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The Finnish Inverse Problems Society organized the Helsinki Tomography Challenge (HTC) in 2022 to reconstruct an image with limited-angle measurements. We participated in this challenge and developed two methods: an Edge Inpainting method and a Learned Primal-Dual (LPD) network. The Edge Inpainting method involves multiple stages, including classical reconstruction using Perona-Malik, detection of visible edges, inpainting invisible edges using a U-Net, and final segmentation using a U-Net. The LPD approach adapts the classical LPD by using large U-Nets in the primal update and replacing the adjoint with the filtered back projection (FBP). Since the challenge only provided five samples, we generated synthetic data to train the networks. The Edge Inpainting Method performed well for viewing ranges above 70 degrees, while the LPD approach performed well across all viewing ranges and ranked second overall in the challenge. [ABSTRACT FROM AUTHOR]

Computed tomography (CT) has become an essential part of modern science and medicine. A CT scanner consists of an X-ray source that is spun around an object of interest. On the opposite end of the X-ray source, a detector captures X-rays that are not absorbed by the object. The reconstruction of an image is a linear inverse problem, which is usually solved by filtered back projection. However, when the number of measurements is small, the reconstruction problem is ill-posed. This is for example the case when the X-ray source is not spun completely around the object, but rather irradiates the object only from a limited angle. To tackle this problem, we present a deep neural network that is trained on a large amount of carefully-crafted synthetic data and can perform limited-angle tomography reconstruction even for only 30° or 40° sinograms. With our approach we won the first place in the Helsinki Tomography Challenge 2022. [ABSTRACT FROM AUTHOR]

Abstract If a spatial-domain function has a finite support, its Fourier transform is an entire function. The Taylor series expansion of an entire function converges at every finite point in the complex plane. The analytic continuation theory suggests that a finite-sized object can be uniquely determined by its frequency components in a very small neighborhood. Trying to obtain such an exact Taylor expansion is difficult. This paper proposes an iterative algorithm to extend the measured frequency components to unmeasured regions. Computer simulations show that the proposed algorithm converges very slowly, indicating that the problem is too ill-posed to be practically solvable using available methods.

Image reconstruction with limited angles projection data is a challenging task in computed tomography (CT). The amount of radiation associated with CT induces health implications to the patient. Besides, image reconstruction with limited-angles projection data distorts the image, thus emasculating the efficiency of diagnosis. Also, the poly-chromatic nature of the X-ray adds beam-hardening artifacts in the reconstruction. The state-of-the-art approaches available in the literature have proposed the solutions for beam-hardening artifacts correction in full span computed tomography. Most of the solutions are hardware based and need extra hardware to remove the beam hardening artifacts. The present manuscript proposes artificial intelligence based software solution for the beam hardening artifacts removal. This manuscript has presented a cascaded encoder-decoder architecture named cascaded deep neural network for image reconstruction (CDNN). The CDNN architecture has convolution neural network blocks that include convolution layers, rectified linear units ReLU, and batch normalization layers. The network has skip-connections for better learning of features between input and output. The network has been designed as a forward model. The stochastic gradient descent optimization method has been used for training the network. Image reconstructed from Fourier transform-based approach has been used as a prior. A novel approach for reduction of beam-hardening artifacts in case of limited-angles computed tomography using CDNN has been presented. The proposed approach is comparable to other hardware/software solutions for aforesaid purpose and does not require any extra hardware. The proposed approach has improved the image quality as compared to U-Net and the other state-of-the-art methods. It has been found from the experiments that the CDNN suppresses the artifacts and improves the reconstruction. The performance of the proposed CDNN has been tested with real-life data having beam hardening artifacts. It has been observed that the CDNN has improved the reconstruction quality by reducing streak, ring artifacts, and beam hardening artifacts and also preserving the profound structures. [ABSTRACT FROM AUTHOR]

