Unsupervised Segmentation of Head Tissues from Multi-modal MR Images for EEG Source Localization

Abstract

In this paper, we present and evaluate an automatic unsupervised segmentation method, hierarchical segmentation approach (HSA)–Bayesian-based adaptive mean shift (BAMS), for use in the construction of a patient-specific head conductivity model for electroencephalography (EEG) source localization. It is based on a HSA and BAMS for segmenting the tissues from multi-modal magnetic resonance (MR) head images. The evaluation of the proposed method was done both directly in terms of segmentation accuracy and indirectly in terms of source localization accuracy. The direct evaluation was performed relative to a commonly used reference method brain extraction tool (BET)–FMRIB’s automated segmentation tool (FAST) and four variants of the HSA using both synthetic data and real data from ten subjects. The synthetic data includes multiple realizations of four different noise levels and several realizations of typical noise with a 20 % bias field level. The Dice index and Hausdorff distance were used to measure the segmentation accuracy. The indirect evaluation was performed relative to the reference method BET-FAST using synthetic two-dimensional (2D) multimodal magnetic resonance (MR) data with 3 % noise and synthetic EEG (generated for a prescribed source). The source localization accuracy was determined in terms of localization error and relative error of potential. The experimental results demonstrate the efficacy of HSA-BAMS, its robustness to noise and the bias field, and that it provides better segmentation accuracy than the reference method and variants of the HSA. They also show that it leads to a more accurate localization accuracy than the commonly used reference method and suggest that it has potential as a surrogate for expert manual segmentation for the EEG source localization problem.

Appendix

A.1Bayesian-Based Adaptive Bandwidth Estimator

The bandwidth is modeled [24] by the a posteriori probability density function p(s|x) of local data spread or variance s given the data (feature) point x. Let M < n (total number of data points) be the number of nearest neighbors to a data sample x _i. We can then define the pseudolikelihood

$$ P\left(s\Big|\mathbf{x}\right)=\prod_{j=1}^NP\left(s\Big|{\mathbf{x}}_{M_j}\right) $$

(10)

where $P\left(s\Big|{\mathbf{x}}_{M_j}\right)$ is the probability of local data spread s depending on the M _j nearest neighborhood samples to ${\mathbf{x}}_{M_j}$ and {M _j | j=1,....,N} is the set of N such neighborhoods of various sizes. The evaluation of these probabilities over the entire set of M _j is then given by

$$ P\left(s\Big|{\mathbf{x}}_{M_j}\right)={\displaystyle \int }P\left(s\Big|{M}_j,{\mathbf{x}}_{M_j}\right)P\left({M}_j\Big|{\mathbf{x}}_{M_j}\right)\ \mathrm{d}{M}_j $$

(11)

Applying Bayes rule we get

$$ P\left({M}_j\Big|{\mathbf{x}}_{M_j}\right)=\frac{P\left({\mathbf{x}}_{M_j}\Big|{M}_j\right)P\left({M}_j\right)}{P\left({\mathbf{x}}_{M_j}\right)} $$

(12)

where $P\left({\mathbf{x}}_{M_j}\Big|{M}_j\right)$ is the probability of the data sample ${\mathbf{x}}_{M_j}$ given the M _j nearest neighborhood. Hereinafter, P(M _j ) is considered to have uniform distribution on the interval [M ₁, M ₂]. Several values are selected for M _j in this interval according to

$$ {M}_j = {M}_1+j\frac{M_2-{M}_1}{N} $$

(13)

For a given M _j , the local variance s _j is computed as

$$ {s}_j=\frac{{\displaystyle {\sum}_{l=1}^{M_j}}\parallel {\mathbf{x}}_{i,l}-{\mathbf{x}}_i{\parallel}^2}{M_j-1}\kern1.25em i=1,2\dots ..\ n\kern1em j=1,2,\dots N $$

(14)

where x _i,j is the lth nearest neighbor to the data point x _i . The distribution of variances is modeled as the Gamma distribution defined as

$$ p\left(s\Big|\alpha, \beta \right)=\frac{\beta^{\alpha }{s}^{\alpha -1}}{\Gamma \left(\alpha \right)}{e}^{-\beta s}\kern1.25em s\ge 0\kern1em \alpha, \beta >0 $$

(15)

where

$$ \varGamma (t)={\displaystyle \underset{0}{\overset{\infty }{\int }}}{r}^{t-1}{e}^{-r}\mathrm{d}r $$

(16)

is the Gamma function, and α and β define the shape and the scale of the Gamma distribution, respectively.

These parameters are estimated using the maximum likelihood approach [24]. The estimate of the adaptive bandwidth is identically the mean of this distribution, i.e.,

$$ \hat{h}\ \left({\mathbf{x}}_i\right)=\hat{\alpha}\hat{\beta}\kern1.25em i=1,2,\dots n $$

(17)