EEG and MEG: forward solutions for inverse methods

A solution of the forward problem is an important component of any method for computing the spatio-temporal activity of the neural sources of magnetoencephalography (MEG) and electroencephalography (EEG) data. The forward problem involves computing the scalp potentials or external magnetic field at a finite set of sensor locations for a putative source configuration. We present a unified treatment of analytical and numerical solutions of the forward problem in a form suitable for use in inverse methods. This formulation is achieved through factorization of the lead field into the product of the moment of the elemental current dipole source with a "kernel matrix" that depends on the head geometry and source and sensor locations, and a "sensor matrix" that models sensor orientation and gradiometer effects in MEG and differential measurements in EEG. Using this formulation and a recently developed approximation formula for EEG, based on the "Berg parameters," we present novel reformulations of the basic EEG and MEG kernels that dispel the myth that EEG is inherently more complicated to calculate than MEG. We also present novel investigations of different boundary element methods (BEM's) and present evidence that improvements over currently published BEM methods can be realized using alternative error-weighting methods. Explicit expressions for the matrix kernels for MEG and EEG for spherical and realistic head geometries are included.     

Methods for Compensating for Variable Electrode Contact in EIT

Electrical impedance tomography (EIT) is an imaging modality that currently shows promise for the detection and characterization of breast cancer. A very significant problem in EIT imaging is the proper modeling of the interface between the body and the electrodes. We have found empirically that it is very difficult, in a clinical setting, to assure that all electrodes make satisfactory contact with the body. In addition, we have observed a capacitive effect at the skin/electrode boundary that is spatially heterogeneous. To compensate for these problems, we have developed a hybrid nonlinear–linear reconstruction algorithm using the complete electrode model in which we first estimate electrode surface impedances, by means of a Levenberg–Marquardt iterative optimization procedure with an analytically computed Jacobian matrix. We, subsequently, use a linearized algorithm to perform a 3-D reconstruction of perturbations in both contact impedances, and in the spatial distributions of conductivity and permittivity. Results show that, with this procedure, artifacts due to electrodes making poor contact can be greatly reduced. If the experimental apparatus physically applies voltages and measures currents, we show that it is preferable to compute the reconstruction with respect to the Dirichlet-to-Neumann map rather than the Neumann-to-Dirichlet map if there is a significant possibility that electrodes will be fully disconnected. Finally, we test our electrode compensation algorithms for a set of clinical data, showing that we can significantly improve the fit of our model to the measurements by allowing the electrode surface impedances to vary.      

The boundary element method in the forward and inverse problem of electrical impedance tomography

In this paper, a new formulation of the reconstruction problem of electrical impedance tomography (EIT) is proposed. Instead of reconstructing a complete two-dimensional picture, a parameter representation of the gross anatomy is formulated, of which the optimal parameters are determined by minimizing a cost function. The two great advantages of this method are that the number of unknown parameters of the inverse problem is drastically reduced and that quantitative information of interest (e.g., lung volume) is estimated directly from the data, without image segmentation steps. The forward problem of EIT is to compute the potentials at the voltage measuring electrodes, for a given set of current injection electrodes and a given conductivity geometry. In this paper, it is proposed to use an improved boundary element method (BEM) technique to solve the forward problem, in which flat boundary elements are replaced by polygonal ones. From a comparison with the analytical solution of the concentric circle model, it appears that the use of polygonal elements greatly improves the accuracy of the BEM, without increasing the computation time. In this formulation, the inverse problem is a nonlinear parameter estimation problem with a limited number of parameters. Variants of Powell's and the simplex method are used to minimize the cost function. The applicability of this solution of the EIT problem was tested in a series of simulation studies. In these studies, EIT data were simulated using a standard conductor geometry and it was attempted to find back this geometry from random starting values. In the inverse algorithm, different current injection and voltage measurement schemes and different cost functions were compared. In a simulation study, it was demonstrated that a systematic error in the assumed lung conductivity results in a proportional error in the lung cross sectional area. It appears that our parametric formulation of the inverse problem leads to a stable minimization problem, with a high reliability, provided that the signal-to-noise ratio is about ten or higher.     