If a spatial-domain function has a finite support, its Fourier transform is an entire function. The Taylor series expansion of an entire function converges at every finite point in the complex plane. The analytic continuation theory suggests that a finite-sized object can be uniquely determined by its frequency components in a very small neighborhood. Trying to obtain such an exact Taylor expansion is difficult. This paper proposes an iterative algorithm to extend the measured frequency components to unmeasured regions. Computer simulations show that the proposed algorithm converges very slowly, indicating that the problem is too ill-posed to be practically solvable using available methods. [ABSTRACT FROM AUTHOR]

In transmission X‐ray microscopy (TXM) systems, the rotation of a scanned sample might be restricted to a limited angular range to avoid collision with other system parts or high attenuation at certain tilting angles. Image reconstruction from such limited angle data suffers from artifacts because of missing data. In this work, deep learning is applied to limited angle reconstruction in TXMs for the first time. With the challenge to obtain sufficient real data for training, training a deep neural network from synthetic data is investigated. In particular, U‐Net, the state‐of‐the‐art neural network in biomedical imaging, is trained from synthetic ellipsoid data and multi‐category data to reduce artifacts in filtered back‐projection (FBP) reconstruction images. The proposed method is evaluated on synthetic data and real scanned chlorella data in 100° limited angle tomography. For synthetic test data, U‐Net significantly reduces the root‐mean‐square error (RMSE) from 2.55 × 10−3 µm−1 in the FBP reconstruction to 1.21 × 10−3 µm−1 in the U‐Net reconstruction and also improves the structural similarity (SSIM) index from 0.625 to 0.920. With penalized weighted least‐square denoising of measured projections, the RMSE and SSIM are further improved to 1.16 × 10−3 µm−1 and 0.932, respectively. For real test data, the proposed method remarkably improves the 3D visualization of the subcellular structures in the chlorella cell, which indicates its important value for nanoscale imaging in biology, nanoscience and materials science. [ABSTRACT FROM AUTHOR]

This paper draws attention to the study of performance of a new Molecular Breast Imaging (MBI) device, whose purpose is the early diagnosis of breast cancer, using Monte Carlo simulations. MBI provides functional and specific information that are more appropriated to dense breasts. Two asymmetric heads with different types of collimators, facing each other in anti-parallel viewing direction, characterize the system. Detectors and phantoms, together with the data taking procedure, are shortly reported. Monte Carlo simulations using the GATE (GEANT4 Application for Tomographic Emission) simulation toolkit have been implemented to evaluate the optimal detector configuration, in terms of sensitivity and spatial resolution, and also to reproduce the real experimental data. The device can be used both in spot compression and in Limited Angle Tomography (LAT); in the latter configuration one detector head with pinhole collimator is able to rotate around the breast in order to diagnose and localized the small tumors. [ABSTRACT FROM AUTHOR]

The total variation (TV) regularization is popular in iterative image reconstruction when the piecewise-constant nature of the image is encouraged. As a matter of fact, the TV regularization is not strong enough to enforce the piecewise-constant appearance. This paper suggests a different regularization function that is able to discourage some smooth transitions in the image and to encourage the piecewise-constant behavior. This new regularization function involves a Gaussian function. We use the limited-angle tomography problem to illustrate the effectiveness of this new regularization function. The limited-angle tomography situation considered in this paper uses a scanning angular range of 40 ° . For two-dimensional parallel-beam imaging, the required angular range is supposed to be 180 ° .

