Direct Reconstruction Algorithm Research Articles

Exact Fan-Beam Reconstruction With Arbitrary Object Translations and Truncated Projections

This article proposes a new method for reconstructing two-dimensional (2D) computed tomography (CT) images from truncated and motion contaminated sinograms. The type of motion considered here is a sequence of rigid translations which are assumed to be known. The algorithm first identifies the sufficiency of angular coverage in each 2D point of the CT image to calculate the Hilbert transform from the local “virtual” trajectory which accounts for the motion and the truncation. By taking advantage of data redundancy in the full circular scan, our method expands the reconstructible region beyond the one obtained with chord-based methods. The proposed direct reconstruction algorithm is based on the Differentiated Back-Projection with Hilbert filtering (DBP-H). The motion is taken into account during backprojection which is the first step of our direct reconstruction, before taking the derivatives and inverting the finite Hilbert transform. The algorithm has been tested in a proof-of-concept study on Shepp-Logan phantom simulations with several motion cases and detector sizes.

Direct Fourier Inversion Reconstruction Algorithm for Computed Laminography.

Synchrotron radiation computed laminography (CL) was developed to complement the conventional computed tomography as a non-destructive 3D imaging method for the inspection of flat thin objects. Recent progress in hardware at synchrotron sources allows one to record internal evolution of specimens at the micrometer scale and sub-second range but also requires increased reconstruction speed to follow structural changes online. A 3D image of the sample interior is usually reconstructed by the well-established filtered backprojection (FBP) approach. Despite of a great success in the reduction of reconstruction time via parallel computations, the FBP algorithm still remains a time-consuming procedure. A promising way to significantly shorten computation time is to directly perform backprojection in frequency domain (a direct Fourier inversion approach). The corresponding algorithms are rarely considered in the literature because of a poor performance or inferior reconstruction quality resulted from inaccurate interpolation in Fourier domain. In this paper, we derive a Fourier-based reconstruction equation designed for the CL scanning geometry. Furthermore, we outline the translation of the continuous solution to a discrete version, which utilizes 3D sinc interpolation. A projection resampling technique allowing for the reduction of the expensive interpolation to its 1D version is proposed. A series of numerical experiments confirms that the resulting image quality is well comparable with the FBP approach while reconstruction time is drastically reduced.

Nonlinear greedy sparsity-constrained algorithm for direct reconstruction of fluorescence molecular lifetime tomography

In order to improve the spatial resolution of time-domain (TD) fluorescence molecular lifetime tomography (FMLT), an accelerated nonlinear orthogonal matching pursuit (ANOMP) algorithm is proposed. As a kind of nonlinear greedy sparsity-constrained methods, ANOMP can find an approximate solution of L0 minimization problem. ANOMP consists of two parts, i.e., the outer iterations and the inner iterations. Each outer iteration selects multiple elements to expand the support set of the inverse lifetime based on the gradients of a mismatch error. The inner iterations obtain an intermediate estimate based on the support set estimated in the outer iterations. The stopping criterion for the outer iterations is based on the stability of the maximum reconstructed values and is robust for problems with targets at different edge-to-edge distances (EEDs). Phantom experiments with two fluorophores at different EEDs and in vivo mouse experiments demonstrate that ANOMP can provide high quantification accuracy, even if the EED is relatively small, and high resolution.

Direct reconstruction method for time-domain fluorescence molecular lifetime tomography.

For the reconstruction of time-domain fluorescence molecular lifetime tomography, conventional methods based on the Laplace or Fourier transform utilize only part of the information from the measurement data, and rely on the selection of transformation factors. To make the best of all the measurement data, a direct reconstruction algorithm is proposed. The fluorescence yield map is first reconstructed with a full-time gate, and then an objective function for the inverse lifetime tomography (instead of the lifetime) is developed so as to avoid dealing with the singularity of the zero points in the lifetime image. Through simulations and physical phantom experiments, the proposed algorithm is demonstrated to have high localization accuracy for fluorescent targets, high quantification accuracy for fluorescence lifetime, and good contrast between different fluorescence targets.

