Core Imaging Library - Part II: multichannel reconstruction for dynamic and spectral tomography.

The newly developed core imaging library (CIL) is a flexible plug and play library for tomographic imaging with a specific focus on iterative reconstruction. CIL provides building blocks for tailored regularized reconstruction algorithms and explicitly supports multichannel tomographic data. In the first part of this two-part publication, we introduced the fundamentals of CIL. This paper focuses on applications of CIL for multichannel data, e.g. dynamic and spectral. We formalize different optimization problems for colour processing, dynamic and hyperspectral tomography and demonstrate CIL's capabilities for designing state-of-the-art reconstruction methods through case studies and code snapshots. This article is part of the theme issue 'Synergistic tomographic image reconstruction: part 2'.

Temporal sparsity exploiting nonlocal regularization for 4D computed tomography reconstruction.

X-ray imaging applications in medical and material sciences are frequently limited by the number of tomographic projections collected. The inversion of the limited projection data is an ill-posed problem and needs regularization. Traditional spatial regularization is not well adapted to the dynamic nature of time-lapse tomography since it discards the redundancy of the temporal information. In this paper, we propose a novel iterative reconstruction algorithm with a nonlocal regularization term to account for time-evolving datasets. The aim of the proposed nonlocal penalty is to collect the maximum relevant information in the spatial and temporal domains. With the proposed sparsity seeking approach in the temporal space, the computational complexity of the classical nonlocal regularizer is substantially reduced (at least by one order of magnitude). The presented reconstruction method can be directly applied to various big data 4D (x, y, z+time) tomographic experiments in many fields. We apply the proposed technique to modelled data and to real dynamic X-ray microtomography (XMT) data of high resolution. Compared to the classical spatio-temporal nonlocal regularization approach, the proposed method delivers reconstructed images of improved resolution and higher contrast while remaining significantly less computationally demanding.

A hybrid multi-particle approach to range assessment-based treatment verification in particle therapy.

Particle therapy (PT) used for cancer treatment can spare healthy tissue and reduce treatment toxicity. However, full exploitation of the dosimetric advantages of PT is not yet possible due to range uncertainties, warranting development of range-monitoring techniques. This study proposes a novel range-monitoring technique introducing the yet unexplored concept of simultaneous detection and imaging of fast neutrons and prompt-gamma rays produced in beam-tissue interactions. A quasi-monolithic organic detector array is proposed, and its feasibility for detecting range shifts in the context of proton therapy is explored through Monte Carlo simulations of realistic patient models and detector resolution effects. The results indicate that range shifts of [Formula: see text] can be detected at relatively low proton intensities ([Formula: see text] protons/spot) when spatial information obtained through imaging of both particle species are used simultaneously. This study lays the foundation for multi-particle detection and imaging systems in the context of range verification in PT.

GREIT: a unified approach to 2D linear EIT reconstruction of lung images.

Electrical impedance tomography (EIT) is an attractive method for clinically monitoring patients during mechanical ventilation, because it can provide a non-invasive continuous image of pulmonary impedance which indicates the distribution of ventilation. However, most clinical and physiological research in lung EIT is done using older and proprietary algorithms; this is an obstacle to interpretation of EIT images because the reconstructed images are not well characterized. To address this issue, we develop a consensus linear reconstruction algorithm for lung EIT, called GREIT (Graz consensus Reconstruction algorithm for EIT). This paper describes the unified approach to linear image reconstruction developed for GREIT. The framework for the linear reconstruction algorithm consists of (1) detailed finite element models of a representative adult and neonatal thorax, (2) consensus on the performance figures of merit for EIT image reconstruction and (3) a systematic approach to optimize a linear reconstruction matrix to desired performance measures. Consensus figures of merit, in order of importance, are (a) uniform amplitude response, (b) small and uniform position error, (c) small ringing artefacts, (d) uniform resolution, (e) limited shape deformation and (f) high resolution. Such figures of merit must be attained while maintaining small noise amplification and small sensitivity to electrode and boundary movement. This approach represents the consensus of a large and representative group of experts in EIT algorithm design and clinical applications for pulmonary monitoring. All software and data to implement and test the algorithm have been made available under an open source license which allows free research and commercial use.

