A Kalman Filter Approach to Track Fast Impedance ... - IEEE Xplore

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 4, APRIL 1998

A Kalman Filter Approach to Track Fast Impedance Changes in Electrical Impedance Tomography Marko Vauhkonen, Pasi A. Karjalainen, Student Member, IEEE, and Jari P. Kaipio,* Associate Member, IEEE

Abstract—In electrical impedance tomography (EIT), an estimate for the cross-sectional impedance distribution is obtained from the body by using current and voltage measurements made from the boundary. All well-known reconstruction algorithms use a full set of independent current patterns for each reconstruction. In some applications, the impedance changes may be so fast that information on the time evolution of the impedance distribution is either lost or severely blurred. In this paper, we propose an algorithm for EIT reconstruction that is able to track fast changes in the impedance distribution. The method is based on the formulation of EIT as a state-estimation problem and the recursive estimation of the state with the aid of the Kalman filter. The performance of the proposed method is evaluated with a simulation of human thorax in a situation in which the impedances of the ventricles change rapidly. We show that with optimal current patterns and proper parameterization, the proposed approach yields significant enhancement of the temporal resolution over the conventional reconstruction strategy. Index Terms— Electrical impedance tomography, Kalman filter, time-varying imaging.

I. INTRODUCTION

I

N electrical impedance tomography (EIT) current patterns are applied to electrodes on the surface of the body and the resulting voltages are measured. An approximation to the resistivity (impedance) distribution inside the body is then made, based on these surface measurements [1], [2]. The spatial resolution of EIT is not comparable to other imaging techniques such as magnetic resonance imaging (MRI) or computed tomography (CT). For this reason, EIT is seldom used in anatomical imaging. The advantage of EIT over these techniques, however, is good temporal resolution, which makes it possible to track relatively fast impedance changes in the human body. One of the most interesting targets is the impedance changes caused by cardiac activity [3], [4]. In order to achieve high temporal resolution, an EIT system with fast data collection is required. The EIT system is usually considered fast if it is capable of measuring data so that at least 25 independent frames (images) are obtained each second. Manuscript received December 17, 1996; revised October 16, 1997. Asterisk indicates corresponding author. M. Vauhkonen is with the Department of Applied Physics, University of Kuopio, FIN-70211 Kuopio, Finland. He is also with Department of Mathematical Sciences, University of Oulu, Linnanmaa, FIN-90571 Oulu, Finland. P. A. Karjalainen is with the Department of Applied Physics, University of Kuopio, P.O. Box 1627, FIN-70211 Kuopio, Finland. *J. P. Kaipio is with the Department of Applied Physics, University of Kuopio, P.O. Box 1627, FIN-70211 Kuopio, Finland (e-mail: [email protected]). Publisher Item Identifier S 0018-9294(98)01799-6.

This is usually enough for the investigation of impedance changes caused by normal cardiac rhythms with the rate of 60 beats/min. There is, however, a need for even faster frame rates, for example, when impedance changes during sport exercise are tracked. In such cases, the heart rate might be near 200 beats/min [5]. One approach to overcome the speed problem is to use shorter time windows in the measurement of voltages. This will, however, increase the noise level of the measurements, which will in turn cause increased errors in each individual reconstruction. Another problem in this approach is that although we assume that the impedances in each pixel inside the volume change relatively fast, the impedance distributions are usually not statistically independent. It would then not be appropriate to consider each reconstruction separately, since if the successive reconstructions are carried out independently, we lose time-sequential information. So far, all the proposed EIT reconstruction algorithms use a full set of independent measurement data (current patterns) in order to obtain an image [2], [6], [7]. For example, with backprojection algorithm with 16 electrodes, 16 different voltage measurements are needed current injections, and for each image. Correspondingly, with trigonometric current patterns with 32 electrodes, 31 current patterns are used for image reconstruction [7]. The dynamical (with the notion of “dynamical” we refer to the interpretation of the impedance distribution as a time-varying quantity whether the reconstruction is static or difference) reconstruction algorithm proposed in this article is based on the state-space representation of the dynamical EIT. , The state of the system, i.e., the impedance distribution is re-estimated after the voltage measurements corresponding to each current pattern. We can thus obtain reconstructions 31 times faster than with the conventional methods (when 31 current patterns with 32 electrodes are used). In addition, the model for the state evolution can be tailored to take into account differences in the dynamical properties and relations of impedance changes between different organs. We evaluate the performance of the proposed algorithm by simulations. We consider a situation related to sport exercise in which the heart rate is as high as 180 beats/min. More precisely, we consider the estimation of the average impedances of the ventricles as a function of time. We assume that the measurements are carried out with the measurement system that acquires data with a rate of 8 frames/s, which would mean less than three reconstructions per cardiac cycle