Direct Parametric Reconstruction Using Anatomical Regularization for Simultaneous PET/MRI Data.

Pharmacokinetic analysis of dynamic positron emission tomography (PET) imaging data maps the measured time activity curves to a set of model-specific pharmacokinetic parameters. Voxel-based parameter estimation via curve fitting is conventionally performed indirectly on a sequence of independently reconstructed PET images, leading to high variance and bias in the parametric images. We propose a direct parametric reconstruction algorithm with raw projection data as input that leverages high-resolution anatomical information simultaneously obtained from magnetic resonance (MR) imaging in a PET/MRI scanner for regularization. The reconstruction problem is formulated in a flexible Bayesian framework with Gaussian Markov Random field modeling of activity, parameters, or both simultaneously. MR information is incorporated through a Bowsher-like prior function. Optimization transfer using an expectation-maximization surrogate and a new Bowsher-like penalty surrogate is applied to obtain a voxel-separable algorithm that interleaves a reconstruction with a fitting step. An analytical input function model is used. The algorithm is evaluated on simulated [(18)F]FDG and clinical [(18)F]FET brain data acquired with a Biograph mMR. The results indicate that direct and simultaneously regularized parametric reconstruction increases image quality. Anatomical regularization leads to higher contrast than conventional distance-weighted regularization.

Influence of Gaussian function index of deformable mirror on iterative algorithm adaptive optical system

Among all kinds of wavefront reconstruction algorithms in adaptive optical systems, the standard and mostly used algorithm is the direct gradient wavefront reconstruction algorithm. As the number of sub-apertures in Shack-Hartmann wavefront sensor and the actuators for deformable mirror increases, the reconstruction matrix in direct gradient wavefront reconstruction algorithm takes too much space and the number of multiplication in the algorithm increases sharply. So, the iterative algorithm is adopted in wavefront reconstruction for the high-resolution adaptive optical system. The number of multiplication and the required space of the iterative algorithm are directly related to the sparseness of both iterative matrix and slope response matrix. In an adaptive optical system, the sparseness of these two matrixes is connected with the system parameters. Therefore, it is necessary to study how to choose the proper parameters for an adaptive optical system when it uses iterative wavefront reconstruction algorithm. In this paper, the sparseness of slope response matrix and iterative matrix are analyzed based on a 613-actuator adaptive optical system. The influence of the Gaussian function index of deformable mirror on the sparsenesses of slope response matrix, iterative matrix, stability and correction qualities of the adaptive optical system are also studied under the condition of constant actuator spacing and coupling coefficient. A larger Gaussian function index results in a lower sparseness of the slope response matrix and the iterative matrix. Too large or too small a Gaussian function index will degrade the stability and the correction quality of an adaptive optical system. Finally, the optimal range of the Gaussian function index is provided by balancing the sparseness of slope response matrix, the correction quality, and the stability of the adaptive optical system.

Application of adaptive kinetic modelling for bias propagation reduction in direct 4D image reconstruction

Parametric imaging in thoracic and abdominal PET can provide additional parameters more relevant to the pathophysiology of the system under study. However, dynamic data in the body are noisy due to the limiting counting statistics leading to suboptimal kinetic parameter estimates. Direct 4D image reconstruction algorithms can potentially improve kinetic parameter precision and accuracy in dynamic PET body imaging. However, construction of a common kinetic model is not always feasible and in contrast to post-reconstruction kinetic analysis, errors in poorly modelled regions may spatially propagate to regions which are well modelled. To reduce error propagation from erroneous model fits, we implement and evaluate a new approach to direct parameter estimation by incorporating a recently proposed kinetic modelling strategy within a direct 4D image reconstruction framework. The algorithm uses a secondary more general model to allow a less constrained model fit in regions where the kinetic model does not accurately describe the underlying kinetics. A portion of the residuals then is adaptively included back into the image whilst preserving the primary model characteristics in other well modelled regions using a penalty term that trades off the models. Using fully 4D simulations based on dynamic [15O]H2O datasets, we demonstrate reduction in propagation-related bias for all kinetic parameters. Under noisy conditions, reductions in bias due to propagation are obtained at the cost of increased noise, which in turn results in increased bias and variance of the kinetic parameters. This trade-off reflects the challenge of separating the residuals arising from poor kinetic modelling fits from the residuals arising purely from noise. Nonetheless, the overall root mean square error is reduced in most regions and parameters. Using the adaptive 4D image reconstruction improved model fits can be obtained in poorly modelled regions, leading to reduced errors potentially propagating to regions of interest which the primary biologic model accurately describes. The proposed methodology, however, depends on the secondary model and choosing an optimal model on the residual space is critical in improving model fits.