A measure of the information content of EIT data.

We ask: how many bits of information (in the Shannon sense) do we get from a set of EIT measurements? Here, the term information in measurements (IM) is defined as: the decrease in uncertainty about the contents of a medium, due to a set of measurements. This decrease in uncertainty is quantified by the change from the inter-class model, q, defined by the prior information, to the intra-class model, p, given by the measured data (corrupted by noise). IM is measured by the expected relative entropy (Kullback-Leibler divergence) between distributions q and p, and corresponds to the channel capacity in an analogous communications system. Based on a Gaussian model of the measurement noise, (Sigma(n)), and a prior model of the image element covariances (Sigma(x)), we calculate IM = 1/2 summation operator log(2)([SNR](i) + 1), where [SNR](i) is the signal-to-noise ratio for each independent measurement calculated from the prior and noise models. For an example, we consider saline tank measurements from a 16 electrode EIT system, with a 2 cm radius non-conductive target, and calculate IM =179 bits. Temporal sequences of frames are considered, and formulae for IM as a function of temporal image element correlations are derived. We suggest that this measure may allow novel insights into questions such as distinguishability limits, optimal measurement schemes and data fusion.

On the Feasibility of Automated Mechanical Ventilation Control Through EIT.

OBJECTIVE: This paper aims to demonstrate the feasibility of coupling electrical impedance tomography (EIT) with models of lung function in order to recover parameters and inform mechanical ventilation control. METHODS: A compartmental ordinary differential equation model of lung function is coupled to simulations of EIT, assuming accurate modeling and movement tracking, to generate time series values of bulk conductivity. These values are differentiated and normalized against the total air volume flux to recover regional volumes and flows. These ventilation distributions are used to recover regional resistance and elastance properties of the lung. Linear control theory is used to demonstrate how these parameters may be used to generate a patient-specific pressure mode control. RESULTS: Ventilation distributions are shown to be recoverable, with Euclidean norm errors in air flow below 9% and volume below 3%. The parameters are also shown to be recoverable, although errors are higher for resistance values than elastance. The control constructed is shown to have minimal seminorm resulting in bounded magnitudes and minimal gradients. CONCLUSION: The recovery of regional ventilation distributions and lung parameters is feasible with the use of EIT. These parameters may then be used in model based control schemes to provide patient-specific care. SIGNIFICANCE: For pulmonary-intensive-care patients mechanical ventilation is a life saving intervention, requiring careful calibration of pressure settings. Both magnitudes and gradients of pressure can contribute to ventilator induced lung injury. Retrieving regional lung parameters allows the design of patient-specific ventilator controls to reduce injury.

Three Dimensional Polarimetric Neutron Tomography of Magnetic Fields.

Through the use of Time-of-Flight Three Dimensional Polarimetric Neutron Tomography (ToF 3DPNT) we have for the first time successfully demonstrated a technique capable of measuring and reconstructing three dimensional magnetic field strengths and directions unobtrusively and non-destructively with the potential to probe the interior of bulk samples which is not amenable otherwise. Using a pioneering polarimetric set-up for ToF neutron instrumentation in combination with a newly developed tailored reconstruction algorithm, the magnetic field generated by a current carrying solenoid has been measured and reconstructed, thereby providing the proof-of-principle of a technique able to reveal hitherto unobtainable information on the magnetic fields in the bulk of materials and devices, due to a high degree of penetration into many materials, including metals, and the sensitivity of neutron polarisation to magnetic fields. The technique puts the potential of the ToF time structure of pulsed neutron sources to full use in order to optimise the recorded information quality and reduce measurement time.

Temporal image reconstruction in electrical impedance tomography.

