Registration of MRI and EEG based on internal and external anatomical similarities

Abstract

Accurate registration of MRI and EEG was obtained by employing anatomical information into the registration process. For this purpose, an optimizing cost function was defined based on the observed internal similarity, e.g. the brain symmetry, and the external similarity, resulting from the likelihood of surface neighbourhood and MRI image intensity histograms. Internal and external similarity was expressed as the sum of Kullback-Leibler divergences between corresponding intensity histograms. The proposed method was evaluated on clinical MRI data with simulated EEG data, yielding mean registration error of 0.48±0.33 mm, while with the real EEG data an average root-mean-square point-to-surface error of 2.27 ± 0.02 mm was obtained.

1 Introduction

Correlating functional information with anatomical localization offers the ability to understand how the brain functions. For in vivo neurophysiological studies of the brain, magnetic resonance imaging (MRI) is usually integrated with functional information like electroencephalography (EEG), magnetoencephalography (MEG) or transcranial magnetic stimulation (TMS). EEG techniques, especially, allow for high-resolution measurements of temporal and, if related to brain anatomy derived from MRI, also spatial dimensions of brain's electrical activity [1]. However, regarding the anatomical precision of clinically relevant locations of brain's electrical activity, accurate registration of MRI and EEG data is needed.

Registration of MRI and EEG is concerned with spatial localization of EEG electrodes in the MRI image. Leaving aside semi-automatic methods that use landmarks and markers, we focus on retrospective registration of MRI and EEG that relies solely on the MRI image data, while mapping EEG physical space to MRI image space. Retrospective registration of MRI and EEG can be efficiently solved by 1) extracting head surface from MRI image, 2) acquiring EEG coordinates in physical space by some digitizer technology and 3) using a surface-matching technique [2] to align the head surface and the digitized coordinates of EEG electrodes. Previously published methods for MRI and EEG/MEG registration [3-7] and similarly for MRI and TMS registration [8] typically utilize the three above-mentioned steps. However, discarding anatomical information by the head surface extraction step renders the registration process an ill-posed problem, especially due to the spherical symmetry of the head shape. Therefore, to get a well-behaved registration process, we propose to include as much anatomical information in the registration of MRI and EEG as possible.

In this paper, the main idea is to use anatomical information to drive the registration of MRI and EEG. Given a set of digitized electrode coordinates or EEG points that resemble the head shape, we pursue to exploit the underlying symmetry of the brain hemispheres by analyzing the topology of the EEG points, which yields the so-called internal similarity. On the other hand, we observe the so-called external similarity by comparing the statistical properties between the local neighbourhoods of the head surface and the whole MRI image. This insight is used to design a cost function yielding a highly accurate and reproducible spatial registration of MRI and EEG.

2 Methods and Materials

In this section, the proposed method for registration of MRI and EEG is presented. The outline of the registration method is depicted in Fig. 1. Given the MRI and EEG input data, pre-registration initialization is done by computing symmetry planes of MRI and EEG data and defining point-to-point correspondences of EEG points, followed by a two-step registration of MRI and EEG: global closed-form registration and local iterative registration. In the following subsections, the MRI and EEG input data, the pre-registration initialization and the global and local two-step registration of MRI and EEG are explained.

Fig. 1.

Proposed method for registration of MRI and EEG: (a) input MRI image and (b) the extracted mid-sagittal plane of the brain [9], (c) input EEG points and (d) the extracted symmetry plane and point-to-point correspondences of EEG points. A global closed-form registration step (e) aligns the extracted symmetry planes of the MRI image and the EEG points, followed by (f) local iterative registration procedure. In (g), the extracted head surface and corresponding MRI coordinate system.

2.1 MRI and EEG datasets

Missing information about the data:

• – MRI scanner. Manufacturer? How many T? Scanning sequence setup? Acquisition setup?

• – MRI image metadata. Voxel sizes? Slice thickness? Scan time?

