Multivariate matching pursuit in optimal Gabor dictionaries: theory and software with interface for EEG/MEG via Svarog

Abstract

Background: Matching pursuit algorithm (MP), especially with recent multivariate extensions, offers unique advantages in analysis of EEG and MEG.

Methods: We propose a novel construction of an optimal Gabor dictionary, based upon the metrics introduced in this paper. We implement this construction in a freely available software for MP decomposition of multivariate time series, with a user friendly interface via the Svarog package (Signal Viewer, Analyzer and Recorder On GPL, http://braintech.pl/svarog), and provide a hands-on introduction to its application to EEG. Finally, we describe numerical and mathematical optimizations used in this implementation.

Results: Optimal Gabor dictionaries, based on the metric introduced in this paper, for the first time allowed for a priori assessment of maximum one-step error of the MP algorithm. Variants of multivariate MP, implemented in the accompanying software, are organized according to the mathematical properties of the algorithms, relevant in the light of EEG/MEG analysis. Some of these variants have been successfully applied to both multichannel and multitrial EEG and MEG in previous studies, improving preprocessing for EEG/MEG inverse solutions and parameterization of evoked potentials in single trials; we mention also ongoing work and possible novel applications.

Conclusions: Mathematical results presented in this paper improve our understanding of the basics of the MP algorithm. Simple introduction of its properties and advantages, together with the accompanying stable and user-friendly Open Source software package, pave the way for a widespread and reproducible analysis of multivariate EEG and MEG time series and novel applications, while retaining a high degree of compatibility with the traditional, visual analysis of EEG.

Figure 1

Sample epoch of sleep EEG. Screenshot displaying an epoch of sleep EEG recording. SWA and sleep spindles were marked by an electroencephalographer as green and gray rectangles, correspondingly (in this case both spindle tags fall inside the epochs marked as SWA). Blue dashed line outlines the epoch from C3 selected for MP decomposition, shown in Figure 2.

Figure 2

Time-frequency distribution of signal’s energy. Results of MP decomposition displayed as an interactive time-frequency of map signal’s energy density in Svarog. Clicking center of a blob (marked by white cross) adds the corresponding function to the reconstruction (bottom signal).

Figure 3

Defining a filter for SWA. Filter defining criteria for waveforms corresponding to SWA in the Svarog interface to MP.

Figure 4

Structures corresponding to SWA. Result of the application of the filter from Figure 3 to the decomposition from Figure 2.

Figure 5

Structures corresponding to sleep spindles. Result of application of the filter defining sleep spindles to the decomposition from Figure 2.

Figure 6

Setting the dictionary parameters for MP decomposition in Svarog. Tab for setting the parameters governing construction of the Gabor dictionary for MP decomposition in Svarog.

Figure 7

Optimal scale factor, frequency step and position step. Optimal scale factor a (top plot—see Eq. (20)), frequency step Δω (middle plot—see Eq. (22)) and position step Δu (see Eq. (24)) as a function of ω for three values of ϵ and scale s=10.