Chapter 7. Conclusion
7.2. Future Works
Nonetheless, this preliminary effort merely marks the beginning of our continuous search for better sparse representations of EEG signals and suitable dictionaries for these representations.
The primary concern is about the biomedical meaning of the decomposition result, most of the pursuit algorithms are not designed to be applied on a specific type of signals.
Due to the sophisticated propagation of EEG signal from their actual brain location to the scalp, through skin, hair and finally to the sensors, the meaningful ingredient might become insignificant and difficult to extract. Always searching for the best fit waveform and justifying the precision of the decomposition of the signals in terms of energy is not always beneficial.
A temporarily possible solution might be the cross reference between the channel and ICA components, with the application of ICA components, one might be able to eliminate the undesired signal components and extract the de facto EEG events well represent the brain activities.
Establishment of the relationship between the result and Compressive Sensing might become a logical next step; during the analyses process of EEG and especially the ERP data, with both the standard dictionary approach and the K-SVD experiments, atoms are found to be clustered into several region and share similar parameter values, further investigation might be carried out to classify the effect, if an jittering and scaling resistive decomposition method can be found, a large number of the single-trial decomposition results can be compressed in advance to save even more data storage resources.
For the dictionary trained by the K-SVD algorithm, an intuitive thought is to discover the proper measurement matrix for the compressive sensing framework described in [12], but still, since the experiment requires an astronomical amount of data to form a proper training signal matrix.
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Mathematical Notations
Spaces
ℝ𝑁 Real signal of size 𝑁 ℂ𝑁 Complex signal of size 𝑁
𝐋2(ℝ) Finite energy continuous functions: ∫ |𝑓(𝑡)|2𝑑𝑡 < ∞ 𝐥2(℞) Finite energy discrete functions: ∑∞𝑛=−∞|𝑓,𝑛-|2 < ∞
𝐇 Hilbert Space
Operators and Operations
𝑧∗ Complex conjugate of 𝑧 ∈ ℂ
‖𝑓‖ 𝐋2 Norm
|𝑓|𝑝 𝐋𝑝 Norm
〈𝑓 𝑔〉 Inner Product (Mathematically varied in different spaces) 𝐴𝑇 Transpose of Matrix
inf*𝑆+ Infimum of set 𝑆 𝜍*𝐴+ Spark of matrix 𝐴
𝑇𝑡0 Shift 𝑡0 in time
𝑀𝜔0 Modulate 𝜔0 in frequency
Transforms and Representations
𝑓̂(𝜔) Fourier transform
𝑓̂,𝑛- Discrete Fourier transform 𝑆𝑓,𝑡 𝜔- Short-time Fourier transform 𝑃𝑆𝑓(𝑡 𝜔) Spectrogram
𝑃𝑉𝑓(𝑡 𝜔) Wigner-Ville distribution
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