Direct Fourier Reconstruction Research Articles

A GPU acceleration of 3-D Fourier reconstruction in cryo-EM

Cryo-electron microscopy is a popular method for macromolecules structure determination. Reconstruction of a 3-D volume from raw data obtained from a microscope is highly computationally demanding. Thus, acceleration of the reconstruction has a great practical value. In this article, we introduce a novel graphics processing unit (GPU)-friendly algorithm for direct Fourier reconstruction, one of the main computational bottlenecks in the 3-D volume reconstruction pipeline for some experimental cases (particularly those with a large number of images and a high internal symmetry). Contrary to the state of the art, our algorithm uses a gather memory pattern, improving cache locality and removing race conditions in parallel writing into the 3-D volume. We also introduce a finely tuned CUDA implementation of our algorithm, using auto-tuning to search for a combination of optimization parameters maximizing performance on a given GPU architecture. Our CUDA implementation is integrated in widely used software Xmipp, version 3.19, reaching 11.4× speedup compared to the original parallel CPU implementation using GPU with comparable power consumption. Moreover, we have reached 31.7× speedup using four GPUs and 2.14×–5.96× speedup compared to optimized GPU implementation based on a scatter memory pattern.

Tomographic imaging of reacting flows in 3D by laser absorption spectroscopy

This paper describes the development of an infrared laser absorption tomography system for the 3D volumetric imaging of chemical species and temperature in reacting flows. The system is based on high-resolution near-infrared tunable diode laser absorption spectroscopy (TDLAS) for the measurement of water vapour above twin, mixed fuel gas burners arranged with an asymmetrical output. Four parallel laser beams pass through the sample region and are rotated rapidly in a plane to produce a wide range of projection angles. A rotation of 180° with 0.5° sampling was achieved in 3.6 s. The effects of changes to the burner fuel flow were monitored in real time for the 2D distributions. The monitoring plane was then moved vertically relative to the burners enabling a stack of 2D images to be produced which were then interpolated to form a 3D volumetric image of the temperature and water concentrations above the burners. The optical transmission of each beam was rapidly scanned around 1392 nm and the spectrum was fitted to find the integrated absorbance of the water transitions and although several are probed in each scan, two of these transitions possess opposite temperature dependencies. The projections of the integrated absorbances at each angle form the sinogram from which the 2D image of integrated absorbance of each line can be reconstructed by the direct Fourier reconstruction based on the Fourier slice theorem. The ratio of the integrated absorbances of the two lines can then be related to temperature alone in a method termed, two-line thermometry. The 2D temperature distribution obtained was validated for pattern and accuracy by thermocouple measurements. With the reconstructed temperature distribution, the temperature-dependent line strengths could be determined and subsequently the concentration distribution of water across the 2D plane whilst variations in burner condition were carried out. These results show that the measurement system based on TDLAS can be used for 2D temporal or 3D volume imaging of temperature and chemical species concentration in reacting flows.

A fast iterative convolution weighting approach for gridding-based direct Fourier three-dimensional reconstruction with correction for the contrast transfer function

We describe a fast and accurate method for the reconstruction of macromolecular complexes from a set of projections. Direct Fourier inversion (in which the Fourier Slice Theorem plays a central role) is a solution for dealing with this inverse problem. Unfortunately, the set of projections provides a non-equidistantly sampled version of the macromolecule Fourier transform in the single particle field (and, therefore, a direct Fourier inversion) may not be an optimal solution. In this paper, we introduce a gridding-based direct Fourier method for the three-dimensional reconstruction approach that uses a weighting technique to compute a uniform sampled Fourier transform. Moreover, the contrast transfer function of the microscope, which is a limiting factor in pursuing a high resolution reconstruction, is corrected by the algorithm. Parallelization of this algorithm, both on threads and on multiple CPU's, makes the process of three-dimensional reconstruction even faster. The experimental results show that our proposed gridding-based direct Fourier reconstruction is slightly more accurate than similar existing methods and presents a lower computational complexity both in terms of time and memory, thereby allowing its use on larger volumes. The algorithm is fully implemented in the open-source Xmipp package and is downloadable from http://xmipp.cnb.csic.es.

