Multi-sheet surface rebinning methods for reconstruction from asymmetrically truncated cone beam projections: I. Approximation and optimality

The mechanical motion of the gantry in conventional cone beam CT scanners restricts the speed of data acquisition in applications with near real time requirements. A possible resolution of this problem is to replace the moving source detector assembly with static parts that are electronically activated. An example of such a system is the Rapiscan Systems RTT80 real time tomography scanner, with a static ring of sources and axially offset static cylinder of detectors. A consequence of such a design is asymmetrical axial truncation of the cone beam projections resulting, in the sense of integral geometry, in severely incomplete data. In particular we collect data only in a fraction of the Tam-Danielsson window, hence the standard cone beam reconstruction techniques do not apply. In this work we propose a family of multi-sheet surface rebinning methods for reconstruction from such truncated projections. The proposed methods combine analytical and numerical ideas utilizing linearity of the ray transform to reconstruct data on multi-sheet surfaces, from which the volumetric image is obtained through deconvolution. In this first paper in the series, we discuss the rebinning to multi-sheet surfaces. In particular we concentrate on the underlying transforms on multi-sheet surfaces and their approximation with data collected by offset multi-source scanning geometries like the RTT. The optimal multi-sheet surface and the corresponding rebinning function are found as a solution of a variational problem. In the case of the quadratic objective, the variational problem for the optimal rebinning pair can be solved by a globally convergent iteration. Examples of optimal rebinning pairs are computed for different trajectories. We formulate the axial deconvolution problem for the recovery of the volumetric image from the reconstructions on multi-sheet surfaces. Efficient and stable solution of the deconvolution problem is the subject of the second paper in this series (Betcke and Lionheart 2013 Inverse Problems 29 115004). © 2013 IOP Publishing Ltd

Direct inversion of the Longitudinal Ray Transform for 2D residual elastic strain fields

We examine the problem of Bragg-edge elastic strain tomography from energy resolved neutron transmission imaging. A new approach is developed for two-dimensional plane-stress and plane-strain systems whereby elastic strain can be reconstructed from its Longitudinal Ray Transform (LRT) as two parts of a Helmholtz decomposition based on the concept of an Airy stress potential. The solenoidal component of this decomposition is reconstructed using an inversion formula based on a tensor filtered back projection algorithm whereas the potential part can be recovered using either Hooke's law or a finite element model of the elastic system. The technique is demonstrated for two-dimensional plane-stress systems in both simulation, and on real experimental data. We also demonstrate that application of the standard scalar filtered back projection algorithm to the LRT in these systems recovers the trace of the solenoidal component of strain and we provide physical meaning for this quantity in the case of 2D plane-stress and plane-strain systems.Comment: 30 pages, 9 figure

Joint image reconstruction method with correlative multi-channel prior for x-ray spectral computed tomography

Rapid developments in photon-counting and energy-discriminating detectors have the potential to provide an additional spectral dimension to conventional x-ray grayscale imaging. Reconstructed spectroscopic tomographic data can be used to distinguish individual materials by characteristic absorption peaks. The acquired energy-binned data, however, suffer from low signal-to-noise ratio, acquisition artifacts, and frequently angular undersampled conditions. New regularized iterative reconstruction methods have the potential to produce higher quality images and since energy channels are mutually correlated it can be advantageous to exploit this additional knowledge. In this paper, we propose a novel method which jointly reconstructs all energy channels while imposing a strong structural correlation. The core of the proposed algorithm is to employ a variational framework of parallel level sets to encourage joint smoothing directions. In particular, the method selects reference channels from which to propagate structure in an adaptive and stochastic way while preferring channels with a high data signal-to-noise ratio. The method is compared with current state-of-the-art multi-channel reconstruction techniques including channel-wise total variation and correlative total nuclear variation regularization. Realistic simulation experiments demonstrate the performance improvements achievable by using correlative regularization methods

Sparsity seeking total generalized variation for undersampled tomographic reconstruction

Here we present a novel iterative approach for tomographic image reconstruction which improves image quality for undersampled and limited view projection measurements. Recently, the Total Generalized Variation (TGV) penalty has been proposed to establish a desirable balance between smooth and piecewise-constant solutions. Piecewise-smooth reconstructions are particularly important for biomedical applications, where the image surface slowly varies. The TGV penalty convexly combines the first and higher order derivatives, which means that for some regions (e.g. uniform background) it can be more challenging to find a sparser solution due to the weight of the higher order term. Therefore we propose a simple heuristic modification over the Chambolle-Pock reconstruction scheme for TGV which consists of adding the wavelet thresholding step which helps to suppress aliasing artifacts and noise while preserve piecewise-smooth appearance. Preliminary numerical results with two piecewise-smooth phantoms show strong improvement of the proposed method over TGV and TV penalties. The resulting images are smooth with sharp edges and fewer artifacts visible

Analysis of the inverse problem for determining nematic liquid crystal director profiles from optical measurements using singular value decomposition.

We use the problem of determining nematic liquid crystal director profiles from optical measurements as an example to illustrate that what is often treated purely as a data fitting problem is really an inverse problem and that useful insights an be obtained by treating it in this way. Specifically we illustrate the analysis of he sufficiency of data and the sensitivity of a solution to measurement errors. We assume a stratified medium where the Berreman method can be used for the optical forward problem and we consider the inverse problem to be the determination of an anisotropic dielectric permittivity tensor from optical data. A numerical Singular Value Decomposition (SVD) analysis reveals that although this inverse problem is severely ill-conditioned it is possible to determine depth-dependent information provided the medium is sufficiently birefringent and that, as one might expect, a larger range of incident angles gives greater information. Analytical solutions of the Berreman equations for general perturbations of an orthorhombic crystal con¯rm uniqueness of solution for the linearized problem and give further insights into the severely ill-posed nature of the inverse problem

Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data

A shape reconstruction method for electrical resistance and capacitance tomography is presented using a level set formulation. In this shape reconstruction approach, the conductivity (or permittivity) values of the inhomogeneous background and the obstacles are assumed to be (approximately) known, but the number, sizes, shapes, and locations of these obstacles have to be recovered from the data. A key point in this shape identification technique is to represent geometrical boundaries of the obstacles by using a level set function. This representation of the shapes has the advantage that the level set function automatically handles the splitting or merging of the objects during the reconstruction. Another key point of the algorithm is to solve the inverse problem of finding the interfaces between two materials using a narrow-band method, which not only decreases the number of unknowns and therefore the computational cost of the inversion, but also tends to improve the condition number of the discrete inverse problem compared to pixel (voxel)-based image reconstruction. Level set shape reconstruction results shown in this article are some of the first ones using experimental electrical tomography data. The experimental results also show some improvements in image quality compared with the pixel-based image reconstruction. The proposed technique is applied to 2D resistance and capacitance tomography for both simulated and experimental data. In addition, a full 3D inversion is performed on simulated 3D resistance tomography data

Time series of EIT chest images using singular value decomposition and Fourier transform

The aim of this study is to propose a useful method for exploring regional ventilation and perfusion in the chest. The paper describes two methods based on singular value decomposition (SVD) and Fourier transform (FT) respectively. This work shows that power spectral density (PSD) and phase images (derived from the Fourier transform) are easier to interpret and more useful tools for exploiting in vivo EIT data in healthy volunteers in order to explore the cardiovascular and respiratory system