Combining sparsity and rotational invariance in EEG/MEG source reconstruction

Stefan Haufe, Vadim V. Nikulin, Andreas Ziehe, Klaus Robert Müller, Guido Nolte

Research output: Contribution to journalArticlepeer-review

102 Citations (Scopus)


We introduce Focal Vector Field Reconstruction (FVR), a novel technique for the inverse imaging of vector fields. The method was designed to simultaneously achieve two goals: a) invariance with respect to the orientation of the coordinate system, and b) a preference for sparsity of the solutions and their spatial derivatives. This was achieved by defining the regulating penalty function, which renders the solutions unique, as a global ℓ1-norm of local ℓ2-norms. We show that the method can be successfully used for solving the EEG inverse problem. In the joint localization of 2-3 simulated dipoles, FVR always reliably recovers the true sources. The competing methods have limitations in distinguishing close sources because their estimates are either too smooth (LORETA, Minimum ℓ1-norm) or too scattered (Minimum ℓ2-norm). In both noiseless and noisy simulations, FVR has the smallest localization error according to the Earth Mover's Distance (EMD), which is introduced here as a meaningful measure to compare arbitrary source distributions. We also apply the method to the simultaneous localization of left and right somatosensory N20 generators from real EEG recordings. Compared to its peers FVR was the only method that delivered correct location of the source in the somatosensory area of each hemisphere in accordance with neurophysiological prior knowledge.

Original languageEnglish
Pages (from-to)726-738
Number of pages13
Issue number2
Publication statusPublished - 2008 Aug 15

Bibliographical note

Funding Information:
This work was supported in part by the Bundesministerium für Bildung und Forschung (16SV2234, 01GQ0415), the Deutsche Forschungsgemeinschaft (MU 987/3-1) and the IST Programme of the European Community, under the PASCAL Network of Excellence (IST-2002-506778). We thank Friederike Hohlefeld and Monika Weber for help in preparing the experiment, and Ryota Tomioka for fruitful discussions.


  • Inverse problem
  • Rotational invariance
  • Second-order cone programming
  • Source localization
  • Sparsity
  • Vector fields
  • ℓ-norm Regularization

ASJC Scopus subject areas

  • Neurology
  • Cognitive Neuroscience


Dive into the research topics of 'Combining sparsity and rotational invariance in EEG/MEG source reconstruction'. Together they form a unique fingerprint.

Cite this