A Fourier reconstruction algorithm in π-scheme short-scan SPECT

In this paper, an approximate analytical algorithm in the form of direct Fourier reconstruction is obtained for the reconstruction of data functions arisen from π -scheme short-scan single-photon emission computed tomography(SPECT) with uniform attenuation, and the modified central slice theorem is developed. Numerical simulations are conducted to demonstrate the effectiveness of the developed method.

Fast direct fourier reconstruction of radial and PROPELLER MRI data using the chirp transform algorithm on graphics hardware

To develop and test a new algorithm for fast direct Fourier transform (DrFT) reconstruction of MR data on non-Cartesian trajectories composed of lines with equally spaced points. The DrFT, which is normally used as a reference in evaluating the accuracy of other reconstruction methods, can reconstruct images directly from non-Cartesian MR data without interpolation. However, DrFT reconstruction involves substantially intensive computation, which makes the DrFT impractical for clinical routine applications. In this article, the Chirp transform algorithm was introduced to accelerate the DrFT reconstruction of radial and Periodically Rotated Overlapping ParallEL Lines with Enhanced Reconstruction (PROPELLER) MRI data located on the trajectories that are composed of lines with equally spaced points. The performance of the proposed Chirp transform algorithm-DrFT algorithm was evaluated by using simulation and in vivo MRI data. After implementing the algorithm on a graphics processing unit, the proposed Chirp transform algorithm-DrFT algorithm achieved an acceleration of approximately one order of magnitude, and the speed-up factor was further increased to approximately three orders of magnitude compared with the traditional single-thread DrFT reconstruction. Implementation the Chirp transform algorithm-DrFT algorithm on the graphics processing unit can efficiently calculate the DrFT reconstruction of the radial and PROPELLER MRI data.

A Fourier reconstruction algorithm with constant attenuation compensation using 180° acquisition data for SPECT

In this paper, we develop an approximate analytical reconstruction algorithm that compensates for uniform attenuation in 2D parallel-beam SPECT with a 180° acquisition. This new algorithm is in the form of a direct Fourier reconstruction. The complex variable central slice theorem is used to derive this algorithm. The image is reconstructed with the following steps: first, the attenuated projection data acquired over 180° are extended to 360° and the value for the uniform attenuator is changed to a negative value. The Fourier transform (FT) of the image in polar coordinates is obtained from the Fourier transform of an analytic function interpolated from an extension of the projection data according to the complex central slice theorem. Finally, the image is obtained by performing a 2D inverse Fourier transform. Computer simulations and comparison studies with a 360° full-scan algorithm are provided.

Direct Fourier Tomographic Reconstruction Image-To-Image Filter

We present an open-source ITK implementation of a direct Fourier method for tomographic reconstruction, applicable to parallel-beam x-ray images. Direct Fourier reconstruction makes use of the central-slice theorem to build a polar 2D Fourier space from the 1D transformed projections of the scanned object, that is resampled into a Cartesian grid. Inverse 2D Fourier transform eventually yields the reconstructed image. Additionally, we provide a complex wrapper to the BSplineInterpolateImageFunction to overcome ITK’s current lack for image interpolators dealing with complex data types. A sample application is presented and extensively illustrated on the Shepp-Logan head phantom. We show that appropriate input zeropadding and 2D-DFT oversampling rates together with radial cubic b-spline interpolation improve 2D-DFT interpolation quality and are efficient remedies to reduce reconstruction artifacts.

A triple-network tricontinuous cubicliquid crystal

Soft matter such as surfactant-water systems, block copolymers or liquid crystals can form periodic structures on nanometre to micrometre scales. This property can be used for templating nanoporous ceramics, surface patterning for electronic devices, or generation of photonic materials. Much attention has been paid to structures appearing between the layer and cylinder phases, the three so-called bicontinuous cubic phases. These are formed by two continuous interpenetrating networks of channels. In this article we describe a related phase, which has the first reported structure consisting of three interpenetrating infinite networks. It is a thermotropic (solvent-free) liquid crystal of cubic symmetry Im3m. The structure is one of the most complex in liquid crystals, and is determined by direct Fourier reconstruction of electron density. We discuss the possible rationale for the existence of such a phase, its structural relationship with the bicontinuous phases, and its position in the phase diagram.