Purpose: The application of traditional machine learning techniques, in the form of regression models based on conventional, "hand-crafted" features, to artifact reduction in limited angle tomography is investigated.Methods: Mean-variation-median (MVM), Laplacian, Hessian, and shift-variant data loss (SVDL) features are extracted from the images reconstructed from limited angle data. The regression models linear regression (LR), multilayer perceptron (MLP), and reduced-error pruning tree (REPTree) are applied to predict artifact images.Results: REPTree learns artifacts best and reaches the smallest root-mean-square error (RMSE) of 29 HU for the Shepp-Logan phantom in a parallel-beam study. Further experiments demonstrate that the MVM and Hessian features complement each other, whereas the Laplacian feature is redundant in the presence of MVM. In fan-beam, the SVDL features are also beneficial. A preliminary experiment on clinical data in a fan-beam study demonstrates that REPTree can reduce some artifacts for clinical data. However, it is not sufficient as a lot of incorrect pixel intensities still remain in the estimated reconstruction images.Conclusion: REPTree has the best performance on learning artifacts in limited angle tomography compared with LR and MLP. The features of MVM, Hessian, and SVDL are beneficial for artifact prediction in limited angle tomography. Preliminary experiments on clinical data suggest that the investigation on more features is necessary for clinical applications of REPTree. [ABSTRACT FROM AUTHOR]

Abstract In this contribution, large SiPM arrays (8 × 8 elements of 6 × 6 mm2 each) are processed with an ASIC-based readout and coupled to a monolithic LYSO crystal to explore their potential use for TOF-PET applications. The aim of this work is to study the integration of this technology in the development of clinical PET systems reaching sub-300 ps coincidence resolving time (CRT). The SiPM and readout electronics have been evaluated first, using a small size 1.6 mm (6 mm height) crystal array (32 × 32 elements). All pixels were well resolved and they exhibited an energy resolution of about 20% (using Time-over-Threshold methods) for the 511 keV photons. Several parameters have been scanned to achieve the optimum readout system performance, obtaining a CRT as good as 330 ± 5 ps FWHM. When using a black-painted monolithic block, the spatial resolution was measured to be on average 2.6 ± 0.5 mm, without correcting for the source size. Energy resolution appears to be slightly above 20%. CRT measurements with the monolithic crystal detector were also carried out. Preliminary results as well as calibration methods specifically designed to improve timing performance, are being analyzed in the present manuscript. Highlights • PET detectors suitable for Limited Angle Tomography (LAT). • TOF-PET detectors based on monolithic scintillators coupled to SiPM photosensors. • Integration of an ASIC-based readout in PET systems. • 330 ps FWHM timing resolution using 5 × 5 cm2 detector blocks. [ABSTRACT FROM AUTHOR]

In the paper we present a novel method of extended depth-of-field limited-angle optical diffraction tomography, in which the change of a focal plane position is performed with a liquid focus-tunable lens. One sinogram is acquired for each state of a focus-tunable lens. After acquisition process is complete, all sinograms are independently reconstructed and stitched to form the final tomographic reconstruction. The presented solution effectively extends the applicability of the Rytov approximation to relatively thick samples and provides uniform resolution of 3D tomographic reconstructions. The method is also combined with Generalized Total Variation Iterative Constraint algorithm, which minimizes distortion of the results due to the limited angular range of acquired projections. The combined solution is dedicated to investigation of transparent and semi-transparent biological micro-structures, like cells and tissue slices. [ABSTRACT FROM AUTHOR]

Limited angle tomography is a challenging task in medical imaging. Due to practical limitations during the image acquisition, the sinogram is recorded incompletely and thus the quality of the reconstruction is deteriorated by streak artifacts. These artifacts are characterized by fast changes of the local intensity gradients and increase the total variation (TV). Generally, an energy functional is optimized which leads to a minimized Total Variation Minimization (TVM). As an outcome, noise and artifacts are reduced while edges are preserved. Anyway, often the orientation of the streak artifacts is not considered at all. Therefore, anisotropic regularization is used to reduce noise and distortions under specific directions.