0018–9294/98$10.00  1998 IEEE

VAUHKONEN et al.: KALMAN-FILTER APPROACH TO TRACK FAST IMPEDANCE CHANGES IN EIT

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with conventional reconstruction methods. With the proposed method, it is possible to obtain 93 reconstructions per cardiac cycle since an estimate for the state is calculated after each current injection. The optimal current pattern selection is also discussed and a method is proposed with which this can be used in dynamical EIT. II. METHODS (a)

A. The Formulation of Dynamical EIT as a State-Estimation Problem Although the formulation could be done in terms of the nonlinear mapping, we consider here only the linearized problem. be the vector Consider first, the time-invariant case. Let containing the voltage measurements corresponding to all current patterns. The linearization of the mapping at is (1) is the impedance distribution that is set to where constant in each finite element method (FEM) discretization is also obtained from element (see Fig. 1). The Jacobian the FEM discretization of the associated partial differential equations; see Section II-B. We will denote the Jacobian . With current patterns we would then have corresponding to the classical frame. We can write (1) in the form .. .

.. .

.. .

(2)

and for all . In (2) the th where block corresponds to the current pattern . In a dynamical situation, the measurements do not correspond to the same impedance distribution since the distribution changes over time during the measurement cycle. We assume, however, that the evolution of the impedance is so slow that the measurements that correspond to a single current pattern can be taken to be approximately from the same distribution. This is a qualitative requirement, in exact terms this would mean that we have an EIT system that is capable of parallel voltage measurements [8], [9]. Let the current pattern at time be so that it is one of the patterns . The . Further, let corresponding measurements are denoted by indicate which of the patterns is used at time . The observation equation at time is then (3) is the impedance distribution at time and where are the measurement errors, whose covariance is denoted by . We describe the evolution of with the discrete-time model (state equation) (4)

(b)

Fig. 1. (a) A finite element grid with 1984 elements and 1025 nodes (dotted lines). These elements are grouped in order to obtain 496 rectangular elements (solid lines) for the inverse calculations. (b) The ROI’s corresponding to the ventricles, lungs, and other tissue.

where

is the state noise process and is the state transition matrix. In particular, we take (the unit matrix) to obtain the so-called randomwalk model. The rate of evolution in this model is governed of . If is by the covariance diagonal, the increments of the impedance evolution of each element are uncorrelated. Equations (3) and (4) constitute a so-called state-space representation of the linearized EIT system. The state-space representation is standard in control and system theory. The based mathematical problem is now to estimate the state on the observations at each time . The recursive estimator of the state is called the Kalman for the state filter. The first step is to obtain an estimate that is based on the observations . The solution to this problem is called the Kalman predictor and it can be expressed in the form (with ) (5) (6) (7) (8) (9) is prediction error, is an estimate for the where covariance of the state estimation error and denotes the (pseudo)inverse. The Kalman filter is completed by adding to the predicted the update based on the observation state (10) In addition to the technical assumptions [10], we have also assumed that the processes and are mutually orthogonal. We will also make the assumption that the state and observation process covariances are diagonal and time invariant with equal variances, that is and . A new reconstruction is, thus, obtained after each observia (5)–(10). We do not review the recursive vation of state estimation problem or the associated interpretations and optimality properties here, since there is a vast literature on

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the subject on more or less advanced level. We refer here only to [10] and [11]. B. The Complete Electrode Model and FEM Discretization We consider here only the estimation and tracking of absolute resistivity distributions. The difference versions (reconstructions of changes in the resistivity) are simple modifications of the presented results. These resistivity changes can be estimated by replacing the where observation equation with the data is the difference between two voltage meaand . surements corresponding to the resistivities In the difference imaging case, the prediction error in the Kalman filter equations becomes then and the difference resistivity estimate at time becomes , while the other equations remain as before. Since we use a 32 electrode model in which the electrodes cover 50% of the boundary, we use the complete electrode model and its FEM approximation in both the generation of the simulation data and the construction of the Jacobian of [12], [13]. For the simulations we have the mapping used a dense grid with 1984 elements and for the Jacobian we have grouped these elements in order to obtain either 496 elements or five elements, see Fig. 1 and Section IV-B for the integrated state variables. The equations of complete electrode model are in