• – EEG measuring system. Manufacturer? (EGI) Model number?

• – EEG sensor hat. Model number? How many electrodes? (128)

• – EEG electrode position digitization. Model number? Number of cameras?

• – Patients. Age? Normal or pathological state?

2.2 Pre-registration initialization

Prior to registration, input MRI and EEG data is pre-processed for two reasons: 1) to get an initial registration of MRI and EEG and 2) to derive from the EEG points the information related to the anatomical symmetry of brain hemispheres. To estimate the initial registration of MRI and EEG, we propose to extract and match the symmetry planes of both MRI and EEG points. Symmetry plane of the MRI image, also called the mid-saggital plane (MSP), is computed by the method of Ardekani et al. [9].

Symmetry plane of the EEG points (ESP) is computed from its 3-D geometrical moments [6], i.e. the gravity center and principal axes derived from inertia matrix of the EEG points. Parameters of the ESP are estimated by using as the plane origin the gravity center of EEG points and as the plane normal a cross-product between two principal axes, corresponding to the smallest and the largest eigenvalue of the inertia matrix. Such estimation of the ESP is valid without loss of generality, since fixed and physically constrained configuration of the EEG sensor array preserves the distribution of EEG points for different patients. The distribution of the EEG points, however, resembles the head shape of the patient, therefore, the ESP also corresponds to the above-mentioned MSP derived from MRI.

Let the ESP plane ^ESP p(x, y, z) = 0 divide the EEG points V = {V_i ; i = 1 … N} in two subsets:

$$V_i\in \{\begin{array}{cc}V1\hfill & \text{if}{}^{\mathit{ESP}}p\left(V_i\right)\ge 0\hfill \\ V2\hfill & \text{otherwise}\hfill \end{array}\phantom{\}}$$ (1)

and construct using {V1, V2} a maximum cardinality bipartite graph [10] according to sorting criterion:

$$\{V{1}_{i\ast },V{2}_{j\ast }\}=\underset{\begin{array}{c}\hfill i=1\dots n_1\hfill \\ \hfill j=1\dots n_2\hfill \end{array}}{\mathrm{argmin}}\Vert V{1}_i^{\perp }-V{2}_j^{\perp }\Vert$$ (2)

where points $V{1}_i^{\perp }$ and $V{2}_j^{\perp }$ correspond to V1_i and V2_j, respectively, projected in the ESP plane and ∥ · ∥ denotes the L₂–norm. In this way, the N_p ≤ N/2 corresponding pairs of points {V1_i*, V2_j*} model the anatomical symmetry of the brain hemispheres.

2.3 Global closed-form registration

Global or initial registration of MRI and EEG is obtained by first aligning the gravity centers of the EEG points and MR image, followed by a rotation that aligns the two normals of MSP and ESP, described in the previous section. The global registration results in a coarse alignment of MRI and EEG, but incorporates a π ambiguity around the z–axis and an undefined rotation around x–axis. The rotation ambiguities are resolved by minimizing, with a discrete angular step at the corresponding axes, a cost function that is used by the local iterative registration.

2.4 Local iterative registration

The distibution of the EEG points incorporates knowledge about the head shape, but also reflects the anatomical symmetry between the brain hemispheres. Our goal is to design a cost function based on the location of the EEG points that would incorporate as much prior knowledge of anatomical symmetry of the brain as possible. For this reason, we use the whole anatomical information provided in the MRI to compute the cost function. To design a suitable cost function, we first observe that general shape of the T1-weighted MR image intensity histogram or global histogram (Fig. 2a) consists of four partially overlapping modes, resulting from the air, cerebrospinal fluid (CSF), grey matter (GM) and white matter (WM). On the other hand, given a general intensity histogram of the head surface neighbourhood or local histogram (Fig. 2b), we observe partially overlapping modes, resulting from the air, CSF, skull, brain tissue, skin and fat. Comparing modes of the global and local intensity histograms reveals the fact, that given a large enough surface neighbourhood for the local histogram, roughly the same information content is represented by both the global and the local histogram.