We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in Rn. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by general pseudodifferential operators. We provide microlocal characterizations of visible and added singularities in RN and define modified versions of reconstruction operators that do not generate added artifac ts. We calculate the symbol of our general reconstruction operators as pseudodifferential operators and provide conditions for the filters under which the reconstruction operators are elliptic for the v isible singularities. If the filters are chosen according to those conditions, we show that almost all visible singularities can be recovered reliably. Our work generalizes the results for the classical line transforms in R² and the classical reconstruction operators (that use specific filters). In our proofs, we employ a general paradigm that is based on the calculus of Fourier integral operators. Since this technique does not rely on explicit expressions of the reconstruction operators, it enables us to anal yze more general imaging situations. [ABSTRACT FROM AUTHOR]

We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in Rn. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by general pseudodifferential operators. We provide microlocal characterizations of visible and added singularities in RN and define modified versions of reconstruction operators that do not generate added artifac ts. We calculate the symbol of our general reconstruction operators as pseudodifferential operators and provide conditions for the filters under which the reconstruction operators are elliptic for the v isible singularities. If the filters are chosen according to those conditions, we show that almost all visible singularities can be recovered reliably. Our work generalizes the results for the classical line transforms in R² and the classical reconstruction operators (that use specific filters). In our proofs, we employ a general paradigm that is based on the calculus of Fourier integral operators. Since this technique does not rely on explicit expressions of the reconstruction operators, it enables us to anal yze more general imaging situations. [ABSTRACT FROM AUTHOR]

We develop a paradigm using microlocal analysis that allows one to characterize the visible and added singularities in a broad range of incomplete data tomography problems. We give precise characterizations for photoacoustic and thermoacoustic tomography and sonar, and provide artifact reduction strategies. In particular, our theorems show that it is better to arrange sonar detectors so that the boundary of the set of detectors does not have corners and is smooth. To illustrate our results, we provide reconstructions from synthetic spherical mean data as well as from experimental photoacoustic data. [ABSTRACT FROM AUTHOR]

Purpose: A first-arrival travel-time sound speed algorithm presented by Tarantola [Inverse Problem Theory and Methods for Model Parameter Estimation (SIAM, Philadelphia, PA, 2005)] is adapted to the medical ultrasonics setting. Through specification of a covariance matrix for the object model, the algorithm allows for natural inclusion of physical a priori information of the object. The algorithm's ability to accurately and robustly reconstruct a complex sound speed distribution is demonstrated on simulation and experimental data using a limited aperture. Methods: The algorithm is first demonstrated generally in simulation with a numerical breast phantom imaged in different geometries. As this work is motivated by the authors' limited aperture dual sided ultrasound breast imaging system, experimental data are acquired with a Verasonics system with dual, 128 element, linear L7-4 arrays. The transducers are automatically calibrated for usage in the eikonal forward model. A priori information such as knowledge of correlated regions within the object is obtained via segmentation of B-mode images generated from synthetic aperture imaging. Results: As one illustration of the algorithm's facility for inclusion of a priori information, physically grounded regularization is demonstrated in simulation. The algorithm's practicality is then demonstrated through experimental realization in limited aperture cases. Reconstructions of sound speed distributions of various complexity are improved through inclusion of a priori information. The sound speed maps are generally reconstructed with accuracy within a few m/s. Conclusions: This paper demonstrates the ability to form sound speed images using two opposed commercial linear arrays to mimic ultrasound image acquisition in the compressed mammographic geometry. The ability to create reasonably good speed of sound images in the compressed mammographic geometry allows images to be readily coregistered to tomosynthesis image volumes for breast cancer detection and characterization studies. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a two-step strategy to provide a quality-predictable image reconstruction. A Pre-computed Back Projection based Penalized-Likelihood (PPL) method is proposed in the strategy to generate consistent image quality. To solve PPL efficiently, relaxed Ordered Subsets (OS) is applied. A training sets based evaluation is performed to quantify the effect of the undetermined parameters in OS, which lets the results as consistent as possible with the theoretical one. [Copyright &y& Elsevier]