Cone-beam and fan-beam image reconstruction algorithms based on spherical and circular harmonics

A cone-beam image reconstruction algorithm using spherical harmonic expansions is proposed. The reconstruction algorithm is in the form of a summation of inner products of two discrete arrays of spherical harmonic expansion coefficients at each cone-beam point of acquisition. This form is different from the common filtered backprojection algorithm and the direct Fourier reconstruction algorithm. There is no re-sampling of the data, and spherical harmonic expansions are used instead of Fourier expansions. As a special case, a new fan-beam image reconstruction algorithm is also derived in terms of a circular harmonic expansion. Computer simulation results for both cone-beam and fan-beam algorithms are presented for circular planar orbit acquisitions. The algorithms give accurate reconstructions; however, the implementation of the cone-beam reconstruction algorithm is computationally intensive. A relatively efficient algorithm is proposed for reconstructing the central slice of the image when a circular scanning orbit is used.

3D-FRP: direct Fourier reconstruction with Fourier reprojection for fully 3-D PET

The direct Fourier method (DFM) for three-dimensional (3-D) reconstruction of a 3-D volume is based on the relationship between the 3-D Fourier transform (FT) of the volume and the two-dimensional (2-D) FT of a parallel-ray projection of the volume. The direct Fourier method has the potential for very fast reconstruction, but a straightforward implementation of the method leads to unsatisfactory results. This paper presents an implementation of the direct Fourier method for fully 3-D positron emission tomography (PET) data with incomplete oblique projections (3D-FRP) that gives results as good as, or better than, those of a much slower 3-D filtered backprojection method (3DRP), and in the same time as a fast but less accurate method using Fourier rebinning (FORE) followed by slice-by-slice reconstruction. In common with 3DRP, 3D-FRP is based on a discretization of an inversion formula, so it is geometrically accurate for large oblique angles, and both methods involve reprojection of an initial image. The critical two steps in the 3D-FRP method are the estimations of the samples of the 3-D transform of the image from the samples of the 2-D transforms of the projections on the planes through the origin of Fourier space, and vice versa for reprojection. These steps use a gridding strategy, combined with new approaches for weighting in the transform and image domains. The authors' experimental results confirm that good image accuracy can be achieved together with a short reconstruction time.

The projection map interpolation in parallel beam gamma-ray computed tomography

In order to avoid the shortcomings of the interpolation step conducted in the Fourier space, an interpolation in the object space is proposed for direct Fourier reconstruction. To implement this idea, the central slice theorem is reconsidered to calculate the density distribution of point weights placed over a square grid. An algorithm is developed and tested on the projection data from a CT scanner with a Cs-137 gamma source. The proposed algorithm provided a better resolution as compared to the standard algorithm.

Data-parallel tomographic reconstruction: A comparison of filtered backprojection and direct Fourier reconstruction

We consider the parallelization of two standard 2D reconstruction algorithms, filtered backprojection and direct Fourier reconstruction, using the data-parallel programming style. The algorithms are implemented on a Connection Machine CM-5 with 16 processors and a peak performance of 2 Gflop/s.

Direct-Fourier reconstruction in tomography and synthetic aperture radar

We investigate the use of direct-Fourier (DF) image reconstruction in computed tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolution-backprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR. We show that the CBP algorithm is equivalent to DF reconstruction using a Jacobian-weighted two-dimensional periodic sinc-kernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms using optimal interpolators may surpass CBP in image quality. We consider use of two types of DF interpolation: a windowed sinc kernel, and the least-squares optimal Yen interpolator. Simulations show that reconstructions using the Yen interpolator do not possess the expected visual quality, because of regularization needed to preserve numerical stability. Next, we show that with a concentric-squares sampling scheme, DF interpolation can be performed accurately and efficiently, producing imagery that is superior to that obtainable by other algorithms. In the case of SAR, we show that the DF method performs very well with interpolators of low complexity. We also study DF reconstruction in SAR for trapezoidal grids. We conclude that the success of the DF method in SAR imaging is due to the nearly Cartesian shape of the sampling grid. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 1–13, 1998

An alternative approach to direct Fourier reconstruction in parallel-beam tomography

In this work, we elaborate on some of the features of the direct Fourier reconstruction method, which has not spurred much interest in the literature. We searched for the possibility of an interpolation in the object space prior to Fourier transformation of the projection. To implement this idea, the central slice theorem based on the finite Fourier transform is reconsidered for a density distribution of point weights placed over a square grid. Some industrial applications carried out on a scanner with a Cs-137 source are given to demonstrate and verify our algorithm. It is observed that the image resolution is improved over that obtainable by the application of the basic direct Fourier transform method.