A conventional method for solving the EIT inverse problem is given in [7]. The method consists of performing one step of the regularized Gauss–Newton iteration of the associated nonlinear least squares problem , where are the measured voltages and are the voltages given by the FE approximation with a given distribution . The regularization is a Tikhonov-type method with nonequal diagonal entries of the regularization matrix. The method can be expressed in the form (16) and is the regularization where is calculated at parameter. In this method a full set of current patterns and voltage measurements are used in the reconstruction of each distribution. D. Parameterization of the State The impedances of the FEM grid can directly be used as the state parameters that are to be estimated. In addition to this, the problem can be stabilized by reducing the dimension of the state estimation problem; see Section IV-B. This can be accomplished by preintegrating FEM elements to produce a smaller amount of parameters. By preintegration we mean that some of the FEM elements are grouped together for the reconstruction. Mathematically, the preintegration can be written in the form (17)

(11) on

(12) (13)

on

C. Conventional Method That Is Used as a Reference

(14)

is electric potential, where is conductivity (assumed to be real), is th electrode, is effective contact impedance between th electrode and tissue (also real), are the measured potentials, are the injected currents and is outward unit normal. Actually the injected currents (and potentials) are time-harmonic functions and are the amplitudes. and After the finite element discretization, the linear matrix equation (15) is the so-called stiffness matrix, is obtained, where contains the injected currents and is the vector of voltages at the FEM nodes and the electrodes. The variational formulation of (11)–(14) that is needed for the construction of and was first presented in [13]. A detailed description of the implementation of the FEM for solving the complete electrode model and the Jacobian can be found in [14].

is the measure (area) of the th FEM element, where is a preintegrated parameter, , and is a parameter vector corresponding to the FEM elements. are called regions of interest The preintegrated domains (ROI’s). In the reconstructions we used either 496 parameters or five parameters; see Fig. 1. The five ROI’s represent the left and right ventricles, the left and right lungs, and the background. In the final results we have shown only the evolutions of mean impedances in these areas. III. THE SIMULATIONS In order to test the dynamical algorithm presented, we simulated the cardiac cycle of human heart. The simulations correspond to a situation in which breath is held so that the volumes of the ventricles and the lungs were assumed to be fixed during the simulation, but the blood volume and, thus, the average impedance of the lungs and the ventricles were made to change slightly. The heart rate was approximately 180 beats/min. The complete electrode model and trigonometric current patterns with 32 electrodes were employed in all the simulations. The trigonometric current patterns are of the form

(18)


Fig. 2. Simulated impedance changes in the heart (two lowest curves; the bold line corresponds to the left ventricle) and in the lungs (two upper curves; the bold line corresponds to the left lung). The impedance of the background 4). is set to constant (

=

where . The contact impedances between the electrodes and the tissue were assumed to be known in the reconstructions. The dynamical algorithm was compared with a conventional method presented in the previous Section. The cardiac cycle was simulated as follows. We changed the impedance of the five ROI’s that are shown in Fig. 1(b) as a function of time and calculated the voltages at each electrode corresponding to these time-varying impedance distributions. The voltage measurements that correspond to a single current pattern were assumed to have been measured simultaneously. However, the impedance distribution changed between the current patterns. These time evolutions of the (averaged) impedances of the different regions (ROI’s) are shown in Fig. 2. The impedance changes in the lungs and in the heart are due to the changes in the blood volume during cardiac cycle. The speed of the measurement system was assumed to be 8 (classical) frames/s, which corresponds to 248 current patterns and eight impedance estimates per second with the conventional approach in which we form an estimate only when the voltages corresponding to all independent current patterns are observed. With the dynamical method we obtain 248 estimates for the impedance distribution.