We analyze representative ill-posed scenarios of tomographic PIV (particle image velocimetry) with a focus on conditions for unique volume reconstruction. Based on sparse random seedings of a region of interest with small particles, the corresponding systems of linear projection equations are probabilistically analyzed in order to determine: (i) the ability of unique reconstruction in terms of the imaging geometry and the critical sparsity parameter, and (ii) sharpness of the transition to non-unique reconstruction with ghost particles when choosing the sparsity parameter improperly. The sparsity parameter directly relates to the seeding density used for PIV in experimental fluids dynamics that is chosen empirically to date. Our results provide a basic mathematical characterization of the PIV volume reconstruction problem that is an essential prerequisite for any algorithm used to actually compute the reconstruction. Moreover, we connect the sparse volume function reconstruction problem from few tomographic projections to major developments in compressed sensing. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the reconstruction problem of limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, electron microscopy, etc. Since the acquired tomographic data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction methods, e.g. filtered backprojection (FBP), do not perform well in such situations. To stabilize the reconstruction procedure additional prior knowledge about the unknown object has to be integrated into the reconstruction process. In this work, we propose the use of the sparse regularization technique in combination with curvelets. We argue that this technique gives rise to an edge-preserving reconstruction. Moreover, we show that the dimension of the problem can be significantly reduced in the curvelet domain. To this end, we give a characterization of the kernel of the limited angle Radon transform in terms of curvelets and derive a characterization of solutions obtained through curvelet sparse regularization. In numerical experiments, we will show that the theoretical results directly translate into practice and that the proposed method outperforms classical reconstructions. [Copyright &y& Elsevier]

We introduce a cone-beam computed tomography with insufficient projections obtained from a limited-angle scan, the socalled digital tomosynthesis, and demonstrate its feasibility for the industrial applications by implementing to the reconstruction of internal slice images of a multilayer printed circuit board. The image reconstruction algorithm is based on the filtered-backprojection approach for the cone-beam geometry with isocentric linear motion in the scanning trajectory of the X-ray source and imaging detector pair. Although the slice image reconstructed at the plane of interest was affected by the structures outside, we could reconstruct the plane of interest with the total scan angle less than 20 degrees and 11 projections. The digital tomosynthesis is expected to be practical for industrial tomography restricted to the limited-angle scan. [ABSTRACT FROM AUTHOR]

Abstract: We describe a cone-beam computed tomography with insufficient projections obtained from a limited angle scan, the so-called cone-beam digital tomosynthesis. Digital tomosynthesis produces cross-sectional images parallel to the axis of rotation from a series of projection images acquired from a planar detector. The image reconstruction algorithm is based on the cone-beam filtered backprojection method. To suppress the out-of-plane artifacts due to the incomplete sampling over a limited angular range, we applied an apodizing filter in the depth direction. We applied the digital tomosynthesis technique to a multilayer printed circuit board possessing thin slab geometry and evaluated its performance with respect to various operation parameters, such as the total scan angle, the step angle and the number of projection images used for reconstruction. The results showed that the image quality of digital tomosynthesis reconstructed for the total scan angle greater than 60 degrees with a step angle as narrow as possible exhibited that it was comparable to that of the computed tomography. The digital tomosynthesis technique is expected to be practical for extracting internal cross-sectional views, parallel to the scan direction, of objects with thin slab geometry. [Copyright &y& Elsevier]

Abstract: This paper aims to reduce the problems of incomplete data in computed tomography, which happens frequently in medical image process and analysis, e.g., when the high-density region of objects can only be penetrated by X-rays at a limited angular range. As the projection data are available only in an angular range, the incomplete data problem can be attributed to the limited angle problem, which is an ill-posed inverse problem. Image reconstruction based on total variation (TV) reduces the problem and gives better performance on edge-preserving reconstruction; however, the artificial parameter can only be determined through considerable experimentation. In this paper, an effective TV objective function is proposed to reduce the inverse problem in the limited angle tomography. This novel objective function provides a robust and effective reconstruction without any artificial parameter in the iterative processes, using the TV as a multiplicative constraint. The results demonstrate that this reconstruction strategy outperforms some previous ones. [Copyright &y& Elsevier]