On combined source solutions for bodies with impedance boundary conditions

Numerical solutions to the impedance boundary condition (IBC) combined source integral equation (CSIE) for scattering from impedance spheres are presented. The CSIE formulation is a well-posed alternative to the IBC electric and magnetic field integral equations which can be contaminated by spurious resonant modes. Compared with the IBC combined field integral equation (CFIE), CSIE solutions have the same accuracy when the combined source coupling admittance is chosen to be the same value as the combined field coupling admittance. However, the CSIE formulation is better suited than the CFIE for creating a general purpose computer code capable of handling aperture radiation problems and/or a scatterer which has a spatially varying surface impedance.     

Transparent boundary conditions for parabolic equation solutions ofradiowave propagation problems

Perfectly transparent boundary conditions are derived for truncating the integration domain when solving radiowave propagation problems with a parabolic equation (PE) method. The boundary conditions are nonlocal: they are expressed as a convolution integral involving the field at all previous ranges. The convolution kernel is matched to the refractive index vertical gradient at the boundary. The boundary conditions include an incoming energy term which can model an arbitrary incident field. In particular, they may be used with plane-wave incidence, or with a point-source located below or above the domain boundary. If required, the solution can be extended to heights above the boundary with a generalized horizontal PE method. Closed-form solutions for the incoming energy term are given for plane-wave incidence and for Gaussian sources when the refractive index above the boundary is constant or linear. The resulting finite-difference algorithms provide efficient solutions to problems involving airborne sources. Numerical examples are given, showing excellent agreement with a pure split-step/Fourier PE algorithm     

Electrical impedance tomography for piecewise constant domains using boundary element shape-based inverse solutions

Shape-based solutions have recently received attention for certain ill-posed inverse problems. Their advantages include implicit imposition of relevant constraints and reduction in the number of unknowns, especially important for nonlinear ill-posed problems. We apply the shape-based approach to current-injection electrical impedance tomography (EIT) reconstructions. We employ a boundary element method (BEM) based solution for EIT. We introduce two shape models, one based on modified B-splines, and the other based on spherical harmonics, for BEM modeling of shapes. These methods allow us to parameterize the geometry of conductivity inhomogeneities in the interior of the volume. We assume the general shape of piecewise constant inhomogeneities is known but their conductivities and their exact location and shape is not. We also assume the internal conductivity profile is piecewise constant, meaning that each region has a constant conductivity. We propose and test three different regularization techniques to be used with either of the shape models. The performance of our methods is illustrated via both simulations in a digital torso model and phantom experiments when there is a single internal object. We observe that in the noisy environment, either simulated noise or real sources of noise in the experimental study, we get reasonable reconstructions. Since the signal-to-noise ratio (SNR) expected in modern EIT instruments is higher than that used in this study, these reconstruction methods may prove useful in practice.     

The effect of measurement conditions on MCG inverse solutions

A magnetic inverse solution that uses a single current dipole in a homogeneous volume conductor with realistic torso shape was tested numerically to establish the effect of magnetic noise, number of measurement points, and torso size on the localization accuracy. Seven different sites of cardiological interest were selected as locations for the source dipole. The three components of the magnetic field were calculated as if measured by second order gradiometers, Gaussian noise was added, and Monte Carlo tests performed for inverse solutions using a single field component, or all three combined. It was found that for any of the single component solutions, and a signal-to-noise ratio of 100, 25 measuring points are sufficient for good accuracy; just 12 points are needed if all three components are used together. If, however, the torso size of the inverse solution is different from that of the field data by 10 or 20%, a larger error occurs, even for 56 measurement points and no noise. In this case, the field component orthogonal to the measurement grid, Bz, yields better results than the other two components, or even all three combined. We conclude that a multichannel system measuring the z component of the magnetic field in about 30 locations would be the best choice to locate a dipolar source, provided the torso of the field data is closely matched by the model used in the inverse solution. To this effect, scaling of the torso model can easily be included in the computation. Imaging techniques could be used to accommodate different torso shapes.     