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at a given time. On the other hand, if we “increase” the covariance, the lag error diminishes but the estimates will be more noisy. The tracking properties of Kalman filter are analyzed for example in [15]. Since the covariance is diagonal, we assume that the increments of the state parameters are mutually orthogonal. In reality, this is not the case, since, for example, the increments of the impedances of the two lungs are clearly correlated. Furthermore, it is clear that we should set the variances corresponding to the background (other muscular tissue) to a smaller value than that of the ventricles. We do not, however, use any ad hoc choice for the state increment covariance (other than the diagonal matrix), since the covariance should be determined with some estimation algorithm that is appropriate for the state-space models, such as the expectation maximization algorithm, that is out of the scope of this paper. 2) The covariance of the observation noise process that corresponds partly to the additive noise in the measurements, was also set to a time-invariant value of . If the additive noise would be the only factor affecting this covariance, the diagonal choice would be justified. However, this term also inherits the approximation errors that correspond to the linearization of the problem. The contribution of the these approximation is clearly not diagonal and the structure is errors to difficult to estimate. However, if the impedance changes are sufficiently small, the observation noise can be assumed to prevail. 3) The initial covariance of the estimation error of the (predicted) state was set to . The effects of this choice die out relatively fast and this choice is usually not crucial. In cases of very short data span, this covariance can be estimated iteratively by successive forward and backward runs of the data. 4) The linearization of the mapping was done at the approximate time averages of the parameters. In this way the mean (with respect to time) approximation errors can be made small. The impedance values were 4 m for the background, 1.78 m for the left heart, 1.85 m for the right heart, 10.35 m for the left lung, and 10.44 m for the right lung; see Fig. 1(b).

IV. RESULTS B. State Parameter Selection A. Choices of Kalman Filter Parameters The parameters of the Kalman filter that were used in the calculations were as follows. that 1) The covariance of the state increment process is related to the speed with which we assume that the impedance values of each state parameter (FEM element or ROI) changes, was set to the time-invariant value of . The qualitative picture is that the smaller the elements of are, the slower the estimates change and the greater the so-called lag error of the algorithm is. By the lag error we mean the error of the true and the (probabilistic) mean of the estimated state

As described above, we can, for example, use a relatively sparse FEM grid or a preintegrated version of this (ROI’s). The ROI version is more stable than the generic FEM grid parameterization since the dimension of the state is much smaller. The stability problems are due to the inversion of a stochastic positive semidefinite matrix in (8) [11]. With a large state dimension, the inversion tends to be unstable. However, we can also use the FEM grid version of the algorithm and post-integrate over the ROI’s. In the post-integration case, we use all the FEM elements as parameters and compute the average of the ROI’s [ in (17)] after the reconstruction. In this case, the computational load of the algorithm can be

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very big, whereas in the preintegration case the complexity is relatively small and real-time implementation is easy to arrange. We mention also that there are two versions of the Kalman filter 1) the covariance and 2) information filters. Which one of these should be used depends on various topics, in particular the dimensions of the state and observation vectors [11]. The second choice is concerned with the choice of current patterns. In conventional EIT we normally use the data corresponding to all independent current patterns. In the proposed approach we do not have to do this. The structure of the Kalman filter is such that the state estimated are formed so that is a sum of the previous estimate the estimate and a “correction” term that is the product of the so-called and the estimation error . Kalman gain matrix In the case of dynamical EIT, the Kalman gain matrix depends on the current pattern that is used at this time. It turns out that the norm of the Kalman gain corresponding to certain current patterns is very small. This means that when these current patterns are used, the state estimate update can be very small even if the true change in the state variables was great. Also, the state update equation can be interpreted and depend on the current as follows: The matrices patterns and for some current patterns the product can have a very small matrix norm. This means that the state update is insignificant for any error vector regardless of the actual changes in the impedance distribution. We could, thus, in principle use just one “meaningful” current pattern each time and the estimates would be driven to the correct direction in the average. How meaningful a current pattern is, depends on the true impedance distribution. The question of optimal current patterns is addressed in Section IV-F. C. The Results Using all Independent Current Patterns We study first the case in which all current patterns are used. The results of the simulations with trigonometric current patterns are shown in Figs. 3 (left ventricle) and 4 (right lung) when the FEM grid choice with 496 parameters (grouped FEM elements) was made; see also, Fig. 1(a). The state evolutions corresponding to the proposed method are post-integrated. The parts of the Kalman filter reconstruction curves in Figs. 3 and 4 with negligible state update correspond to the “meaningless” current patterns. It can be seen that the results with the proposed approach are not better than with the conventional method. The corresponding results when the five ROI’s were used as the state parameters, are shown in Figs. 5 (left ventricle) and 6 (right lung); see also Fig. 1(b). The ROI version of the algorithm is clearly more stable and tracks the true evolution with good precision. This applies to the tracking of both ventricle and lung evolutions. It is, thus, clearly advantageous to use preintegration of ROI’s instead of post-integration. The results for the right ventricle and left lung are similar and are not shown. The phase lag of the conventional method in Figs. 3 and 5 are a result of the initial distribution and the phase of the

Fig. 3. Results from the left ventricle with 496 FEM element parameters and 31 current patterns. Simulated impedance (bold line), conventional method with linear interpolation (medium line), and the proposed method (weak line).