Electrical impedance tomography (EIT) calculates images of the body from body impedance measurements. While the spatial resolution of these images is relatively low, the temporal resolution of EIT data can be high. Most EIT reconstruction algorithms solve each data frame independently, although Kalman filter algorithms track the image changes across frames. This paper proposes a new approach which directly accounts for correlations between images in successive data frames. Image reconstruction is posed in terms of an augmented image x and measurement vector y, which concatenate the values from the d previous and future frames. Image reconstruction is then based on an augmented regularization matrix R, which accounts for a model of both the spatial and temporal correlations between image elements. Results are compared for reconstruction algorithms based on independent frames, Kalman filters and the proposed approach. For low values of the regularization hyperparameter, the proposed approach performs similarly to independent frames, but for higher hyperparameter values, it uses adjacent frame data to reduce reconstructed image noise.

Validation of a finite-element solution for electrical impedance tomography in an anisotropic medium.

Electrical impedance tomography is an imaging method, with which volumetric images of conductivity are produced by injecting electrical current and measuring boundary voltages. It has the potential to become a portable non-invasive medical imaging technique. Until now, implementations have neglected anisotropy even though human tissues such as bone, muscle and brain white matter are markedly anisotropic. We present a numerical solution using the finite-element method that has been modified for modelling anisotropic conductive media. It was validated in an anisotropic domain against an analytical solution in an isotropic medium after the isotropic domain was diffeomorphically transformed into an anisotropic one. Convergence of the finite element to the analytical solution was verified by showing that the finite-element error norm decreased linearly related to the finite-element size, as the mesh density increased, for the simplified case of Laplace's equation in a cubic domain with a Dirichlet boundary condition.

Absolute conductivity reconstruction in magnetic induction tomography using a nonlinear method.

Magnetic induction tomography (MIT) attempts to image the electrical and magnetic characteristics of a target using impedance measurement data from pairs of excitation and detection coils. This inverse eddy current problem is nonlinear and also severely ill posed so regularization is required for a stable solution. A regularized Gauss-Newton algorithm has been implemented as a nonlinear, iterative inverse solver. In this algorithm, one needs to solve the forward problem and recalculate the Jacobian matrix for each iteration. The forward problem has been solved using an edge based finite element method for magnetic vector potential A and electrical scalar potential V, a so called A, A - V formulation. A theoretical study of the general inverse eddy current problem and a derivation, paying special attention to the boundary conditions, of an adjoint field formula for the Jacobian is given. This efficient formula calculates the change in measured induced voltage due to a small perturbation of the conductivity in a region. This has the advantage that it involves only the inner product of the electric fields when two different coils are excited, and these are convenient computationally. This paper also shows that the sensitivity maps change significantly when the conductivity distribution changes, demonstrating the necessity for a nonlinear reconstruction algorithm. The performance of the inverse solver has been examined and results presented from simulated data with added noise.

A narrow-band level set method applied to EIT in brain for cryosurgery monitoring.

In this paper, we investigate the feasibility of applying a novel level set reconstruction technique to electrical imaging of the human brain. We focus particularly on the potential application of electrical impedance tomography (EIT) to cryosurgery monitoring. In this application, cancerous tissue is treated by a local freezing technique using a small needle-like cryosurgery probe. The interface between frozen and nonfrozen tissue can be expected to have a relatively high contrast in conductivity and we treat the inverse problem of locating and monitoring this interface during the treatment. A level set method is used as a powerful and flexible tool for tracking the propagating interfaces during the monitoring process. For calculating sensitivities and the Jacobian when deforming the interfaces we employ an adjoint formula rather than a direct differentiation technique. In particular, we are using a narrow-band technique for this procedure. This combination of an adjoint technique and a narrow-band technique for calculating Jacobians results in a computationally efficient and extremely fast method for solving the inverse problem. Moreover, due to the reduced number of unknowns in each step of the narrow-band approach compared to a pixel- or voxel-based technique, our reconstruction scheme tends to be much more stable. We demonstrate that our new method also outperforms its pixel-/voxel-based counterparts in terms of image quality in this application.

Uses and abuses of EIDORS: an extensible software base for EIT.