Fig. 2.

Intensity histograms of (a) MRI image and (b) head surface neighbourhood. Modes of MRI image intensity histogram are represented by air, cerebrospinal fluid (CSF), grey matter (GM) and white matter (WM), while modes of head surface neighbourhood histogram are generally represented by air, CSF, brain tissue, skin and fat.

Resulting from the above findings, we define the notion of external and internal similarity. Registered EEG points should ideally lie on the head surface, thus the local histograms of those registered points exhibit high similarity to the global histogram, which we call the external similarity. In contrast to the external similarity, we define the internal similarity as the similarity between the local histograms of corresponding pairs of points {V1_i*, V2_j*}, as defined in section 2.2. Therefore, optimal registration of MRI and EEG should yield a maximum of both external and internal similarity.

Instead of measuring the similarity, we measure the dissimilarity of two histograms H₁ and H₂ by the Kullback-Leibler divergence or relative entropy [11]:

$$\mathit{KL}(H_1\Vert H_2)=\underset{i}{\Sigma }{p}_i\log \frac{p_i}{q_i}$$ (3)

where p_i and q_i are probability distributions from H₁ and H₂, respectively. Local histograms are obtained by sampling a sphere of radius r with step s along the x–, y– and z– axes at the corresponding 3-D locations in the MRI image. Partial volume interpolation [12] is used to obtain the local histograms, which results in a smooth cost function:

$$CF=\underset{k=1}{\overset{N}{\Sigma }}KL(H\left(V_k\right)\Vert H_g)+\frac{N}{N_p}\underset{k=1}{\overset{N_p}{\Sigma }}KL(H\left(V{1}_{i\ast \left(k\right)}\right)\Vert H\left(V{2}_{j\ast \left(k\right)}\right))$$ (4)

with H_g being the global histogram and H(V_k) being the local histogram computed at point V_k. In (4), the first sum measures the external dissimilarity, while the second sum measures the internal dissimilarity. Finally, optimal parameters for rigid registration of MRI and EEG are found by iteratively minimizing the cost function (4) using Powell's multi-dimensional directional set method and Brent's one-dimensional optimization algorithm [13].

3 Experiments and Results

Performances of the presented method for registration of MRI and EEG were evaluated on real data and by Monte Carlo simulations, as suggested by Singh et al. [14]. For this purpose, the head surface was extracted from MRI by interactive thresholding and manual correction (Fig. 1g). From the extracted head surface, 128 points were uniformly sampled to obtain the simulated EEG data. Transforming the simulated EEG points with a known rigid transformation and running the local iterative registration enables the estimation of mean registration error (MRE), computed as a mean distance between the true and the registered position of the simulated EEG points. By combined random translations and rotations of the simulated EEG, initial MREs were generated in the range of [0, 30] mm, with 10 MREs for each 1 mm subinterval, yielding 300 starting positions. After the local iterative registration, the obtained average MRE was 0.77 ± 0.36, 0.39 ± 0.38 and 0.28 ± 0.24 mm for each of the three datasets. Simulation results for the three datasets are shown in Fig. 3.

Fig. 3.

Registration results with simulated EEG data for the three datasets, showing mean registration error (MRE) of simulated EEG points before and after registration. By random translation and rotation of the simulated EEG points, 300 initial MREs were generated in the range of [0, 30] mm, with 10 MREs for each 1 mm subinterval.

Using real EEG data, the global closed-form registration was first executed. Next, 300 displacements of the obtained globally registered EEG points were generated by a combined random translation and rotation in the range of [−30, 30] mm and [−30, 30] degrees, respectively, followed by the local iterative registration. Root-mean-square error (RMS) was computed between the extracted head surface and the registered EEG data, yielding 2.41 ± 0.04, 2.26 ± 0.02 and 2.14 ± 0.01 mm for each of the three datasets. Computing the overall average RMS enabled us to compare our results to the previously published results (table 1).