Abstract: This paper describes a study investigating the potential of tomosynthesis as a post check-in baggage scanning system. A laboratory system has been constructed consisting of a moveable source and detector, arranged around a mini 90° bend conveyor system, from which multiple projection images can be collected. Simulation code has been developed to allow the optimum source and detector positions to be determined. Reconstruction methods are being developed to modify the Shift-And-Add (SAA) algorithm to accommodate the non-typical imaging geometry. [Copyright &y& Elsevier]

Abstract: The effective high neutron scattering absorption coefficient of hydrogen (48.5cm2/g) due to the scattering allows neutrons to reveal hydrocarbon structures with more contrast than X-rays, but at the same time limits the sample size and thickness that can be investigated. Many planar shaped objects, particularly wood samples, are sufficiently thin to allow thermal neutrons to transmit through the sample in a direction perpendicular to the planar face but not in a parallel direction, due to increased thickness. Often, this is an obstacle that prevents some tomographic reconstruction algorithms from obtaining desired results because of inadequate information or presence of distracting artifacts due to missing projections. This can be true for samples such as the distribution of glue in glulam (boards of wooden layers glued together), or the course of partially visible annual rings in trees where the features of interest are parallel to the planar surface of the sample. However, it should be possible to study these features by rotating the specimen within a limited angular range. In principle, this approach has been shown previously in a study with fast neutrons . A study of this kind was performed at the Antares facility of FRM II in Garching with a 2.6×107/cm2 s thermal neutron beam. The limit of penetration was determined for a wooden step wedge carved from a 2cm×4cm block of wood in comparison to other materials such as heavy metals and Lucite as specimens rich in hydrogen. The depth of the steps was 1cm, the height 0.5cm. The annual ring structures were clearly detectable up to 2cm thickness. Wooden specimens, i.e. shivers, from a sunken old ship have been subjected to tomography. Not visible from the outside, clear radial structures have been found that are typical for certain kinds of wood. This insight was impaired in a case where the specimen had been soaked with ethylene glycol. In another large sample study, a planar board made of glulam has been studied to show the glued layers. This study shows not only the limits of penetration in wood but also demonstrates access to structures perpendicular to the surface in larger planar objects by tomography with fast neutrons, even with incomplete sets of projection data that covers an angular range of only 90° or even 60°. [Copyright &y& Elsevier]

Abstract: We assess the performance of filtered backprojection (FBP), the simultaneous algebraic reconstruction technique (SART) and the maximum likelihood (ML) algorithm for digital breast tomosynthesis (DBT) under variations in key imaging parameters, including the number of iterations, number of projections, angular range, initial guess, and radiation dose. This is the first study to compare these algorithms for the application of DBT. We present a methodology for the evaluation of DBT reconstructions, and use it to conduct preliminary experiments investigating trade-offs between the selected imaging parameters. This investigation includes trade-offs not previously considered in the DBT literature, such as the use of a stationary detector versus a C-arm imaging geometry. A real breast CT volume serves as a ground truth digital phantom from which to simulate X-ray projections under the various acquisition parameters. The reconstructed image quality is measured using task-based metrics, namely signal CNR and the AUC of a Channelised Hotelling Observer with Laguerre–Gauss basis functions. The task at hand is the detection of a simulated mass inserted into the breast CT volume. We find that the image quality in limited view tomography is highly dependent on the particular acquisition and reconstruction parameters used. In particular, we draw the following conclusions. First, we find that optimising the FBP filter design and SART relaxation parameter yields significant improvements in reconstruction quality from the same projection data. Second, we show that the convergence rate of the maximum likelihood algorithm, optimised with paraboloidal surrogates and conjugate gradient ascent (ML-PSCG), can be greatly accelerated using view-by-view updates. Third, we find that the optimal initial guess is algorithm dependent. In particular, we obtained best results with a zero initial guess for SART, and an FBP initial guess for ML-PSCG. Fourth, when the exposure per view is constant, increasing the total number of views within a given angular range improves the reconstruction quality, albeit with diminishing returns. When the total dose of all views combined is constant, there is a trade-off between increased sampling using a larger number of views and increased levels of quantum noise in each view. Fifth, we do not observe significant differences when testing various access ordering schemes, presumably due to the limited angular range of DBT. Sixth, we find that adjusting the z-resolution of the reconstruction can improve image quality, but that this resolution is best adjusted by using post-reconstruction binning, rather than by declaring lower-resolution voxels. Seventh, we find that the C-arm configuration yields higher image quality than a stationary detector geometry, the difference being most outspoken for the FBP algorithm. Lastly, we find that not all prototype systems found in the literature are currently being run under the best possible system or algorithm configurations. In other words, the present study demonstrates the critical importance (and reward) of using optimisation methodologies such as the one presented here to maximise the DBT reconstruction quality from a single scan of the patient. [Copyright &y& Elsevier]

