CHAPTER 2. BASICS OF EEG-BASED BRAIN IMAGING 27
3.4 Concluding Remarks
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 48
Laplace’s PDE because we want to reconstruct the field without using any source modeling and relying only on the sensor’s measurements. After 20 iterations, when the Kaczmarz method has converged to the minimum norm solution, the 3D-VFT and ART-based solutions are comparable, since in this case the 3D-VFT approxima- tion performs better by less than 1% and 3◦. Hence, 3D-VFT regularizes the ill-posed problem by discretization and can yield a solution by applying a direct solver, instead of solving it through an iterative method.
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 49
but tolerant to the sensors displacements. This is rather encouraging, considering that some of the noise in the measurements can be tackled using filtering techniques.
On the other hand, if a set of sensors is displaced during a practical application, as is often the case for EEG electrodes, the true values cannot be retrieved. Moreover, when the same problem is solved using Laplace’s PDE, the numerical solution errors are lower when the source lies outside the field, but much higher when the sources approach or lie within the domain. The suggested method yields comparable results to the ART’s ones.
After the presentation of the theoretical framework for reconstructing electric fields, we will focus on the application of the method to the inverse EEG problem.
The voltages recorded by the electrodes will serve as the method’s boundary mea- surements. The modeling of the field within the head as bounded and irrotational complies with the physical properties adopted for the formulation of the method.
Moreover, due to the nature of the method, we do not need to use any source mod- eling, or modeling of the propagation properties of the field. Instead, we will assume that we have a uniform bounded space. In the next chapter, the formulation of the method to comply with a more realistic setting that resembles the one of the inverse EEG problem will be described.
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 50
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(iii) RME (%) (iv) AE(o)
Figure 3.4: A simulation example for the recovery of an electrostatic field, caused by a point source located at (15, 15, 15). Here, U=4, P=1 and ∆r was taken equal to 0.1.
(i) The field, as estimated from Coulomb’ s law. (ii) The reconstructed field, using only the values provided by Coulomb’ s law on the locations of the point sensors. (iii) The relative errors in the magnitude between the theoretical and the reconstructed field. (iv) The corresponding orientation errors. In both cases, (a)-(h) present the errors arranged in one slice of the z-axis, for better visibility. Finally, (v) and (vi) present the distributions of the two kinds of errors, i.e., RME (%) and AE (◦), denoted in the corresponding figure legend respectively.
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 51
10−4 10−3
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(ii)Mean(AE◦)
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∆r
(iii)Conditionnumber
Figure 3.5: The errors in the solution of the simulation of Fig. 3.4 for different values of the sampling step ∆r. (i) Shows the increase of the mean relative magnitude error, (ii) depicts the mean orientation error, while (iii) shows the condition number of the system’s matrix.
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 52
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(iii) RME (%) (iv) AE(o)
Figure 3.6: As in Fig. 3.4, but here the vector field is created by 20 point sources, distributed on the planez=0, along the circle with radius equal to 12.
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 53
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Figure 3.7: As in Figs. 3.4, 3.6, but here the vector is field generated by two point sources lying within the bounded domain, localized at (2, 2, 2) and (-2, -2, -2). The blue asterisks indicate the sources positions.
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 54
1 3 5 7
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(d) All Sensors Perturbed
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Figure 3.8: Errors in the solution recovered by the 3D-VFT method for simulation #1, when noise was added to the sensors measurements as a percentage of the true values. The errors are tested for a range of different sampling step ∆r. (i), (ii) Correspond to the mean RME and mean AE, respectively. In both cases, (a):(d) present the errors when different percentages of sensors are perturbed, starting from 25% of the sensors and concluding with the total of them.
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 55
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Figure 3.9: Reconstruction errors when perturbations, expressed as a percentage of the true location, was added to the sensors positions (simulation #2), for different ∆r values.
(i), (ii) Correspond to the mean RME and mean AE, respectively. In both cases, (a):(d) show the errors for various percentages of sensors perturbed, ranging from 25% to 100%.
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 56
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Additive Noise (%) Sampling Step
Figure 3.10: Errors in the reconstruction when the sensors are perturbed with both measurement additive noise and displacement of sensors (simulation #3). The results are also a function of a changing sampling step ∆r. As in Figs. 3.8, 3.9, (i), (ii) correspond to the mean RME and mean AE, respectively, while (a):(d) show the errors for percentages of 25% to 100% sensors perturbed.
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 57
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Figure 3.11: Comparison of field reconstructions by the (i) proposed 3D-VFT method, (ii) ART for w=20, and (iii) Laplace’s equation, where the domain parameters are U = 4 and P = 1. In all three cases, (a) presents the recovery on the slice z=1.5 of a field generated by a point source placed at (10, 10, 10), and (b) shows the recovery on the same slice when the monopole is located at (1, 1, 1). The asterisks show the location of the source. In (c), the mean RME and mean AE for the case (a) and the particular slice are depicted. (d) As in (c), but for the case of (b). (e) Shows the distribution of magnitude and angular errors for the whole reconstruction domain for the case (a). (f) As in (e), but for the case (b).
Chapter 4
Implementing 3D-VFT to solve the inverse EEG problem
In this chapter, we use the methodology presented in Chapter 3 to propose an alterna- tive solution for the inverse EEG problem. Instead of using models for the underlying sources and trying to estimate the lead field matrix, we reconstruct the electric fields inside the head generated by neural activity using a finite set of discretized longitu- dinal line integrals. The quasistatic approximation holds for these bioelectric fields, which can be expressed as the negative gradients of the scalar potentials. Hence, the line integral between two sensors can be written as the difference of the potential values measured at the sensors. Then, we can define the regions of activation as the locations of high values of the reconstructed electric field. We modify 3D-VFT to accommodate the realistic EEG problem, by using an ellipsoid 3D domain as the reconstruction domain and the locations of the 256 electrodes of EGI 300 GES as the positions of the sensors. We implement and test the method within Brainstorm, an open source application for EEG/MEG data processing to facilitate analysis and visualization. However, because of the realistic space and sensor locations, further regularization is required. The chapter is organized as follows. Section 4.1 describes the formulation of 3D-VFT using additional constraints to acquire a good solution in the realistic EEG problem. Section 4.2 describes the implementation details within Brainstorm with information about the brain template, the reconstruction space and
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