Fig. 4. Results from the right lung with 496 FEM element parameters and 31 current patterns. Simulated impedance (bold line), conventional method with linear interpolation (medium line), and the proposed method (weak line).

Fig. 5. Results from the left ventricle with five ROI parameters and 31 current patterns. Simulated impedance (bold line), conventional method with linear interpolation (medium line), and the proposed method (weak line).

“organ evolution” at the initialization. Note that there is no phase lag in Figs. 4 and 6.


Fig. 6. Results from the right lung with five ROI parameters and 31 current patterns. Simulated impedance (bold line), conventional method with linear interpolation (medium line), and the proposed method (weak line).

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Fig. 8. Results from the left ventricle with 496 FEM element parameters and p 5 optimal current patterns. Simulated impedance (bold line), the proposed method with 31 current patterns (medium line), and with five optimal patterns (weak line).

=

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 7. Three current patterns that are sensitive to impedance changes in the left ventricle. The boldest line corresponds to the current pattern giving the maximal sensitivity.

D. Spatial Resolution and Effect of the Observation Noise We computed the solutions for the inverse problem with different noise levels. These noise levels were 0.5%, 1.5%, and 3% of the maximum of the measured voltage values. For each noise level we also computed the normalized root mean square errors of the reconstructed images compared to the true resistivity distribution. As a result, it can be said that with the Kalman filter approach with the given choice of the parameters the reconstruction error accumulated after 1.5% of noise, whereas with the conventional method these noise levels had only small effect on the reconstructed images. However, when we increased the value of the observation noise variances , we obtained stable reconstructions also in the covariance with the Kalman filter approach. In order to obtain a better view of the quality of the reconstructed images, we computed the reconstructions starting from the uniform resistivity value of 5.8 (average of the time average). Noise of 1.5% was added to the simulated voltages . The evolution of the Kalman and the covariance filter reconstructions until 93 current patterns (three frames

Fig. 9. Results from the reconstructions with the Kalman filter approach and the conventional approach when 1.5% maximum noise is added to the simulated voltages. (a) 1, (b) 17, (c) 32, (d) 63, (e) 93 reconstructions with the Kalman filter approach, and (f) the third reconstruction (current patterns 63–93 were used for the reconstruction) with the conventional method.

when conventional method is used) is shown in Fig. 9. As a comparison, the conventional solution reconstructed using the current patterns 63–93 is also shown in Fig. 9 (the last image

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in the second column). The convergence of the Kalman filter approach is clearly visible in Fig. 9. Until the convergence, the spatial resolution is seen to be only moderately worse than the spatial resolution of the conventional method. E. Effect of the State Noise Covariance As explained earlier, we can modify the “speed” of the . reconstruction by modifying the state noise covariance The effect of this modification is shown in Fig. 10. Fig. 10 shows the results of the reconstructions with different values from the simulated data with 1.5% noise added. The of were [Fig. 10(a)] and [Fig. 10(b)]. covariances The effect of different choices for is clearly visible in Fig. 10. Bigger variances result in more noisy state evolutions with small lag error and smaller variances result in smoother state evolutions with bigger lag error.