A VLSI architecture for lifting-based forward and inverse wavelettransform

We propose an architecture that performs the forward and inverse discrete wavelet transform (DWT) using a lifting-based scheme for the set of seven filters proposed in JPEG2000. The architecture consists of two row processors, two column processors, and two memory modules. Each processor contains two adders, one multiplier, and one shifter. The precision of the multipliers and adders has been determined using extensive simulation. Each memory module consists of four banks in order to support the high computational bandwidth. The architecture has been designed to generate an output every cycle for the JPEG2000 default filters. The schedules have been generated by hand and the corresponding timings listed. Finally, the architecture has been implemented in behavioral VHDL. The estimated area of the proposed architecture in 0.18-μ technology is 2.8 nun square, and the estimated frequency of operation is 200 MHz     

Improving the forward solver for the complete electrode model in EIT using algebraic multigrid

Image reconstruction in electrical impedance tomography is an ill-posed nonlinear inverse problem. Linearization techniques are widely used and require the repeated solution of a linear forward problem. To account correctly for the presence of electrodes and contact impedances, the so-called complete electrode model is applied. Implementing a standard finite element method for this particular forward problem yields a linear system that is symmetric and positive definite and solvable via the conjugate gradient method. However, preconditioners are essential for efficient convergence. Preconditioners based on incomplete factorization methods are commonly used but their performance depends on user-tuned parameters. To avoid this deficiency, we apply black-box algebraic multigrid, using standard commercial and freely available software. The suggested solution scheme dramatically reduces the time cost of solving the forward problem. Numerical results are presented using an anatomically detailed model of the human head.     

EIT forward problem parallel simulation environment with anisotropic tissue and realistic electrode models

Electrical impedance tomography (EIT) is an imaging technology based on impedance measurements. To retrieve meaningful insights from these measurements, EIT relies on detailed knowledge of the underlying electrical properties of the body. This is obtained from numerical models of current flows therein. The nonhomogeneous and anisotropic electric properties of human tissues make accurate modeling and simulation very challenging, leading to a tradeoff between physical accuracy and technical feasibility, which at present severely limits the capabilities of EIT. This work presents a complete algorithmic flow for an accurate EIT modeling environment featuring high anatomical fidelity with a spatial resolution equal to that provided by an MRI and a novel realistic complete electrode model implementation. At the same time, we demonstrate that current graphics processing unit (GPU)-based platforms provide enough computational power that a domain discretized with five million voxels can be numerically modeled in about 30 s.     

High performance architecture for unified forward and inverse transform of HEVC

High efficiency video coding (HEVC) transform algorithm for residual coding uses 2-dimensional (2D) 4×4 transforms with higher precision than H.264's 4×4 transforms, resulting in increased hardware complexity. In this paper, we present a shared architecture that can compute the 4×4 forward discrete cosine transform (DCT) and inverse discrete cosine transform (IDCT) of HEVC using a new mapping scheme in the video processor array structure. The architecture is implemented with only adders and shifts to an area-efficient design. The proposed architecture is synthesized using ISE14.7 and implemented using the BEE4 platform with the Virtex-6 FF1759 LX550T field programmable gate array (FPGA). The result shows that the video processor array structure achieves a maximum operation frequency of 165.2 MHz. The architecture and its implementation are presented in this paper to demonstrate its programmable and high performance.     

Application of electromagnetic inverse boundary conditions to profile characteristics inversion of conducting spherical shapes

An inverse problem of continuous wave electromagnetic scattering is considered. It is assumed that the incident and the scattered fields are given everywhere and that the material surface properties satisfy the Leontovich boundary condition. Applying the concept of electromagnetic inverse boundary conditions it is shown how the shape and the averaged local surface impedance for spherical monobody and two-body scattering geometries can exactly be recovered. To enable accurate inversion for the multibody or general spherical case, analytical continuation methods in three dimensions are introduced.     

