3-D electrical impedance tomography for piecewise constant domains with known internal boundaries

Electrical impedance tomography (EIT) is a badly posed inverse problem, but can be stabilized if one assumes that the conductivity is piecewise constant, with a relatively small number of distinct regions, and that the region boundaries are known, for example from prior anatomical imaging. With this assumption, we introduce a three-dimensional (3-D) boundary element method (BEM) model for the forward EIT map from injected currents to measured voltages, and 3-D inverse solutions for both BEM and the finite element method (FEM) which explicitly take into account the parameterization implied by the known boundary locations. We develop expressions for the Jacobians for both methods, since they are nonlinear, to more rapidly solve the inverse problem. We show simulation results in a torso geometry with the heart and lungs as inhomogeneities. In a simulation study, we could reconstruct the conductive values of some internal organs of a human torso with more than 92% accuracy even with inaccurate internal boundary locations, a randomized rather than constant conductivity profile (with the standard deviation of the Gaussian-distributed conductivities set to 20% of their mean values), signal to measurement noise of 50 dB, and with different meshes used for the forward and inverse problems. BEM and FEM perform similarly, leading to the conclusion that the choice between them should be based on secondary considerations such as computational efficiency or the need to model conductivity anisotropies     

Electrode boundary conditions and experimental validation for BEM-based EIT forward and inverse solutions

In this paper, we present theoretical developments and experimental results for the problem of estimating the conductivity map inside a volume using electrical impedance tomography (EIT) when the boundary locations of any internal inhomogeneities are known. We describe boundary element method (BEM) implementations of advanced electrode models for the forward problem of EIT. We then use them in the inverse problem with known internal boundaries and derive the associated Jacobians. We report on the results of two EIT phantom studies, one using a homogeneous cubical tank, and one using a cylindrical tank with agar conductivity inhomogeneities. We test both the accuracy of our BEM forward model, including the electrode models, as well as our inverse solution, against the measured data. Results show good agreement between measured values and both forward-computed tank voltages and inverse-computed conductivities; for instance, in a phantom experiment, we reconstructed the conductivities of three agar objects inside a cylindrical tank with an error less than 2% of their true value.     

Electrical impedance tomography: regularized imaging and contrast detection

Dynamic electrical impedance tomography (EIT) images changes in the conductivity distribution of a medium from low frequency electrical measurements made at electrodes on the medium surface. Reconstruction of the conductivity distribution is an under-determined and ill-posed problem, typically requiring either simplifying assumptions or regularization based on a priori knowledge. This paper presents a maximum a posteriori (MAP) approach to linearized image reconstruction using knowledge of the noise variance of the measurements and the covariance of the conductivity distribution. This approach has the advantage of an intuitive interpretation of the algorithm parameters as well as fast (near real time) image reconstruction. In order to compare this approach to existing algorithms, the authors develop figures of merit to measure the reconstructed image resolution, the noise amplification of the image reconstruction, and the fidelity of positioning in the image. Finally, the authors develop a communications systems approach to calculate the probability of detection of a conductivity contrast in the reconstructed image as a function of the measurement noise and the reconstruction algorithm used.     

Combining learning-based intensity distributions with nonparametric shape priors for image segmentation

Integration of shape prior information into level set formulations has led to great improvements in image segmentation in the presence of missing information, occlusion, and noise. However, most shape-based segmentation techniques incorporate image intensity through simplistic data terms. A common underlying assumption of such data terms is that the foreground and the background regions in the image are homogeneous, i.e., intensities are piecewise constant or piecewise smooth. This situation makes integration of shape priors inefficient in the presence of intensity inhomogeneities. In this paper, we propose a new approach for combining information from shape priors with that from image intensities. More specifically, our approach uses shape priors learned by nonparametric density estimation and incorporates image intensity distributions learned in a supervised manner. Such a combination has not been used in previous work. Sample image patches are used to learn the intensity distributions, and segmented training shapes are used to learn the shape priors. We present an active contour algorithm that takes these learned densities into account for image segmentation. Our experiments on synthetic and real images demonstrate the robustness of the proposed approach to complicated intensity distributions, and occlusions, as well as the improvements it provides over existing methods.     

Optimal experiments in electrical impedance tomography

Electrical impedance tomography (EIT) is a noninvasive imaging technique which aims to image the impedance within a body from electrical measurements made on the surface. The reconstruction of impedance images is a ill-posed problem which is both extremely sensitive to noise and highly computationally intensive. The authors define an experimental measurement in EIT and calculate optimal experiments which maximize the distinguishability between the region to be imaged and a best-estimate conductivity distribution. These optimal experiments can be derived from measurements made on the boundary. The analysis clarifies the properties of different voltage measurement schemes. A reconstruction algorithm based on the use of optimal experiments is derived. It is shown to be many times faster than standard Newton-based reconstruction algorithms, and results from synthetic data indicate that the images that it produces are comparable.     

