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This paper presents two new computationally efficient direct methods for fitting n-dimensional ellipsoids to noisy data. They conduct the fitting by minimizing the algebraic distance in subject to suitable quadratic constraints. The hyperellipsoid-specific (HES) method is an elaboration of existing ellipse and 3D ellipsoid-specific fitting methods. It is shown that HES is ellipsoid-specific in n-dimensional space. A limitation of HES is that it may provide biased fitting results with data originating from an ellipsoid with a large ratio between the longest and shortest main axis. The sum-of-discriminants (SOD) method does not have such a limitation. The constraint used by SOD rejects a subset of non-ellipsoidal quadrics, which enables a high tendency to produce ellipsoidal solutions. Moreover, a regularization technique is presented to force the solutions towards ellipsoids with SOD. The regularization technique is compatible also with several existing 2D and 3D fitting methods. The new methods are compared through extensive numerical experiments with n-dimensional variants of three commonly used direct fitting approaches for quadratic surfaces. The results of the experiments imply that in addition to the superior capability to create ellipsoidal solutions, the estimation accuracy of the new methods is better or equal to that of the reference approaches.
RESUMO
OBJECTIVES: Transcranial magnetic stimulation (TMS) and high-resolution electroencephalography (EEG) were used to study the spreading of cortical activation in 6 healthy volunteers. METHODS: Five locations in the left sensorimotor cortex (within 3cm(2)) were stimulated magnetically, while EEG was recorded with 60 scalp electrodes. A frameless stereotactic method was applied to determine the anatomic locus of stimulation and to superimpose the results on magnetic resonance images. Scalp potential and cortical current-density distributions were derived from averaged electroencephalographic (EEG) data. RESULTS: The maxima of the ipsilateral activation were detected at the gyrus precentralis, gyrus supramarginalis, and lobulus parietalis superior, depending on the subject. Activation over the contralateral cortex was observed in all subjects, appearing at 22plus minus2ms (range 17--28); the maxima were located at the gyrus precentralis, gyrus frontalis superior, and the lobulus parietalis inferior. Contralateral EEG waveforms showed consistent changes when different sites were stimulated: stimulation of the two most medial points evoked the smallest responses fronto-parietally. CONCLUSIONS: With the combination of TMS, EEG, and magnetic resonance imaging, an adequate spatiotemporal resolution may be achieved for tracing the intra- and interhemispheric spread of activation in the cortex caused by a magnetic pulse.