A performance study of 3D reconstruction algorithms for positron emission tomography

The authors investigate the statistical and systematic accuracy of five three-dimensional reconstruction algorithms for multi-ring PET scanners operated without septa: the reprojection method, the direct Fourier reconstruction, the FAVOR algorithm, and the single-slice and multi-slice rebinning algorithms. Simulated data of a uniform cylinder, of Gaussian sources, and of spherical sources are used to compare respectively the noise properties, the modulation transfer function, and the recovery coefficients of the algorithms. Brain scans reconstructed with the different algorithms are compared by calculating the linear regression of the mean values within regions of interest. The most significant observations are a slight loss of transaxial resolution with the reprojection algorithm in the external slices of the scanner, and increased noise in the images reconstructed using multi-slice rebinning.

On the possibility of direct Fourier reconstruction from divergent-beam projections

A great number of tomographic systems, especially those equipped with fast data acquisition techniques, scan the objects investigated by divergent (fan) X-ray beams. Fan-beam projection data require special reconstruction techniques to be implemented. Among reconstruction techniques from parallel projection data, the direct Fourier method (DFM) proved to be one of the most promising ones, especially for high-speed image reconstruction in the high-end 3-D and dynamic tomographic systems. The goal of this work is to answer the topical question: how would direct use of the DFM influence the quality of image reconstruction from the fan-beam data? The formula describing the error caused by such an approximation is derived. The conclusions deduced from the formula are confirmed by computer simulations. The boundary values of data acquisition geometry parameters have been estimated for the case of using the DFM without recalculating the fan-beam data.

A study of reconstruction algorithms and filters for an industrial X-ray tomography system

Computed tomography (CT) has been incorporated in an industrial Diode-Array Digital Radiography (DADR) system. An input data size of 512 pixel points×400 projections yielded a 400×400 output image matrix. Reconstruction algorithms used are the filtered backprojection (FBP) and the direct Fourier reconstruction (DFR). Various filters were used in the FBP reconstruction process and their effects on image quality were evaluated. A spatial resolution of 100 μm was measured with a block of plates and a minimum detectable feature size in the range of 10–100 μm was measured using thin wires. Industrial specimens imaged have included ceramic samples, ball bearings and integrated circuits. A number of engineering problems have been solved, such as adjustment of the X-ray source, centering of the rotator spindle in the view field and beam-hardening corrections.

Interpolation from Samples on a Linear Spiral Scan

An interpolation method useful for reconstructing an image from its Fourier plane samples on a linear spiral scan trajectory is presented. This kind of sampling arises in NMR imaging. We first present a theorem that enables exact interpolation from spiral samples to a Cartesian lattice. We then investigate two practical implementations of the theorem in which a finite number of interpolating points are used to calculate the value at a new point. Our experimental results confirm the theorem's validity and also demonstrate that both practical implementations yield very good reconstructions. Thus, the theorem and/or its practical implementations suggest the possibility of using direct Fourier reconstruction from linear spiral-scan NMR imaging.

Direct Fourier Reconstruction in Fan-Beam Tomography

We consider the problem of reconstructing tomographic imagery from fan-beam projections using the direct Fourier method (DFM). Previous DFM reconstructions from parallel-beam projections produced images of quality comparable to that of filtered convolution back-projection. Moreover, the number of operations using DFM in the parallel-beam case is proportional to N2 log N versus N3 for back projection [3]. The fan-beam case is more complicated because additional interpolation of the nonuniformly spaced rebinned data is required. We derive bounds on the detector spacing in fan-beam CT that enable direct Fourier reconstruction and describe the full algorithm necessary for processing the fan-beam data. The feasibility of the method is demonstrated with an example. A key result of this paper is that high-quality imagery can be reconstructed from fan-beam data using the DFM in 0 (N2 log N) operations.

Authors' reply to "Comments on 'direct Fourier reconstruction in computer tomography'"

In their comments, Fan and Sanz discuss an interpolation formula used by us in our papers on direct Fourier transform tomography [1], [2]. They raise questions relating to: 1) the relation between our work and others'; and 2) the optimality of the interpolation procedure in direct Fourier reconstruction in CT. We show that: 1) most of their points are irrelevant; and 2) other points follow from an error which was already corrected by the authors in [3], prior to the publication of the comments.