Bayesian image reconstruction in SPECT using higher order mechanical models as priors

While the ML-EM algorithm for reconstruction for emission tomography is unstable due to the ill-posed nature of the problem. Bayesian reconstruction methods overcome this instability by introducing prior information, often in the form of a spatial smoothness regularizer. More elaborate forms of smoothness constraints may be used to extend the role of the prior beyond that of a stabilizer in order to capture actual spatial information about the object. Previously proposed forms of such prior distributions were based on the assumption of a piecewise constant source distribution. Here, the authors propose an extension to a piecewise linear model-the weak plate-which is more expressive than the piecewise constant model. The weak plate prior not only preserves edges but also allows for piecewise ramplike regions in the reconstruction. Indeed, for the authors' application in SPECT, such ramplike regions are observed in ground-truth source distributions in the form of primate autoradiographs of rCBF radionuclides. To incorporate the weak plate prior in a MAP approach, the authors model the prior as a Gibbs distribution and use a GEM formulation for the optimization. They compare quantitative performance of the ML-EM algorithm, a GEM algorithm with a prior favoring piecewise constant regions, and a GEM algorithm with their weak plate prior. Pointwise and regional bias and variance of ensemble image reconstructions are used as indications of image quality. The authors' results show that the weak plate and membrane priors exhibit improved bias and variance relative to ML-EM techniques.     

Electric lead field for a piecewise homogeneous volume conductormodel of the head

A new method is presented for computing the electric lead field of a realistic head shape model which has piecewise homogenous conductivity. The basic formulae are derived using the well-known reciprocity theorem. Previously described methods are also based upon this theorem, but these first calculate the electric potential inside the head by a scalar boundary element method (BEM), and then approximate the ohmic current density by some sort of gradient. In contrast, this paper proposes the direct evaluation of the ohmic current density by discretizing the vector Green's second identity which leads to a rector version of BEM. This approach also allows the derivation of the same equations for the three concentric spheres model as obtained by Rush and Driscoll (1969). The results of simulations on a spherical head model indicate that the use of a vector BEM leads to an improvement of accuracy in the computation of the ohmic current density with respect to those reported previously, in term of different measures of error     

A common formalism for the integral formulations of the forward EEG problem

The forward electroencephalography (EEG) problem involves finding a potential V from the Poisson equation inverted Delta x (sigma inverted Delta V) f, in which f represents electrical sources in the brain, and sigma the conductivity of the head tissues. In the piecewise constant conductivity head model, this can be accomplished by the boundary element method (BEM) using a suitable integral formulation. Most previous work uses the same integral formulation, corresponding to a double-layer potential. In this paper we present a conceptual framework based on a well-known theorem (Theorem 1) that characterizes harmonic functions defined on the complement of a bounded smooth surface. This theorem says that such harmonic functions are completely defined by their values and those of their normal derivatives on this surface. It allows us to cast the previous BEM approaches in a unified setting and to develop two new approaches corresponding to different ways of exploiting the same theorem. Specifically, we first present a dual approach which involves a single-layer potential. Then, we propose a symmetric formulation, which combines single- and double-layer potentials, and which is new to the field of EEG, although it has been applied to other problems in electromagnetism. The three methods have been evaluated numerically using a spherical geometry with known analytical solution, and the symmetric formulation achieves a significantly higher accuracy than the alternative methods. Additionally, we present results with realistically shaped meshes. Beside providing a better understanding of the foundations of BEM methods, our approach appears to lead also to more efficient algorithms.     

静态阻抗断层图像重建新方法

阻抗断层图像重建是一个严重病态的非线性的逆问题,特别是在静态阻抗断层成像中,由于其图像重建模型误差和测量噪声的影响更为严重,因此常用的基于目标函数梯度信息不断迭代的改进的Newton-Raphson类重建算法,即使使用正则化技术,其稳定性仍较差,甚至发散.本文提出一种全新的静态阻抗断层图像重建方法,它利用基于生物自然选择与遗传机理的遗传算法去搜索阻抗图像重建问题的最优解,无需正则化技术,也不会象改进的Newton-Raphson类算法那样易陷入局部最优解.实验结果也表明基于遗传算法的图像重建方法重建的静态阻抗断层图像,其成像精度和空间分辨率都大大好于改进的Newton-Raphson类重建算法.     