Table 1.

Registration results of the simulations and the patient's studies from the present and the previously published methods.

Method	Simulation	Patient's study
Schwartz el al. [3]	0.37±0.15 mm
Huppertz et al. [4]		3.39±0.24 mm
Brinkmann et al. [5]		3.36±0.91 mm
Kozinska el al. [6]	0.73−1.22 mm	1.55±0.07 mm
Lamm et al. [7]	0.61±0.26 mm	2.43±0.22 mm
Noirhomme et al. [8]	0.17±0.30 mm	1.17±0.38 mm
Present method	0.48±0.33 mm	2.27±0.02 mm

Histogram computation

Local histograms were obtained from a spherically shaped neighbourhood with radius r = 10 mm and sampling s = 1 mm. All histograms had 32 bins. Typical plots of the cost function (4) around the registered position are shown in Fig. 4.

Fig. 4.

Plot of the cost function (4) with respect to x−, y and z− translation (top) and rotation (bottom) from the registered position in the range of [−30, 30] mm and [−30, 30] degrees, respectively.

4 Discussion

Accurate registration of MRI and EEG was obtained by employing anatomical information into the registration process. The proposed method (Fig. 1) finds the corresponding EEG points, thereby modelling the anatomical symmetry of the brain, which yields the internal similarity. The external similarity was observed by comparing the global or MRI histogram with the local or surface neighbourhood histogram (Fig. 2). Internal and external similarity was expressed as the Kullback-Leibler divergence [11] between corresponding local and between local and global intensity histograms, respectively. The contributions of the internal and external similarity were summed (4), resulting in the smooth cost function (Fig. 4) that was minimized with respect to parameters of rigid registration to obtain the final registration of MRI and EEG.

Results with simulated EEG data gave average MRE of 0.48 ± 0.33 mm, while the results on the real EEG data gave an average RMS of 2.27 ± 0.02 mm. It should be noted, that with our method the evaluation measures (MRE, RMS) are independent of the optimizing cost function (4), contrary to the previously published, competing methods [3-8]. However, as can be seen from table 1, the results obtained by these competing methods are comparable to the results obtained with our method. The competing methods are exclusively based on the free-form surface matching techniques [2], that rely on proper segmentation of the head surface from MRI. According to Noirhomme et al. [8], registration errors due to segmentation of MRI can amount to as much as 1 mm in MRE and RMS. Due to the spherical symmetry of the head shape, more points than there is EEG electrodes has to be obtained, using either spline interpolation of EEG points [7], acquiring more than 1000 virtual EEG points [3-6] or applying a special point pattern during point acquisition [8], resulting in 200–800 virtual EEG points. Generally, the less points used for the free-form surface matching, the poorer the registration accuracy. Moreover, reducing the set of EEG points can lead to local minima in the distance measure, used by the free-form surface matching methods in [3-8]. On the other hand, using as much points as there are EEG electrodes and exploiting the anatomical information yields accurate registration of MRI and EEG. Besides, no local minima was observed (Fig. 4), thus resulting in a large capturing range (> 30 mm, > 30 degrees) of the proposed registration method.

To conclude, a fully automatic anatomy-driven method for registration of MRI and EEG was presented and evaluated on simulated and real EEG data. By incorporating anatomical information in the registration process, the segmentation of MRI can be omitted without deteriorating the registration accuracy (table 1). Moreover, using the framework of the proposed method (Fig. 1), other anatomical (CT) or functional (positron emission tomography or PET and functional MRI) information can be correlated with modern point-based functional methods (EEG, MEG, TMS). In this way, the clinical neurophysiology would benefit from the ability of matching any tomographic data with the point-based functional methods [1], thus reducing cost and time of patient care.