber- Brown [4]. The algorithm was arranged based on the assumption of circular geometry using the adjacent collecting data methods. We proposed the forward problem solution from FEM methods as back projection operator to implement back projection algorithm to various object element model and collecting data. ELECTRICAL IMPEDANCE TOMOGRAPHY A. Forward Problem The forward problem of EIT is to estimate the potential distribution on the surface of a given object when the current density is injected with the known inner conductivity distribution. If there is no current source and the conductivity distribution are given, the potential distribution inside the object will be agree with the laplace equation, in (1) With boundary conditions of potential and current density on the surface, on (2)

2 on (3) where is the conductivity, is the potential distribution within the medium, 0 is the boundary potential and J 0 is the boundary current density and n denotes the normal unit vector pointed outward on the boundary, respectively. Equations (1),(2) and (3) can be solved by FEM methods, by dividing the object into small trianguler element and assuming that the electrical properties are homogeneous and isotropic. The FEM can yield a system of linear algebraic equation, (4) where Y is the admitance matrix, which denotes geometry and the conductivity distribution functions, U is the potential distribution vector and C is the current vector, respectively. The whole potentials can be obtain by manipulating the equation (4) tobe, (5) The potential boundary data of the model can be calculated as follows, (6) where T r denotes a transformation matrix. The equation (6) shows that the potensial boundary is the non linier function of conductivity. B. Invers Problem The invers problem calculates the conductivity distribution of object by measuring potential and current density on boundary. Several methods have been proposed and can generally be grouped into two kinds, those based on optimization dan linearization [2]. The optimizatoin methods can yield a static image that provides information about the absolut conductivity distribution. The success of the optimization methods are determined by the proper between the geometry model and forward problem which used toward the real geometry and the potential boundary data [5]. The method was time-consuming because it requires the iteration, but it will produce an accurate image reconstruction. Gauss-Newton and Newton Raphson are example of the optimization methods. They solve the invers problem by iterating from the non linear relationship between the conductivity and the potential data [6]. Methods based on linearization can yield images that provide information about relative image which reflects the change of physiological function [7]. Linear solution assumes that a small change of conductivity distribution will change linearly the boundary potential [5][8]. The reconstruction method used to solve the linear solution is known as back projection algorithm [6]. The algorithm is efficient and fast. It is developed firstly by Barber and Brown [4]. The algorithm is based on the adjacent collecting data and the equipotential lines in circular geometry. It is required two boundary potential data. There are reference data set V ref, measured from uniformly assumed conductivity distribution and the other data set V is measured after a change has occured in the conductivity distribution. Back projection algorithm based on equipotential lines was written in matrix as follows [4][9][10], (7) Where p is amount of the elements, q is amount of the electrodes, [B] is the weighted back projection operator, [F] is representation of the filter matrix, [ V n ] is the normalized change of the boundary voltage measurement, and [ n ] is the normalized variation of conductivity. PROPOSED METHODS Linearization method assumes that the changes of the boundary potential is the linear function of the conductivity change [5][8]. The relationship between these changes is given in vector form by (8) Where [ V] is the change of the boundary potential, [S] is the sensitivity matrices, and [ ] is the change of conductivity distribution. The sensitivity matrices [S] can be obtained by performing variation of on the whole elements in the equation (6) by the following steps, a. assume a known conductivity at each one of p elements as b. use solution forward problem in equation (6) to get the boundary potential c. change the conductivity of the i-th element into 0 + d. form the vector (px1) all elements are 0 but the i-th element contains 0 + e. repeat step (b) to get the boundary potential of i-th element f. repeat step (c) untill (e) of all p elements g. form a matrix which is equal to a matrix Matrix is equal to matrix because. Matrix [S] does not depend on 0 or, because 0 and will only result in the different weight to the matrix [S]. For simplicity, the study used 0 = 1 and = 1. The method can be used to the whole object model and the methods of collecting data if the forward problem solution is obtained. Once the matrix [S] is found, by algebraic manipulation, the equation (8) can be solved.

3 However, the matrix [S] is not square so [ ] can not be solved directly. (9) (10) Generally, [S] T [S] is singular matrix. Therefore, it has no inverse. Tikhonov regularitation can be used to solve the problem, so that the solution of the linearized regularization problem is given by, (11) where is the regularization parameter and I is the identity matrix. In equation (11), It appears that [S] T and ([S] T [S]+ I) -1 are identical to [B] and [F] in equation (7). The reconstruction is just an operation of matrix multiplication. The reconstruction of equation (11) can be solved quickly because the matrix [S] can be obtained in advance and it get only from memory. The research was conducted by simulating the numerical model of a circular object. The model was composed of 248 elements and 141 nodes with 16 electrodes. Circular objects represent 4 conditions, those are model A representing the single anomali in the centre of object, model B representing the single anomali in the side of object, model C representing the anomali of high frequency object and model D representing the anomali of low frequency object. The used methods of collecting data are neighboring-1 (adjacent), shift of neighboring-1, neighboring-2, neighboring-3, neighboring-4, neighboring-5, neighboring-6, neighboring-7, neighboring-8 (opposite), multirefference, combination of neighboring-1 until neighboring-8, and combination of neighboring-1 and shift of neighboring-1. Neighboring-1 means that the current electrode and ground electrode has a difference of one electrode, neighboring-2 means that between the current electrode and ground electrode has a difference of two electrodes and so on. Shift of neighboring -1 means that the position of neighboring-1 electrodes is moved to the position of the origin where there is no electrodes. The whole various methods of collecting data are compared and analyzed numerically by NRMSE (Normalized Root Mean Square Error). NRMSE calculates the similarity between two data, the smaller the NMRSE, the better the data, because two data are more identical. The NRMSE formulation can be expressed, (12) Where max is the maximum of both data and min is the minimum of both data. (13) Where n is amount of the data, A is the data and B is the reference data RESULT AND DISCUSSION FIGURE 1. The Element mesh model of circular geometry with 248 elements and 141 nodes on Neighboring and shift of neighboring collecting data FIGURE 2. The circular model objects from left to right are A (single anomali in the centre of object), model B (single anomali in the side of object), model C (anomali of high frequency object) and model D (anomali of low frequency object) FIGURE 3. From up to down, reconstruction image of Back Projection on adjacent, opposite and multireference collecting data Regularization parameter was chosen to get the reconstruction image from the filtered back projection are 1x10-7, 1x10-7, 5x10-3, dan 1x10-4 respectively.

