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A new multidipole estimation method which gives a sparse solution of the biomagnetic inverse problem is proposed. This solution is extracted from the basic feasible solutions of linearly independent data equations. These feasible solutions are obtained by selecting exactly as many dipole-moments as the number of magnetic sensors. By changing the selection, we search for the minimum-norm vector of selected moments. As a result, a practically sparse solution is obtained; computer-simulated solutions for Lp-norm (p = 2, 1, 0.5, 0.2) have a small number of significant moments around the real source-dipoles. In particular, the solution for L1-norm is equivalent to the minimum-L1-norm solution of the original inverse problem. This solution can be uniquely computed by using Linear Programming.
A new multidipole estimation method which gives a sparse solution of the biomagnetic inverse problem is proposed. This solution is extracted from the basic feasible solutions of linearly independent data equations. These feasible solutions are obtained by selecting exactly as many dipole-moments as the number of magnetic sensors. By changing the selection, we search for the minimum-norm vector of selected moments. As a result, a practically sparse solution is obtained; computer-simulated solutions for Lp-norm (p = 2, 1, 0.5, 0.2) have a small number of significant moments around the real source-dipoles. In particular, the solution for L1-norm is equivalent to the minimum-L1-norm solution of the original inverse problem. This solution can be uniquely computed by using Linear Programming.
Inequalty constraints are introduced to a normalized minimum-L1-norm estimator, which gives a sparse solution of the biomagnetic inverse problem. The constraints have a numeric tolerance to take into account the measurement ambiguity caused by noise. Computer simulation and phantom-data analysis show how the solution is improved by the constraints with a moderate tolerance; the improvement is examined in noisy conditions such that signal-to-noise ratios (SNR's) are lower than 10 dB.
~ IntmdllctlonWe repolt here on a method of magnetic source imaning based on a method without a priori parametric source model. The method we propose uses curretlt element distribution [I]. Thd algorithm takes a group of current elements to represent any kind of source current patterns. In previous papers [I] [Z], we have discussed the method with computer simulated d f a k Simulated magnetic sources were detected by the distribution of current elements, which converged to a specific pattern (Fig. 1). In (his paper. we apply this method to practical magnetoencephalograpby (MEG), to estimate the focal region of human brain activities. MEG MeasurementSomatosensory evoked fields were measured with a 160 ch SQUID system manufactured by Yokogawa Electric Cop 131. The pick up coil of the sensa is a coaxial 1st-order gradiometer whose baseline length is 50 min. The MEG signals from the gradiometers were filtered with a 3 -500 Hz bandpass filter and sampled at 2 kHz. 400 epochs were averaged for improvement of S/N.A digital low pass filter at 30 Hz was applied on the averaged signals for further analysis. The subject was a healthy male whose median nerve of the right wrist was electrically stimulated.AnMRI of the subject's brain was obtained and a concemed region G in tbe magnetic source imaging was determined from the MRI. Fig. 2. The ME0 signals from 16Och ms. A large peak a10 ms isan ami m noise fm a snnauaensay sumulam. Theflnt prak d evoked field appeared a1 m.5 ms after SUrndUE. . -. Yoshiah Adaclu Fig. 3. An isalield mnlDu map d the data se1 at 205 ms of lalency. Dots means locauom of s e m . Top and bmm o mspmd D I n e M and accipiol region. respectively. h e step corresponds 10 IO IT change. h k gray lincp rcpecnl o u i p ingfieldsofthehdbghtgray linea represen1 mmrmng field-. R e d t s and Analvsis Figure 2 shows superimposed evoked MEG signals from all 160 sensors. A contour map composed by the data at latency 23.5 ms is shown in Fig. 3. There are two poles of magnetic fluxintensity around the parietal area of the right hemisphere. We applied the current element distribution method on this set of data.X K ) current elements were distributed at random into G (Fig. 4a). Figure 4b shows distribution of the elements after 2,ooO iterations were cmied out. Stabilized condition [21 for optimizing the distribution of current elements was used. The result of conventional equivalent dipole fitting computed from the same set of data was superimposed on Fig. 4b. DiscussionFigure 4 shows that the current elements are concentrated around the corresponding position found by the conventional equivalent current dipole, around the somatosensory area. They formed not only a dipole-like structure, but also a secondary current distribution. Thus our method can, in principle, be applied to any pattern with a magnetic source, is not restricted to dipole-shaped sources. This suggests a great number of potential applications of the current element distribution method to MEG. Unlike in a simulation, the limitations of arrangement and n...
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