Estimation and statistical bounds for three-dimensional polar shapes in diffuse optical tomography

Voxel-based reconstructions in diffuse optical tomography (DOT) using a quadratic regularization functional tend to produce very smooth images due to the attenuation of high spatial frequencies. This then causes difficulty in estimating the spatial extent and contrast of anomalous regions such as tumors. Given an assumption that the target image is piecewise constant, we can employ a parametric model to estimate the boundaries and contrast of an inhomogeneity directly. In this paper, we describe a method to directly reconstruct such a shape boundary from diffuse optical measurements. We parameterized the object boundary using a spherical harmonic basis, and derived a method to compute sensitivities of measurements with respect to shape parameters. We introduced a centroid constraint to ensure uniqueness of the combined shape/center parameter estimate, and a projected Newton method was utilized to optimize the object center position and shape parameters simultaneously. Using the shape Jacobian, we also computed the Cramér-Rao lower bound on the theoretical estimator accuracy given a particular measurement configuration, object shape, and level of measurement noise. Knowledge of the shape sensitivity matrix and of the measurement noise variance allows us to visualize the shape uncertainty region in three dimensions, giving a confidence region for our shape estimate. We have implemented our shape reconstruction method, using a finite-difference-based forward model to compute the forward and adjoint fields. Reconstruction results are shown for a number of simulated target shapes, and we investigate the problem of model order selection using realistic levels of measurement noise. Assuming a signal-to-noise ratio in the amplitude measurements of 40 dB and a standard deviation in the phase measurements of 0.1 degrees , we are able to estimate an object represented with an eighth-order spherical harmonic model having an absorption contrast of 0.15 cm(-1) and a volume of 4.82 cm(3) with errors of less than 10% in object volume and absorption contrast. We also investigate the robustness of our shape-based reconstruction approach to a violation of the assumption that the medium is purely piecewise constant.     

An efficient forward solver in electrical impedance tomography by spectral element method

In electrical impedance tomography (EIT), a forward solver capable of predicting the voltages on electrodes for a given conductivity distribution is essential for reconstruction. The EIT forward solver is normally based on the conventional finite element method (FEM). One of the major problems of three-dimensional (3-D) EIT is its high demand in computing power and memory since high precision is required for obtaining a small secondary field which is typical for a small anomaly. This accuracy requirement is also set by the level of noise in the real data; although currently the noise level is still an issue, future EIT systems should significantly reduce the noise level to be capable of detecting very small anomalies. To accurately simulate the forward solution with the FEM, a mesh with large number of nodes and elements is usually needed. To overcome this problem, we proposed the spectral element method (SEM) for EIT forward problem. With the introduction of SEM, a smaller number of nodes and hence less computational time and memory are needed to achieve the same or better accuracy in the forward solution than the FEM. Numerical results demonstrate the efficiency of the SEM in 3-D EIT simulation.     

Elliptic cylinder geometry for distinguishability analysis in impedance tomography

Electrical impedance tomography (EIT) is a technique that computes the cross-sectional impedance distribution within the body by using current and voltage measurements made on the body surface. It has been reported that the image reconstruction is distorted considerably when the boundary shape is considered to be more elliptical than circular as a more realistic shape for the measurement boundary. This paper describes an alternative framework for determining the distinguishability region with a finite measurement precision for different conductivity distributions in a body modeled by elliptic cylinder geometry. The distinguishable regions are compared in terms of modeling error for predefined inhomogeneities with elliptical and circular approaches for a noncircular measurement boundary at the body surface. Since most objects investigated by EIT are noncircular in shape, the analytical solution for the forward problem for the elliptical cross section approach is shown to be useful in order to reach a better assessment of the distinguishability region defined in a noncircular boundary. This paper is concentrated on centered elliptic inhomogeneity in the elliptical boundary and an analytic solution for this type of forward problem. The distinguishability performance of elliptical cross section with cosine injected current patterns is examined for different parameters of elliptical geometry.     

Shape from focus using multilayer feedforward neural networks

The conventional shape-from-focus (SFF) methods have inaccuracies because of piecewise constant approximation of the focused image surface (FIS). We propose a scheme for SFF based on representation of three-dimensional (3-D) FIS in terms of neural network weights. The neural networks are trained to learn the shape of the FIS that maximizes the focus measure.     