4 FIGURE 6. From up to down, reconstruction image of the Filtered Back Projection from the combination of Neighboring-1 until Neighboring-8 collecting data and combination between Neighboring-1 and shift of Neighboring-1 collecting data TABLE I. NRMS of several methods of collecting data on circular objects Collecting NRMS (%) Methods A B C D Neighboring Shift of Neighboring-1 Neighboring Neighboring Neighboring Neighboring Neighboring Neighboring Neighboring Multireference Combination of Neighboring-1 until Neighboring-8 Combination of Neighboring-1 and Shift of Neighboring FIGURE 4. From up to down, reconstruction image of the Filtered Back Projection from Neighboring-1, shift of Neighboring-1, Neighboring-2, Neighboring-3, Neighboring- 4, Neighboring-5, Neighboring-6, Neighboring-7 dan Neighboring-8 collecting data FIGURE 5. The reconstruction image of Filtered Back Projection from multireference collecting data Figure 3 shows that the methods of back projection can result in blur image on several methods of collecting data, such as adjacent and opposite, but it can not result in image on the method of multirefference collecting data. By adding a filter, it can result in sharp image on several methods of collecting data, such as Neighboring-1 (adjacent), shift of Neighboring-1, Neighboring-2, Neighboring-3, Neighboring-4, Neighboring-5, Neighboring-6, Neighboring-7, Neighboring-8 (opposite), combination of Neighboring-1 until Neighboring-8, combination of Neighboring-1 and shift of Neighboring-1, but it can not result in good image on multirefference, as shown in figure 5. The method of adjacent and shift of Neighboring-1 collecting data is the best among the whole collecting data, as shown in figure 4. Combination of Neighboring-1 until Neighboring-8 result in only image as such as the adjacent collecting

5 method, as shown in figure 6. It reveals that eight methods of collecting data are not complementary. Combination of Neighboring-1 and shift of Neighboring-1 can result in the better image than Neighboring-1 or shift of Neighboring-1 alone, as shown in figure 6. It reveals that Neighboring-1 and shift Neighboring-1 are complementary. Table 1, shows numerically that the adjacent and combination between Neighboring-1 and shift of Neighboring-1 collecting data can result in the better image than others. CONCLUSION AND FUTURE WORK The operator of back projection and the filter can be obtained from the forward problem solution. The technique can solve several methods of collecting data e.g. adjacent, opposite, and combination fastly and accurately results, but, it can not solve the method of multireference collecting data. The Neighboring-1 and shift of Neighboring-1 collecting data are complementary such that they can enhance the quality of reconstruction image. Further research, the combination of Neighboring-1 and shift of Neighboring-1 data can be used to improve the image quality on a non-linear method. REFERENCES 1. Shuai Zhang, Guizhi Xu, Jianjun Zhang, Hongbin Wang, Duyan Geng, Xueqin Shen, Weili Yan, Fugui Liu, Xiaoguang Yang, Three Dimensional Detection and Imaging for Human Lung Based on Node Back- Projection Algorithm with a 64 Electrodes EIT System, Li Yaqin, A Novelty Dynamic Image Reconstruction Algorithm in Electrical Impedance Tomography Based on Nachman Theory, International Forum on Information Technology and Applications, Jan C. de Munck, Theo J.C. Faes and Rob M. Heethaar, The Use of the Boundary Element Method in the forward Problem of Electrical Impedance Tomography, Proceedings - 19th International Conference - IEEE/EMBS Oct Nov. 2, 1997 Chicago, IL. USA. 4. Santosa, F. and Vogelius, M., Backprojection algorithm for electrical impedance imaging, SIAM Journal on Applied Mathematics, vol.50, No. 1, pp , Andrea Borsic, Regularisation Methods for Imaging from Electrical Measurements, A thesis of Philosophy Doctor, Oxford Brookes University, R.H. Bayford, Bioimpedance Tomography (Electrical Impedance Tomography), Annual Review of Biomedical Engineering, Vol. 8, J. Zhang, R. P. Patterson, A. V. Koqenevsky, Comparison and Analysis of Electrical Impedance Tomographic Images Reconstructed Using Two Algorithms, Proceedings of the Second Joint EMBS/BMES Conference, X.Y. Chen, H.X. Wang, J. C. Newell, Lung Ventilation Reconstruction by Electrical Impedance Tomography Based on Physical Information, Third International Conference on Measuring Technology and Mechatronics Automation, IEEE Computer Society, Guizhi Xu, Shuai Zhang, Huanli Wu, Shuo Yang, Duyan Geng, Weili Yan and Mingshi Wang, The Acquisition Hardware System with Direct Digital Synthesis and Filtered Back-Projection Imaging in Electrical Impedance Tomography, Proceedings of the 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference, Shanghai, China, B.M. Eyuboglu, B.H. Brown and D.C. Barber, Limitations to SV Determination from APT Images, IEEE Engineering in Medicine and Biology Society 1 th Annual International Conference, 1989.