(a)

F. Optimal Current Patterns Although the preintegration of the elements give good estimates for resistivities in the preselected areas (ROI’s), sometimes we do not want to use prior information about the structure of the object. In these cases, we may want to keep all FEM elements as state parameters. The problem involved with the nonmeaningful current patterns can then be reduced by using so-called optimal current patterns. The meaningfulness of the current patterns is clearly visible , in which case the for example in Fig. 3, algorithm does not considerably change the state estimates, although the estimation error has the same sign during this whole period. On the other hand, at certain instants there are clear jumps to the correct direction. These jumps occur at times when the current pattern is with small. Thus, the state estimates are most sensitive to measurements corresponding to current patterns with small spatial differences in the current values. We could, thus, select a few sinusoidal current patterns with small and use these only. However, it is possible to select a ROI (an organ) and determine the current patterns so that the sensitivity of the measurements with respect to changes in this ROI is maximized by using these current patterns. The selection is based on the definition of distinguishability and was first presented in [16] and further developed in [17] and [18]. Here we have used a modification of the principle. The selection is based on the maximization of the sensitivity in some region, that is, we want to find a current pattern that is the solution to (19) is the impedance of the region (ROI). The solution where can be found by noting that

(b) Fig. 10. Effect of the state noise covariance

0 v = 2 :8 I .

0v . (a) 0v = 0:008I

and (b)

In Fig. 7 the three eigenvectors corresponding to the three that maximizes the sensitivity in the largest eigenvalues of left ventricle. The greatest currents and their spatial changes occur near electrodes 3–6, which are situated nearest to the left ventricle. We select optimal current patterns and simulate the measurements so that only these current patterns are used successively. This means that these five current patterns were rotated such that after the fifth current pattern we started again from the first one and so on. The time evolution of the impedance distribution was as in the case when the sinusoidal current patterns were used. The results with reduced amount of current patterns with 496 estimated parameters are shown in Fig. 8. As can be seen, there is a slight improvement in the results and the estimate is not so stepwise anymore. It must be noted, however, that if we maximize the sensitivity in a specified ROI, we are very likely to obtain reduced sensitivity in other ROI’s. In the case of the particular situation (thorax), we obtain reasonably good tracking properties for all ROI’s when the sinusoidal current patterns with small are used.

(20) V. DISCUSSION so that the solution for (19) is the eigenvector that corresponds to the largest eigenvalue of [16]. We will, thus, select the eigenvectors corresponding to the largest eigenvalues of .

In dynamical EIT applications it is sometimes necessary to track fast impedance changes. Adequate tracking can usually be obtained with fast and accurate measurement system design.


However, there are situations, for example in sports medicine, in which conventional reconstruction strategies fall short of the tracking requirements. In this paper we have proposed a method that makes it possible to obtain reconstructions that are capable of faster tracking of impedance changes than conventional strategies. The method is based on the formulation of the EIT inverse problem as a state estimation problem and the recursive solution to this problem with the Kalman filter. The method was compared to a conventional EIT reconstruction method in which the impedance distribution is assumed to be time-invariant within the measurement cycle. There are many possible extensions for the state-space model presented here. For example, the random-walk model (state evolution model) can be argued not to be a proper model. A more appropriate, but somewhat more complicated, model is, e.g., a modified (time-varying) compartmental model. With this extension to the proposed method, we could take into account the dynamics of the blood circulation between the organs [19]. Another possibility is to use higher-order state evolution models (hypermodels); see [20]. The state evolutions corresponding to these models are solved with so-called multistep algorithms instead of (their special case) the Kalman filter. With the higher-order hypermodels, we can obtain state evolutions that can be very fast, but smooth with respect to higher-order time differences of the estimates. Further extensions include the extended Kalman filter (the nonlinearity extension) and nontrivial covariance models for state noise. When real-time estimation is not necessary, which is usually the case, the employment of the Kalman smoother will usually yield better estimates than the Kalman filter. The implementation of the Kalman smoother is, however, considerably more tedious; see [21], [11] for Kalman smoothers based on different data sets and [22] and [23] for stable implementations.