EIDORS is an open source software suite for image reconstruction in electrical impedance tomography and diffuse optical tomography, designed to facilitate collaboration, testing and new research in these fields. This paper describes recent work to redesign the software structure in order to simplify its use and provide a uniform interface, permitting easier modification and customization. We describe the key features of this software, followed by examples of its use. One general issue with inverse problem software is the difficulty of correctly implementing algorithms and the consequent ease with which subtle numerical bugs can be inadvertently introduced. EIDORS helps with this issue, by allowing sharing and reuse of well-documented and debugged software. On the other hand, since EIDORS is designed to facilitate use by non-specialists, its use may inadvertently result in such numerical errors. In order to address this issue, we develop a list of ways in which such errors with inverse problems (which we refer to as 'cheats') may occur. Our hope is that such an overview may assist authors of software to avoid such implementation issues.

Characterization of objects by electrosensing fish based on the first order polarization tensor.

Weakly electric fish generate electric current and use hundreds of voltage sensors on the surface of their body to navigate and locate food. Experiments (von der Emde and Fetz 2007 J. Exp. Biol. 210 3082-95) show that they can discriminate between differently shaped conducting or insulating objects by using electrosensing. One approach to electrically identify and characterize the object with a lower computational cost rather than full shape reconstruction is to use the first order polarization tensor (PT) of the object. In this paper, by considering experimental work on Peters' elephantnose fish Gnathonemus petersii, we investigate the possible role of the first order PT in the ability of the fish to discriminate between objects of different shapes. We also suggest some experiments that might be performed to further investigate the role of the first order PT in electrosensing fish. Finally, we speculate on the possibility of electrical cloaking or camouflage in prey of electrosensing fish and what might be learnt from the fish in human remote sensing.

Krylov subspace iterative techniques: on the detection of brain activity with electrical impedance tomography.

In this paper, we review some numerical techniques based on the linear Krylov subspace iteration that can be used for the efficient calculation of the forward and the inverse electrical impedance tomography problems. Exploring their computational advantages in solving large-scale systems of equations, we specifically address their implementation in reconstructing localized impedance changes occurring within the human brain. If the conductivity of the head tissues is assumed to be real, the pre-conditioned conjugate gradients (PCGs) algorithm can be used to calculate efficiently the approximate forward solution to a given error tolerance. The performance and the regularizing properties of the PCG iteration for solving ill-conditioned systems of equations (PCGNs) is then explored, and a suitable preconditioning matrix is suggested in order to enhance its convergence rate. For image reconstruction, the nonlinear inverse problem is considered. Based on the Gauss-Newton method for solving nonlinear problems we have developed two algorithms that implement the PCGN iteration to calculate the linear step solution. Using an anatomically detailed model of the human head and a specific scalp electrode arrangement, images of a simulated impedance change inside brain's white matter have been reconstructed.

Generation of anisotropic-smoothness regularization filters for EIT.

In the inverse conductivity problem, as in any ill-posed inverse problem, regularization techniques are necessary in order to stabilize inversion. A common way to implement regularization in electrical impedance tomography is to use Tikhonov regularization. The inverse problem is formulated as a minimization of two terms: the mismatch of the measurements against the model, and the regularization functional. Most commonly, differential operators are used as regularization functionals, leading to smooth solutions. Whenever the imaged region presents discontinuities in the conductivity distribution, such as interorgan boundaries, the smoothness prior is not consistent with the actual situation. In these cases, the reconstruction is enhanced by relaxing the smoothness constraints in the direction normal to the discontinuity. In this paper, we derive a method for generating Gaussian anisotropic regularization filters. The filters are generated on the basis of the prior structural information, allowing a better reconstruction of conductivity profiles matching these priors. When incorporating prior information into a reconstruction algorithm, the risk is of biasing the inverse solutions toward the assumed distributions. Simulations show that, with a careful selection of the regularization parameters, the reconstruction algorithm is still able to detect conductivities patterns that violate the prior information. A generalized singular-value decomposition analysis of the effects of the anisotropic filters on regularization is presented in the last sections of the paper.

EIT reconstruction algorithms: pitfalls, challenges and recent developments.

We review developments, issues and challenges in electrical impedance tomography (EIT) for the 4th Conference on Biomedical Applications of Electrical Impedance Tomography, held at Manchester in 2003. We focus on the necessity for three-dimensional data collection and reconstruction, efficient solution of the forward problem, and both present and future reconstruction algorithms. We also suggest common pitfalls or 'inverse crimes' to avoid.