Direct 2-D reconstructions of conductivity and permittivity from EIT data on a human chest.

A novel direct D-bar reconstruction algorithm is presented for reconstructing a complex conductivity distribution from 2-D EIT data. The method is applied to simulated data and archival human chest data. Permittivity reconstructions with the aforementioned method and conductivity reconstructions with the previously existing nonlinear D-bar method for real-valued conductivities depicting ventilation and perfusion in the human chest are presented. This constitutes the first fully nonlinear D-bar reconstructions of human chest data and the first D-bar permittivity reconstructions of experimental data. The results of the human chest data reconstructions are compared on a circular domain versus a chest-shaped domain.

Quality and Precision of Parametric Images Created From PET Sinogram Data by Direct Reconstruction: Proof of Concept

We have previously implemented the direct reconstruction of dense kinetic model parameter images ("parametric images") from sinogram data, and compared it to conventional image domain kinetic parameter estimation methods . Although it has been shown that the direct reconstruction algorithm estimates the kinetic model parameters with lower root mean squared error than the conventional image domain techniques, some theoretical obstacles remain. These obstacles include the difficulty of evaluating the accuracy and precision of the estimated parameters. In image domain techniques, the reconstructed time activity curve (TAC) and the model predicted TAC are compared, and the goodness-of-fit is evaluated as a measure of the accuracy and precision of the estimated parameters. This approach cannot be applied to the direct reconstruction technique as there are no reconstructed TACs. In this paper, we propose ways of evaluating the precision and goodness-of-fit of the kinetic model parameters estimated by the direct reconstruction algorithm. Specifically, precision of the estimates requires the calculation of variance images for each parameter, and goodness-of-fit is addressed by reconstructing the difference between the measured and the fitted sinograms. We demonstrate that backprojecting the difference from sinogram space to image space creates error images that can be examined for goodness-of-fit and model selection purposes. The presence of nonrandom structures in the error images may indicate an inadequacy of the kinetic model that has been incorporated into the direct reconstruction algorithm. We introduce three types of goodness-of-fit images. We propose and demonstrate a number-of-runs image as a means of quantifying the adequacy or deficiency of the model. We further propose and demonstrate images of the F statistic and the change in the Akaike Information Criterion as devices for identifying the statistical advantage of one model over another at each voxel. As direct reconstruction to parametric images proliferates, it will be essential for imagers to adopt methods such as those proposed herein to assess the accuracy and precision of their parametric images.

Conductivity reconstructions using real data from a new planar electrical impedance tomography device

In this paper, we present results of reconstructions using real data from a new planar electrical impedance tomography device developed at the Institut für Physik, Johannes Gutenberg Universität, Mainz, Germany. The prototype consists of a planar sensing head of circular geometry, and it was designed mainly for breast cancer detection. There are 12 large outer electrodes arranged on a ring of radius cm where the external currents are injected, and a set of 54 point-like high-impedance inner electrodes where the induced voltages are measured. Two direct (i.e. non-iterative) reconstruction algorithms are considered: one is based on a discrete resistor model, and the other one is an integral equation approach for smooth conductivity distributions.

