Matching Pursuit And Unification In EEG Analysi...

This paper presents a summary of time-frequency analysis of the electrical activity of the brain (EEG). It covers in details two major steps: introduction of wavelets and adaptive approximations. Presented studies include time-frequency solutions to several standard research and clinical problems, encountered in analysis of evoked potentials, sleep EEG, epileptic activities, ERD/ERS and pharmaco-EEG. Based upon these results we conclude that the matching pursuit algorithm provides a unified parametrization of EEG, applicable in a variety of experimental and clinical setups. This conclusion is followed by a brief discussion of the current state of the mathematical and algorithmical aspects of adaptive time-frequency approximations of signals.

Matching Pursuit and Unification in EEG Analysi...

where γ i i = 1..M represents the indices of the chosen functions gγi. Finding such an optimal approximation is an NP-hard problem. A suboptimal expansion can be found by means of an iterative procedure, such as the matching pursuit algorithm (MP) proposed by Mallat and Zhang [7].

1The resolution of the matching pursuit is hard to define in general, since the procedure is nonlinear and signal-dependent. It should be related to the distance between neighboring dictionary waveforms available for decomposition. In the described procedure, this distance does not exceed twice the value of the corresponding parameter (Δt, Δω or Δs).

Other, more theoretical examples of MP failures are given in [16] and [17]. Some of these cases can be properly solved by the orthogonalized matching pursuit [18], at a cost of increased computational requirements and a possibility of introducing numerical instabilities [19]. Another modification of the MP algorithm, discussed in [20], relies on a modification of the similarity function used in each step to choosed the "best fit". Other works [21] indicate that global optimalization of the l1 norm of expansion's coefficients might be the best choice [17], but, in spite of recent advances in linear programming, computational complexity is still very high. 041b061a72