Uniqueness and reconstruction in magnetic resonance-electrical impedance tomography (MR-EIT).

Magnetic resonance-electrical impedance tomography (MR-EIT) was first proposed in 1992. Since then various reconstruction algorithms have been suggested and applied. These algorithms use peripheral voltage measurements and internal current density measurements in different combinations. In this study the problem of MR-EIT is treated as a hyperbolic system of first-order partial differential equations, and three numerical methods are proposed for its solution. This approach is not utilized in any of the algorithms proposed earlier. The numerical solution methods are integration along equipotential surfaces (method of characteristics), integration on a Cartesian grid, and inversion of a system matrix derived by a finite difference formulation. It is shown that if some uniqueness conditions are satisfied, then using at least two injected current patterns, resistivity can be reconstructed apart from a multiplicative constant. This constant can then be identified using a single voltage measurement. The methods proposed are direct, non-iterative, and valid and feasible for 3D reconstructions. They can also be used to easily obtain slice and field-of-view images from a 3D object. 2D simulations are made to illustrate the performance of the algorithms.

Multimodal Image Reconstruction Using Supplementary Structural Information in Total Variation Regularization.

In this paper, we propose an iterative reconstruction algorithm which uses available information from one dataset collected using one modality to increase the resolution and signal-to-noise ratio of one collected by another modality. The method operates on the structural information only which increases its suitability across various applications. Consequently, the main aim of this method is to exploit available supplementary data within the regularization framework. The source of primary and supplementary datasets can be acquired using complementary imaging modes where different types of information are obtained (e.g. in medical imaging: anatomical and functional). It is shown by extracting structural information from the supplementary image (direction of level sets) one can enhance the resolution of the other image. Notably, the method enhances edges that are common to both images while not suppressing features that show high contrast in the primary image alone. In our iterative algorithm we use available structural information within a modified total variation penalty term. We provide numerical experiments to show the advantages and feasibility of the proposed technique in comparison to other methods.

Shape deformation in two-dimensional electrical impedance tomography.

Electrical impedance tomography (EIT) uses measurements from surface electrodes to reconstruct an image of the conductivity of the contained medium. However, changes in measurements result from both changes in internal conductivity and changes in the shape of the medium relative to the electrode positions. Failure to account for shape changes results in a conductivity image with significant artifacts. Previous work to address shape changes in EIT has shown that in some cases boundary shape and electrode location can be uniquely determined for isotropic conductivities; however, for geometrically conformal changes, this is not possible. This prior work has shown that the shape change problem can be partially addressed. In this paper, we explore the limits of compensation for boundary movement in EIT using three approaches. First, a theoretical model was developed to separate a deformation vector field into conformal and nonconformal components, from which the reconstruction limits may be determined. Next, finite element models were used to simulate EIT measurements from a domain whose boundary has been deformed. Finally, an experimental phantom was constructed from which boundary deformation measurements were acquired. Results, both in simulation and with experimental data, suggest that some electrode movement and boundary distortions can be reconstructed based on conductivity changes alone while reducing image artifacts in the process.

Impact of model shape mismatch on reconstruction quality in electrical impedance tomography.

Electrical impedance tomography (EIT) is a low-cost, noninvasive and radiation free medical imaging modality for monitoring ventilation distribution in the lung. Although such information could be invaluable in preventing ventilator-induced lung injury in mechanically ventilated patients, clinical application of EIT is hindered by difficulties in interpreting the resulting images. One source of this difficulty is the frequent use of simple shapes which do not correspond to the anatomy to reconstruct EIT images. The mismatch between the true body shape and the one used for reconstruction is known to introduce errors, which to date have not been properly characterized. In the present study we, therefore, seek to 1) characterize and quantify the errors resulting from a reconstruction shape mismatch for a number of popular EIT reconstruction algorithms and 2) develop recommendations on the tolerated amount of mismatch for each algorithm. Using real and simulated data, we analyze the performance of four EIT reconstruction algorithms under different degrees of shape mismatch. Results suggest that while slight shape mismatch is well tolerated by all algorithms, using a circular shape severely degrades their performance.