Direct Estimation of Kinetic Parametric Images for Dynamic PET

Dynamic positron emission tomography (PET) can monitor spatiotemporal distribution of radiotracer in vivo. The spatiotemporal information can be used to estimate parametric images of radiotracer kinetics that are of physiological and biochemical interests. Direct estimation of parametric images from raw projection data allows accurate noise modeling and has been shown to offer better image quality than conventional indirect methods, which reconstruct a sequence of PET images first and then perform tracer kinetic modeling pixel-by-pixel. Direct reconstruction of parametric images has gained increasing interests with the advances in computing hardware. Many direct reconstruction algorithms have been developed for different kinetic models. In this paper we review the recent progress in the development of direct reconstruction algorithms for parametric image estimation. Algorithms for linear and nonlinear kinetic models are described and their properties are discussed.

Learned Shrinkage Approach for Low-Dose Reconstruction in Computed Tomography

We propose a direct nonlinear reconstruction algorithm for Computed Tomography (CT), designed to handle low-dose measurements. It involves the filtered back-projection and adaptive nonlinear filtering in both the projection and the image domains. The filter is an extension of the learned shrinkage method by Hel-Or and Shaked to the case of indirect observations. The shrinkage functions are learned using a training set of reference CT images. The optimization is performed with respect to an error functional in the image domain that combines the mean square error with a gradient-based penalty, promoting image sharpness. Our numerical simulations indicate that the proposed algorithm can manage well with noisy measurements, allowing a dose reduction by a factor of 4, while reducing noise and streak artifacts in the FBP reconstruction, comparable to the performance of a statistically based iterative algorithm.

Distributed Field Reconstruction in Wireless Sensor Networks Based on Hybrid Shift-Invariant Spaces

We use the theory and algorithms developed for so-called shift-invariant spaces to develop a novel distributed architecture for sampling and reconstructing time-varying non-bandlimited physical fields in wireless sensor networks. We introduce hybrid shift-invariant spaces that generalize conventional shift-invariant spaces and can adapt to local smoothness properties of the field. Using shift-invariant spaces with compactly supported generator functions allows us to split the global reconstruction into several smaller local problems that can be solved independently. Capitalizing on the sparsity of the matrices involved in the reconstruction, we propose direct and iterative reconstruction algorithms whose complexity per time slot scales only linearly with the number of sensor nodes. We furthermore analyze the impact of sensor localization errors on the mean square error of the reconstructed field. Numerical simulations illustrate that the proposed field reconstruction scheme performs close to bandlimited reconstruction and is less sensitive to sensor location errors while providing a significant reduction in computational complexity.

A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2D

A direct reconstruction algorithm for complex conductivities in W2, ∞(Ω), where Ω is a bounded, simply connected Lipschitz domain in , is presented. The framework is based on the uniqueness proof by Francini (2000 Inverse Problems 6 107–19), but equations relating the Dirichlet-to-Neumann to the scattering transform and the exponentially growing solutions are not present in that work, and are derived here. The algorithm constitutes the first D-bar method for the reconstruction of conductivities and permittivities in two dimensions. Reconstructions of numerically simulated chest phantoms with discontinuities at the organ boundaries are included.

Radial Imaging With Multipolar Magnetic Encoding Fields

We present reconstruction methods for radial magnetic resonance imaging (MRI) data which were spatially encoded using a pair of orthogonal multipolar magnetic fields for in-plane encoding and parallel imaging. It is shown that a direct method exists in addition to iterative reconstruction. Standard direct projection reconstruction algorithms can be combined with a previously developed direct reconstruction for multipolar encoding fields acquired with Cartesian trajectories. The algorithm is simplified by recasting the reconstruction problem into polar coordinates. In this formulation distortion and aliasing become separate effects. Distortion occurs only along the radial direction and aliasing along the azimuthal direction. Moreover, aliased points are equidistantly distributed in this representation, and, consequently, Cartesian SENSE is directly applicable with highly effective unfolding properties for radio-frequency coils arranged with a radial symmetry. The direct and iterative methods are applied to simulated data to analyze basic properties of the algorithms and for the first time also measured in vivo data are presented. The results are compared to linear spatial encoding using a radial trajectory and quadrupolar encoding using a Cartesian trajectory. The direct reconstruction gives good results for fully sampled datasets. Undersampled datasets, however, show star-shaped artifacts, which are significantly reduced with the iterative reconstruction.