Generalized fast mixed-radix algorithm for the computation of forward and inverse MDCTs

This paper presents a generalized mixed-radix decimation-in-time (DIT) fast algorithm for computing the modified discrete cosine transform (MDCT) of the composite lengths N=2×q^m, m≥2, where q is an odd positive integer. The proposed algorithm not only has the merits of parallelism and numerical stability, but also needs less multiplications than that of type-IV discrete cosine transform (DCT-IV) and type-II discrete cosine transform (DCT-II) based MDCT algorithms due to the optimized efficient length-(N/q) modules. The computation of MDCT for composite lengths N=q^m×2ⁿ, m≥2, n≥2, can then be realized by combining the proposed algorithm with fast radix-2 MDCT algorithm developed for N=2ⁿ. The combined algorithm can be used for the computation of length-12/36 MDCT used in MPEG-1/-2 layer III audio coding as well as the recently established wideband speech and audio coding standards such as G.729.1, where length-640 MDCT is used. The realization of the inverse MDCT (IMDCT) can be obtained by transposing the signal flow graph of the MDCT.     

Absorbing boundary conditions for convex object-conformableboundaries

Absorbing boundary conditions (ABCs) are developed that can be applied on object-conformable outer boundaries. The new ABCs are based on the local enforcement of the Nth order Bayliss-Turkel boundary conditions where a scattering center is defined for each outer boundary node. A demonstration of the effectiveness of the new construction is provided by considering representative numerical experiments using the finite-elements method. Results show that the new ABCs provide accuracy that compares very favorably with the method of moments solution     

Relaxation method for mixed boundary conditions

The numerical solution of Laplace's equation in 2-dimensional Cartesian co-ordinates with mixed boundary conditions is described using the successive-overrelaxation method for solution, with the acceleration parameter decided by a method proposed formerly. Its effectiveness is compared with that of other methods.     

Effective boundary conditions for syncytial tissues

This study derives effective boundary conditions for potentials and currents on the interface between syncytial tissue and a surrounding volume conductor. The derivation is based on an idealized representation of the syncytium as a network of interconnected cells arranged periodically in space. The microscopic model of an interface assumes that the extracellular fluid is in direct contact with the outside volume conductor and that the inside of the cells is separated from the outside by the membrane. From this microscopic model, a homogenization process and boundary layer analysis derive effective boundary conditions applicable to macroscopic volume-averaged potentials. These effective boundary conditions call for the extracellular potential and current density to be continuous with the potential and current density in the volume conductor, and for the intracellular current to vanish. Hence, the long-debated appropriate boundary conditions for the bidomain model are established     

Thermal design and experimental validation for optical transmitters

This paper describes a thermal design methodology for a 2.5 Gbps optical transmitter that mainly comprises a laser array of 12 VCSELs and a laser driver module. An integrated heat sink design was performed and optimized through modeling and simulation. Temperature regulation of the laser array has been performed through design and optimization of the thermal path (cavity and heat spreader) and separations (wire bonding lengths). Detailed module simulation was performed after the heat sink design and the temperature regulations. To validate the simulation results, a test vehicle of 2.5 Gbps transmitter was built up and tested under various thermal conditions. The airflow rate and ambient temperature were controlled by a wind tunnel. It has shown that the experimental and detailed module simulation results are comparable. A cooling solution with natural convection has been achieved so that the case temperature can be kept under 70 °C without using a fan. The modeling and simulation were done by using a computational fluid dynamics (CFD) program.     

Sufficient conditions for inventive solutions

Engineering systems are developed through numerous modification cycles. Most of them arise from routine engineering processes such as optimization or by adopting new technologies. On occasion though, a system is modified through a qualitatively different process: a creative process. We call this type of modification a design invention. The paper presents an objective metric for the evaluation of the inventiveness of an engineering solution. This metric can be used not only to evaluate a known solution but also to guide the search toward new design inventions. The paper presents a formal treatment of the sufficient conditions with illustrative examples and case studies. An empirical study demonstrates that solutions which satisfy the conditions tend to score high on creativity evaluation by field experts     

Digital realisation of forward and inverse models of electromagnetic scattering

A recent geometrical treatment of forward and inverse scattering problems, posed by electromagnetic wave illuminating a one-dimensional dielectric inhomogeneity, is interpreted within the framework of digital signal analysis. Quantitative inverse solutions are described, and are illustrated by an example in the context of real-time nondestructive evaluation.<>