Three-dimensional attenuation map reconstruction using geometrical models and free-form deformations

We address the issue of using deformable models to reconstruct an unknown attenuation map of the torso from a set of transmission scans. We assume the three-dimensional (3-D) distribution of attenuation coefficients to be piecewise uniform. We represent the unknown distribution by a set of closed surfaces defining regions having the same attenuating properties. The methods of reconstruction published so far tend to directly deform the surfaces, the parameters being the surface elements. Rather than deforming the surfaces, we explore the possibility of deforming the space in which the geometrical primitives are contained. We focus on the use of free-form deformations (FFD's) to describe the continuous transformation of space used to match a set of transmission measurements. We illustrate this approach by reconstructing realistically simulated transmission scans of the torso with various noise levels and compare the results to standard reconstruction methods.     

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.     

Blind separation of piecewise stationary non-Gaussian sources

We address independent component analysis (ICA) of piecewise stationary and non-Gaussian signals and propose a novel ICA algorithm called Block EFICA that is based on this generalized model of signals. The method is a further extension of the popular non-Gaussianity-based FastICA algorithm and of its recently optimized variant called EFICA. In contrast to these methods, Block EFICA is developed to effectively exploit varying distribution of signals, thus, also their varying variance in time (nonstationarity) or, more precisely, in time-intervals (piecewise stationarity). In theory, the accuracy of the method asymptotically approaches Cramér–Rao lower bound (CRLB) under common assumptions when variance of the signals is constant. On the other hand, the performance is practically close to the CRLB even when variance of the signals is changing. This is demonstrated by comparing our algorithm with various methods that are asymptotically efficient within ICA models based either on the non-Gaussianity or the nonstationarity. The benefit of our algorithm is demonstrated by examples with real-world audio signals.     

Estimation of optimal PDE-based denoising in the SNR sense.

This paper is concerned with finding the best partial differential equation-based denoising process, out of a set of possible ones. We focus on either finding the proper weight of the fidelity term in the energy minimization formulation or on determining the optimal stopping time of a nonlinear diffusion process. A necessary condition for achieving maximal SNR is stated, based on the covariance of the noise and the residual part. We provide two practical alternatives for estimating this condition by observing that the filtering of the image and the noise can be approximated by a decoupling technique, with respect to the weight or time parameters. Our automatic algorithm obtains quite accurate results on a variety of synthetic and natural images, including piecewise smooth and textured ones. We assume that the statistics of the noise were previously estimated. No a priori knowledge regarding the characteristics of the clean image is required. A theoretical analysis is carried out, where several SNR performance bounds are established for the optimal strategy and for a widely used method, wherein the variance of the residual part equals the variance of the noise.     

Improved filter sharpening

We propose a natural extension to Kaiser-Hamming (1977) filter sharpening methods to allow for a piecewise linear desired amplitude change function (ACF). The primary advantages of the proposed ACF over piecewise constant ACFs is that we obtain better control of selective improvement (or degradation) in either the passband or stopband or both, and we are not restricted to applying our methods to filters with piecewise constant pass and stopbands, since linear segments of slope 1 can be used to retain existing filter performance in either passband or stopband. The proposed ACF approximating polynomial (AP) is easy to compute, may be constrained to have simple (or integer) coefficients, and may be expressed as the AP of Kaiser and Hamming plus a correction polynomial. We also provide applications for motivation     

A Variational Model for Staircase Reduction in Image Denoising

We propose a new variational model to reduce the staircase that often appears in Total variation (TV) based models in image denoising. The model uses BV-seminorm and Besov-seminorm to measure the piece-wise constant component and piecewise smooth component of the image, respectively. We discuss the nontrivial prop-erty of the proposed model and introduce an alternating iteration algorithm that combines the dual projection al-gorithm with Wavelet soft thresholding (WST) algorithm to solve the model numerically. The experimental results show that the proposed model is effective for noise removal and staircase reduction, while the contour can be preserved in the denoised images. Furthermore, compared with two classical staircase reduction models, CEP2 and TGV, the proposed model is much faster than these two models.     

Joint Segmentation of Piecewise Constant Autoregressive Processes by Using a Hierarchical Model and a Bayesian Sampling Approach

We propose a joint segmentation algorithm for piecewise constant autoregressive (AR) processes recorded by several independent sensors. The algorithm is based on a hierarchical Bayesian model. Appropriate priors allow us to introduce correlations between the change locations of the observed signals. Numerical problems inherent to Bayesian inference are solved by a Gibbs sampling strategy. The proposed joint segmentation methodology yields improved segmentation results when compared with parallel and independent individual signal segmentations. The initial algorithm is derived for piecewise constant AR processes whose orders are fixed on each segment. However, an extension to models with unknown model orders is also discussed. Theoretical results are illustrated by many simulations conducted with synthetic signals and real arc-tracking and speech signals