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Background: Transcranial magnetic stimulation (TMS) plays an important role in treatment of mental and neurological illnesses, and neurosurgery. However, it is difficult to target specific brain regions accurately because the complex anatomy of the brain substantially affects the shape and strength of the electric fields induced by the TMS coil. A volume conductor model can be used for determining the accurate electric fields; however, the construction of subject-specific anatomical head structures is timeconsuming. Objective: The aim of this study is to propose a method to estimate electric fields induced by TMS from only T1 magnetic resonance (MR) images, without constructing a subject-specific anatomical model. Methods: Very large sets of electric fields in the brain of subject-specific anatomical models, which are constructed from T1 and T2 MR images, are computed by a volume conductor model. The relation between electric field distribution and T1 MR images is used for machine learning. Deep neural network (DNN) models are applied for the first time to electric field estimation. Results: By determining the relationships between the T1 MR images and electric fields by DNN models, the process of electric field estimation is markedly accelerated (to 0.03 s) due to the absence of a requirement for anatomical head structure reconstruction and volume conductor computation. Validation shows promising estimation accuracy, and rapid computations of the DNN model are apt for practical applications. Conclusion: The study showed that the DNN model can estimate the electric fields from only T1 MR images and requires low computation time, suggesting the possibility of using machine learning for realtime electric field estimation in navigated TMS.

Let us consider a case where all of the elements in some continuous slices are missing in tensor data. In this case, the nuclear-norm and total variation regularization methods usually fail to recover the missing elements. The key problem is capturing some delay/shift-invariant structure. In this study, we consider a low-rank model in an embedded space of a tensor. For this purpose, we extend a delay embedding for a time series to a "multi-way delay-embedding transform" for a tensor, which takes a given incomplete tensor as the input and outputs a higher-order incomplete Hankel tensor. The higher-order tensor is then recovered by Tucker-based low-rank tensor factorization. Finally, an estimated tensor can be obtained by using the inverse multiway delay embedding transform of the recovered higherorder tensor. Our experiments showed that the proposed method successfully recovered missing slices for some color images and functional magnetic resonance images.

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