An experimental study of induced-current electrical impedance tomography verifies that image quality is enhanced by employing six rather than three induction coils by increasing the number of independent measurements. However, with an increasing number of coils, the inverse problem becomes more sensitive to measurement noise. Using 16 electrodes to measure surface voltages, it is possible to collect 6/spl times/15=90 independent measurements. For comparison purposes, images of two-dimensional conductivity perturbations are reconstructed by using the data for three and six coils with the truncated pseudoinverse algorithm. By searching for the optimal truncation index that minimizes the noise error plus the resolution error, the signal-to-noise ratio of the data acquisition system was established as 58 db. Images obtained with this six-coil system reveal the sizes and locations of the conductivity perturbations. This system also provides images within the central region of the object space, a capability not achieved in previous experimental studies using only three circular coils. Nevertheless, the three-coil system can identify the conductivity perturbations near the periphery. However, it displays shifts in the locations and spread in the sizes of perturbations near the center of the object.

Electrical impedance tomography: induced-current imaging achieved with a multiple coil system

A new imaging modality is introduced to image electrical conductivity of biological tissues via contactless measurements. This modality uses magnetic excitation to induce currents inside the body and measures the magnetic fields of the induced currents. In this study, the mathematical basis of the methodology is analyzed and numerical models are developed to simulate the imaging system. The induced currents are expressed using the A/spl I.oarr/-/spl phi/ formulation of the electric field where A/spl I.oarr/ is the magnetic vector potential and /spl phi/ is the scalar potential function. It is assumed that A/spl I.oarr/ describes the primary magnetic vector potential that exists in the absence of the body. This assumption considerably simplifies the solution of the secondary magnetic fields caused by induced currents. In order to solve /spl phi/ for objects of arbitrary conductivity distribution a three-dimensional (3-D) finite-element method (FEM) formulation is employed. A specific 7/spl times/7-coil system is assumed nearby the upper surface of a 10/spl times/10/spl times/5-cm conductive body. A sensitivity matrix, which relates the perturbation in measurements to the conductivity perturbations, is calculated. Singular-value decomposition of the sensitivity matrix shows various characteristics of the imaging system. Images are reconstructed using 500 voxels in the image domain, with truncated pseudoinverse. The noise level is assumed to produce a representative signal-to-noise ratio (SNR) of 80 dB. It is observed that it is possible to identify voxel perturbations (of volume 1 cm/sup 3/) at 2 cm depth. However, resolution gradually decreases for deeper conductivity perturbations.

Electrical conductivity imaging via contactless measurements

A method for obtaining a practical inverse for the distribution of neural activity in the human cerebral cortex is developed for electric, magnetic, and bimodal data to exploit their complementary aspects. Intracellular current is represented by current dipoles uniformly distributed on two parallel sulci joined by a gyrus. Linear systems of equations relate electric, magnetic, and bimodal data to unknown dipole moments. The corresponding lead-field matrices are characterized by singular value decomposition (SVD). The optimal reference electrode location for electric data is chosen on the basis of the decay behavior of the singular values. The singular values of these matrices show better decay behavior with increasing number of measurements, however, that property is useful depending on the noise in the measurements. The truncated SVD pseudo-inverse is used to control noise artifacts in the reconstructed images. Simulations for single-dipole sources at different depths reveal the relative contributions of electric and magnetic measures. For realistic noise levels the performance of both unimodal and bimodal systems do not improve with an increase in the number of measurements beyond /spl sim/100. Bimodal image reconstructions are generally superior to unimodal ones in finding the center of activity.

Differential characterization of neural sources with the bimodal truncated SVD pseudo-inverse for EEG and MEG measurements

We introduce a theoretical lower error bound for solutions to magnetostatic linear inverse problems and we propose it as a figure of merit for the comparative evaluation of sensor arrays. With the help of the proposed error bound, we demonstrate the superiority of three-axial biomagnetic sensor arrays applying truncated singular value decomposition analysis to a kernel matrix computed from boundary-element-method (BEM) models of the human torso for a biomagnetic application. In simulations, we found that, for a more complex five-compartment BEM model, the advantage of using three-axial measurements is more pronounced, compared to a three-compartment BEM model

Theoretical Lower Error Bound for Comparative Evaluation of Sensor Arrays in Magnetostatic Linear Inverse Problems

In this paper, a constrained signal extrapolation algorithm is proposed and discussed for the image reconstruction from limited spectral informations. The proposed algorithm is imposed by the additional constraints based on a priori knowledge of original image in order to stabilize the extrapolation process. Improvement is obtained over the conventional extrapolation method in the presence of noise and other errors. Experimental results showing image reconstruction from the limited frequency spectrum due to incomplete observation are presented.

An image reconstruction from incomplete observation by constrained spectrum extrapolation

The problem of extrapolating a band-limited signal in discrete-time is viewed as one of solving an underdetermined system of linear equations. The matrix to be inverted is also generally ill-conditioned. Singular value decomposition (SVD) provides both a means for implementing the inverse and a method for improving the numerical stability of the problem. An expression for the mean-square error incurred in solving a system of linear equations via SVD is derived. This expression can be used to estimate the number of singular values needed to form the inverse. Further examination of the expression indicates that, in the case of an oversampled signal, decimation can be applied without significantly degrading the extrapolation. The results developed for the one-dimensional case can be extended to higher dimensions, as illustrated by the two-dimensional case. Examples of the SVD approach to extrapolation are given, along with examples using other extrapolation techniques for comparison. The SVD approach is seen to yield the best extrapolation of the known minimum-norm least-squares methods.

B. Sullivan

Papers

Extrapolation of discrete-time band-limited signals using singular value decomposition with decimation