Abstract: This paper focuses on the problem of incomplete data in the applications of the circular cone-beam computed tomography. This problem is frequently encountered in medical imaging sciences and some other industrial imaging systems. For example, it is crucial when the high density region of objects can only be penetrated by X-rays in a limited angular range. As the projection data are only available in an angular range, the above mentioned incomplete data problem can be attributed to the limited angle problem, which is an ill-posed inverse problem. This paper reports a modified total variation minimisation method to reduce the data insufficiency in tomographic imaging. This proposed method is robust and efficient in the task of reconstruction by showing the convergence of the alternating minimisation method. The results demonstrate that this new reconstruction method brings reasonable performance. [ABSTRACT FROM AUTHOR]

Abstract: This paper studied incomplete data problems of computed tomography that frequently occur in medical or industrial imaging, for example, when the high-density region of objects can only be penetrated by X-rays at a limited angular range. When projection data are available only in an angular range, the incomplete data problem can be attributed to the limited angle problem, which is a severely ill-posed inverse problem. In this paper, a numerical method for the treatment of inverse problems based on an adaptive wavelet-Galerkin method is introduced and investigated. The paper focuses especially on how to avoid inverse crimes in numerical simulations. The method used here combines numerical simplicity and characteristics of adapting to the unknown smoothness of a reconstructed image, which leads to significant reduction in the computational cost. The reconstruction strategy has a comparable performance with a significant reduction in computational time. [Copyright &y& Elsevier]

Filtered back-projection and weighted back-projection have long been the methods of choice within the electron microscopy community for reconstructing the structure of macromolecular assemblies from electron tomography data. Here, we describe electron lambda-tomography, a reconstruction method that enjoys the benefits of the above mentioned methods, namely speed and ease of implementation, but also addresses some of their shortcomings. In particular, compared to these standard methods, electron lambda-tomography is less sensitive to artifacts that come from structures outside the region that is being reconstructed, and it can sharpen boundaries. [ABSTRACT FROM AUTHOR]

A new iterative algorithm of tomographic reconstruction of objects on the basis of projection data available in a limited range of angles only is proposed. The algorithm is based on calculating artificial projections in those directions where projection data are unavailable. By means of numerical simulations, it is verified that the algorithm developed ensured high quality of reconstruction up to the angular interval of 45–60°. [ABSTRACT FROM AUTHOR]

A novel approach to the X-ray tomography problem with sparse projection data is proposed. Nonnegativity of the X-ray attenuation coefficient is enforced by modelling it as max{F(x), 0}, where F is a smooth function. The function F is computed as the equilibrium solution of a nonlinear evolution equation analogous to the equations used in level set methods. The reconstruction algorithm is applied to (a) simulated full and limited angle projection data of the Shepp-Logan phantom with sparse angular sampling and (b) measured limited angle projection data of in vitro dental specimens. The results are significantly better than those given by traditional backprojection-based approaches, and similar in quality (but faster to compute) compared to the algebraic reconstruction technique. [ABSTRACT FROM AUTHOR]