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[11] B. D. O. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. [12] K.-S. Cheng, D. Isaacson, J. C. Newell, and D. G. Gisser, “Electrode models for electric current computed tomography,” IEEE Trans. Biomed. Eng., vol. 36, pp. 918–924, 1989. [13] E. Somersalo, M. Cheney, and D. Isaacson, “Existence and uniqueness for electrode models for electric current computed tomography,” SIAM J. Appl. Math., vol. 52, pp. 1023–1040, 1992. [14] M. Vauhkonen, D. Vadász, J. P. Kaipio, and P. A. Karjalainen, “Tikhonov regularization and prior information in electrical impedance tomography,” Univ. Kuopio, Dept. Applied Physics, Tech. Rep. 3/96, 1996. [15] H.-F. Chen and L. Guo, Identification and Stochastic Adaptive Control. Boston, MA: Birkhäuser, 1991. [16] D. Isaacson, “Distinguishability of conductivities by electric current computed tomography,” IEEE Trans. Med. Imag., vol. MI-5, pp. 91–95, 1986. [17] P. Hua, E. J. Woo, J. G. Webster, and W. J. Tompkins, “Improved methods to determine optimal currents in electrical impedance tomography,” IEEE Trans. Med. Imag., vol. 11, pp. 488–495, 1992. [18] M. Cheney and D. Isaacson, “Distinguishability in impedance imaging,” IEEE Trans. Biomed. Eng., vol. 39, pp. 852–860, 1992. [19] J. Strackee and N. Westerhof, The Physics of Heart and Circulation. Bristol: Inst. Physics, 1993. [20] A. Benveniste, M. Metivier, and P. Priouret, Adaptive Algorithms and Stochastic Approximations. New York: Springer-Verlag, 1990. [21] C. K. Chui and G. Chen, Kalman Filtering. New York: SpringerVerlag, 1987. [22] P. G. Park and T. Kailath, “Square-root Bryson-Frazier smoothing algorithms,” IEEE Trans. Automat. Contr., vol. 40, pp. 761–766, 1995. [23] P. G. Park and T. Kailath, “New square-root smoothing algorithms,” IEEE Trans. Automat. Contr., vol. 41, pp. 727–733, 1996.

Marko Vauhkonen received the Ph.D. degree in 1997 from the University of Kuopio, Kuopio, Finland. He is currently a Researcher in the Department of Applied Physics, University of Kuopio. His research interests are in the reconstruction and medical applications of impedance imaging, including regularization techniques and finite element computations.

REFERENCES [1] A. P. Calderon, “On an inverse boundary value problem,” in Proc. Brazilian Math. Society, Seminar on Numerical Analysis and Its Applications to Continuum Physics, W. H. Meyer and M. A. Raupp, Eds., Rio de Janeiro, 1980, pp. 65–73. [2] D. C. Barber and B. H. Brown, “Applied potential tomography,” J. Phys. E: Sci. Instrum., vol. 17, pp. 723–733, 1984. [3] B. H. Brown, D. C. Barber, and A. D. Seagar, “Applied potential tomography: Possible clinical applications,” Clin. Phys. Physiol. Meas., vol. 6, pp. 109–121, 1985. [4] B. M. Eyüboglu, B. H. Brown, and D. C. Barber, “In vivo imaging of cardiac related impedance changes,” IEEE Eng. Med. Biol. Mag., vol. 8, pp. 39–45, 1989. [5] J. H. Wilmore and D. L. Costill, Physiology of Sport and Exercise. Champaign, IL: Human Kinetics, 1994. [6] T. J. Yorkey, J. G. Webster, and W. J. Tompkins, “Comparing reconstruction algorithms for electrical impedance tomography,” IEEE Trans. Biomed. Eng., vol. BME-34, pp. 843–852, 1987. [7] M. Cheney, D. Isaacson, J. C. Newell, S. Simske, and J. Goble, “NOSER: An algorithm for solving the inverse conductivity problem,” Int. J. Imag. Syst., Technol., vol. 2, pp. 66–75, 1990. [8] A. D. Seagar and B. H. Brown, “Limitations and in hardware design in impedance imaging,” Clin. Phys. Physiol. Meas., Suppl. A, vol. 8, pp. 85–90, 1987. [9] J. G. Webster, Electrical Impedance Tomography. London, U.K.: Adam Hilger, 1990. [10] P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods. New York: Springer-Verlag, 1991.

Pasi A. Karjalainen (S’88) received the Ph.D. degree in 1997 from the University of Kuopio, Kuopio, Finland. Since 1988 he has been working in University of Kuopio as Assistant and in Kuopio University Hospital as Physicist. He is currently an Associate Professor in the Department of Applied Physics. His research areas include Bayesian estimation and regularization theory and their applications to biomedical inverse problems.

Jari P. Kaipio (S’84–A’96) received the Ph.D. degree from the University of Kuopio, Kuopio, Finland. Since 1987, he has held various teaching positions in the Department of Applied Physics and the Department of Computer Science and Applied Mathematics, University of Kuopio. He is currently a Professor in the Department of Applied Physics. His research and teaching are concerned with the regularization theory of inverse problems and time series analysis. The applications of this research are focused on problems in medical physics, biophysics, and biomedical engineering.