A direct numerical reconstruction algorithm for the 3D Calderón problem

In three dimensions Calderón's problem was addressed and solved in theory in the 1980s in a series of papers, but only recently the numerical implementation of the algorithm was initiated. The main ingredients in the solution of the problem are complex geometrical optics solutions to the conductivity equation and a (non-physical) scattering transform. The resulting reconstruction algorithm is in principle direct and addresses the full non-linear problem immediately. In this paper we will outline the theoretical reconstruction method and describe how the method can be implemented numerically. We will give three different implementations, and compare their performance on a numerical phantom.

Direct numerical reconstruction of conductivities in three dimensions using scattering transforms

A direct three-dimensional EIT reconstruction algorithm based on complex geometrical optics solutions and a nonlinear scattering transform is presented and implemented for spherically symmetric conductivity distributions. The scattering transform is computed both with a Born approximation and from the forward problem for purposes of comparison. Reconstructions are computed for several test problems. A connection to Calderón's linear reconstruction algorithm is established, and reconstructions using both methods are compared.

An efficient numerical algorithm for crack reconstruction in elastic media by the circular US scanning

In the present work, a direct numerical reconstruction algorithm is proposed, which is applied to the identification of the position and shape of cracks detected in elastic solid samples, in the case when the input data available for the reconstruction is provided by a circular ultrasonic (US) scanner. The US probe gives the far-field back scattered pattern over the full interval of the incident polar angle. The problem is first reduced to a system of basic boundary integral equations. Then the inverse reconstruction problem is formulated as a minimization problem for a certain strongly nonlinear functional. The proposed numerical algorithm is tested on some examples of complex-shaped cracks. Then the influence of the error in the input data on the precision of the reconstruction is studied in detail.

Initial Evaluation of Direct 4D Parametric Reconstruction with Human PET Data.

Previously, we presented a direct EM method for producing kinetic parameter images from list mode PET data, where the time-activity curve for each voxel is described by a one-tissue compartment model (1T). The initial evaluations were performed with simulations, without motion, randoms, or scatter effects included. By extension of our previous frame-based physics correction methods, a practical direct 4D parametric reconstruction algorithm is now proposed and implemented for human data. Initial evaluations were performed using 3 human subjects with the serotonin transporter tracer [(11)C]AFM. Comparisons with the 2-step approach (frame-based reconstruction followed by voxel-by-voxel parameter estimation) provided encouraging initial results. Regional analysis showed that the 2-step and 4D methods have similar K(1) and V(T) values, but with a consistent difference. Visual analysis showed some noise reduction in 4D. These initial results suggest that direct 4D parametric reconstruction can be performed with real data, and offers the potential for improved accuracy and precision over the 2-step frame method.

Generalized Algorithms for Direct Reconstruction of Parametric Images From Dynamic PET Data

Indirect and direct methods have been developed for reconstructing parametric images from dynamic positron emission tomography (PET) data. Indirect methods are simple and easy to implement because reconstruction and kinetic modeling are performed in two separate steps. Direct methods estimate parametric images directly from dynamic PET sinograms and, in theory, can be statistically more efficient, but the algorithms are often difficult to implement and are very specific to the kinetic model being used. This paper presents a class of generalized algorithms for direct reconstruction of parametric images that are relatively easy to implement and can be adapted to different kinetic models. The proposed algorithms use optimization transfer principle to convert the maximization of a penalized likelihood into a pixel-wise weighted least squares (WLS) kinetic fitting problem at each iteration. Thus, it can employ existing WLS algorithms developed for kinetic models. The proposed algorithms resemble the empirical iterative implementation of the indirect approach, but converge to a solution of the direct formulation. Computer simulations showed that the proposed direct reconstruction algorithms are flexible and achieve a better bias-variance tradeoff than indirect reconstruction methods.