1.1. Motivation
Electroencephalograms (EEGs) have gained its popularity in neurological research and clinical applications due to their mostly non-invasive recording procedures, simple and inexpensive operations and high time resolutions. Originally, the clinical application of EEG analysis is to distinguish epileptic seizures from normal brain activities such as syncope, sub-cortical movement disorders and migraine variants. Advanced signal processing techniques such as independent component analysis (ICA) [1,2] and event related spectral perturbation (ERSP) analysis [3,4] have also become increasingly popular in clinical brain research. In recent years, researchers began to view multichannel EEG recordings as correlated signals with quasi-sparse representations and analyze them with the powerful arsenal of digital signal processing techniques.
However, due to the fact that the recording of EEG for a meaningful and accurate analysis is often necessarily time consuming; the relatively high time resolution and the characteristic of concurrent multichannel recording will produce enormous amounts of data needed to be stored and transported over media. Therefore, interests in searching for sparse and compressible representations of EEG have rose considerably during recent decades.
1.2. Recent Difficulties and Solutions
Unfortunately, determining whether a signal can be optimally approximated and decomposed with a linear expansion over a redundant (over-complete) dictionary of atomic waveforms is an NP-complete problem, furthermore, finding such a solution is an NP-hard one [5]. However, an approximate method that has already been discovered long before is found to be capable enough to tackle this problem. It was the Matching Pursuit algorithm [6]
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proposed by S. Mallat and Z. Zhang in 1993, which is a simple and economical way to produce a sub-optimal signal expansion by iteratively choosing the atom that best matches the signal structure and remove the atom ingredient from the signal until certain stopping condition is met. This algorithm has already been well studied and improved in the most recent decade, such as the randomized variant – the Stochastic Matching Pursuit (SMP) – pioneered by Piotr J. Durka in 2001.
Despite these efforts, sparse representation remains an impalpable objective. It’s well-known that the sparse decomposition result produced by SMP is possible, however unlikely, to be inconsistent. The dominant atoms in the decomposed sequences for one signal can be significantly different between multiple instances of SMP operations.
1.3. Objectives
The ultimate objective of this research is to identify the dictionary suitable for sparse representation of the EEG ERP signals and components, then design a Memetic Algorithm adopting the belief of Matching Pursuit, which is able to factorize a given EEG signal into a linear combination of only a small number of time-frequency atoms (Gabor functions, for instance) over a redundant dictionary, meanwhile preserving a large portion of signal energy.
1.4. Research Approach
The sparsity of data assumed the ability of representing signals as linear combinations of only a small amount of atoms in a redundant, pre-specified dictionary. This requires the prior knowledge on the signal characteristic and a properly chosen dictionary.
In the aspect of prior knowledge to the signals, researchers performing brain activity analysis based on the equivalent electric dipole modeling [7,8] believed that the stimulus and response events may be well modeled as equivalent dipoles oscillating locally at certain
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time period and frequency band. So far, on the aspect of proper selection of the dictionaries, Ron Rubinstein et al. indicated that there are two mainstream methods for searching for the proper redundant dictionary [9] currently.
1.4.1. Dictionary Chosen According to Mathematical Model of Signals
The build-up of dictionaries according to mathematical model is mainly functions with closed form mathematical expressions such as Gabor functions and wavelet atoms, which can be easily parameterized.
Further, about the searching method for the representation in the various types of standard dictionaries, while Durka’s Stochastic MP algorithm focuses mainly on bias elimination and generates only sequences of atoms with integral multiplications of sampling steps over the time and frequency domain, a new method employing advanced optimization algorithms is proposed to improve the accuracy and efficiency of parameter searching in each MP iterative step by regarding the parameters as continuous variables and providing an even sparser parameterized representation.
1.4.2. Dictionary Evolved Based on Training Data
This type of dictionaries emerges, in Ron’s words [9], from a given set of realizations of the data. This set serves as the examples of the signals to be modeled; with a given initial dictionary, the training algorithm will evolve the dictionary atoms iteratively to make the resulted dictionary performs better on the training set. In such case, an initialization method with minimal mutual coherence based on the Grassmannian frame for the dictionary is proposed to enhance the learning procedures.
1.5. Contributions
The major contribution for the standard dictionary approach is an efficient algorithm to
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find a decomposition sequence in the scalable Gabor dictionary, which improves the Matching Pursuit by extending the search space into continuous real number domain and with one more degree of freedom with variable scaling factors. With the behaviors of the parameters of the Gabor standard dictionary are well understood, the signal can therefore be decomposed into merely a small number of parameter sets with minimal energy losses.
Another contribution is the successful application of the K-SVD on the EEG signal integrated with the concept of Grassmannian frames; the resulted dictionary contains several fingerprints of the event related potential of the EEG signal, which might be used further to identify different stimuli with the help of classification methods.
1.6. Thesis Outline
Along the roadmap, in Chapter 2, the fundamental tools and prior knowledge will be introduced. Each succeeding chapters will focus on each milestone of major discoveries.
In Chapter 3, the identification of Gabor atoms as the most suitable type of waveform and an intuitive, however stable and consistent method based on Matching Pursuit to find the representation are described here, including some first-stage experiment results.
In Chapter 4, a better optimization method based on Natural Gradient and a sub-optimal uniform sampling method over Gabor parameter space using the Grassmannian concept will be introduced. The experiment results will be compared with their counterpart mentioned in Chapter 3. The Chapter 3 and Chapter 4 will cover the approach of realizing sparse representation via overcomplete standard waveform dictionaries.
Chapter 6 describes the improvements on the K-SVD algorithm via a new dictionary initialization method using the Grassmannian frame; the experiment result will be compared to the default initialization method using the normalized random Gaussian noise.
Chapter 7 includes the achievements and the future work of this research.
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