Three-dimensional reconstruction from cone-beam data in O(N3logN) time

Low-dose cone-beam computed tomography reconstruction through a fast three-dimensional compressed sensing method based on the three-dimensional pseudo-polar fourier transform

Background:

Reconstruction of high quality two dimensional images from fan beam computed tomography (CT) with a limited number of projections is already feasible through Fourier based iterative reconstruction method. However, this article is focused on a more complicated reconstruction of three dimensional (3D) images in a sparse view cone beam computed tomography (CBCT) by utilizing Compressive Sensing (CS) based on 3D pseudo polar Fourier transform (PPFT).

Method:

In comparison with the prevalent Cartesian grid, PPFT re gridding is potent to remove rebinning and interpolation errors. Furthermore, using PPFT based radon transform as the measurement matrix, reduced the computational complexity.

Results:

In order to show the computational efficiency of the proposed method, we compare it with an algebraic reconstruction technique and a CS type algorithm. We observed convergence in <20 iterations in our algorithm while others would need at least 50 iterations for reconstructing a qualified phantom image. Furthermore, using a fast composite splitting algorithm solver in each iteration makes it a fast CBCT reconstruction algorithm. The algorithm will minimize a linear combination of three terms corresponding to a least square data fitting, Hessian (HS) Penalty and l1 norm wavelet regularization. We named it PP-based compressed sensing-HS-W. In the reconstruction range of 120 projections around the 360° rotation, the image quality is visually similar to reconstructed images by Feldkamp-Davis-Kress algorithm using 720 projections. This represents a high dose reduction.

Conclusion:

The main achievements of this work are to reduce the radiation dose without degrading the image quality. Its ability in removing the staircase effect, preserving edges and regions with smooth intensity transition, and producing high-resolution, low-noise reconstruction results in low-dose level are also shown.

An Exact and Fast CBCT Reconstruction via Pseudo-Polar Fourier Transform based Discrete Grangeat's Formula

The recent application of Fourier Based Iterative Reconstruction Method (FIRM) has made it possible to achieve high-quality 2D images from a fan beam Computed Tomography (CT) scan with a limited number of projections in a fast manner. The proposed methodology in this article is designed to provide 3D Radon space in linogram fashion to facilitate the use of FIRM with cone beam projections (CBP) for the reconstruction of 3D images in a sparse view angles Cone Beam CT (CBCT). For this reason, in the first phase, the 3D Radon space is generated using CBP data after discretization and optimization of the famous Grangeat's formula. The method used in this process involves fast Pseudo Polar Fourier transform (PPFT) based on 2D and 3D Discrete Radon Transformation (DRT) algorithms with no wraparound effects. In the second phase, we describe reconstruction of the objects with available Radon values, using direct inverse of 3D PPFT. The method presented in this section eliminates noises caused by interpolation from polar to Cartesian space and exhibits no thorn, V-shaped and wrinkle artifacts. This method reduces the complexity to for images of size n × n × n The Cone to Radon conversion (Cone2Radon) Toolbox in the first phase and MATLAB/ Python toolbox in the second phase were tested on three digital phantoms and experiments demonstrate fast and accurate cone beam image reconstruction due to proposed.

Recent Advances in X-ray Cone-beam Computed Laminography

X-ray computed tomography is an established volume imaging technique used routinely in medical diagnosis, industrial non-destructive testing, and a wide range of scientific fields. Traditionally, computed tomography uses scanning geometries with a single axis of rotation together with reconstruction algorithms specifically designed for this setup. Recently there has however been increasing interest in more complex scanning geometries. These include so called X-ray computed laminography systems capable of imaging specimens with large lateral dimensions or large aspect ratios, neither of which are well suited to conventional CT scanning procedures. Developments throughout this field have thus been rapid, including the introduction of novel system trajectories, the application and refinement of various reconstruction methods, and the use of recently developed computational hardware and software techniques to accelerate reconstruction times. Here we examine the advances made in the last several years and consider their impact on the state of the art.

Compensating the intensity fall-off effect in cone-beam tomography by an empirical weight formula

The Feldkamp-David-Kress (FDK) algorithm is widely adopted for cone-beam reconstruction due to its one-dimensional filtered backprojection structure and parallel implementation. In a reconstruction volume, the conspicuous cone-beam artifact manifests as intensity fall-off along the longitudinal direction (the gantry rotation axis). This effect is inherent to circular cone-beam tomography due to the fact that a cone-beam dataset acquired from circular scanning fails to meet the data sufficiency condition for volume reconstruction. Upon observations of the intensity fall-off phenomenon associated with the FDK reconstruction of a ball phantom, we propose an empirical weight formula to compensate for the fall-off degradation. Specifically, a reciprocal cosine can be used to compensate the voxel values along longitudinal direction during three-dimensional backprojection reconstruction, in particular for boosting the values of voxels at positions with large cone angles. The intensity degradation within the z plane, albeit insignificant, can also be compensated by using the same weight formula through a parameter for radial distance dependence. Computer simulations and phantom experiments are presented to demonstrate the compensation effectiveness of the fall-off effect inherent in circular cone-beam tomography.

Hyperfast parallel-beam and cone-beam backprojection using the cell general purpose hardware

Tomographic image reconstruction, such as the reconstruction of computed tomography projection values, of tomosynthesis data, positron emission tomography or SPECT events, and of magnetic resonance imaging data is computationally very demanding. One of the most time-consuming steps is the backprojection. Recently, a novel general purpose architecture optimized for distributed computing became available: the cell broadband engine (CBE). To maximize image reconstruction speed we modified our parallel-beam backprojection algorithm [two dimensional (2D)] and our perspective backprojection algorithm [three dimensional (3D), cone beam for flat-panel detectors] and optimized the code for the CBE. The algorithms are pixel or voxel driven, run with floating point accuracy and use linear (LI) or nearest neighbor (NN) interpolation between detector elements. For the parallel-beam case, 512 projections per half rotation, 1024 detector channels, and an image of size 512(2) was used. The cone-beam backprojection performance was assessed by backprojecting a full circle scan of 512 projections of size 1024(2) into a volume of size 512(3) voxels. The field of view was chosen to completely lie within the field of measurement and the pixel or voxel size was set to correspond to the detector element size projected to the center of rotation divided by square root of 2. Both the PC and the CBE were clocked at 3 GHz. For the parallel backprojection of 512 projections into a 512(2) image, a throughput of 11 fps (LI) and 15 fps (NN) was measured on the PC, whereas the CBE achieved 126 fps (LI) and 165 fps (NN), respectively. The cone-beam backprojection of 512 projections into the 512(3) volume took 3.2 min on the PC and is as fast as 13.6 s on the cell. Thereby, the cell greatly outperforms today's top-notch backprojections based on graphical processing units. Using both CBEs of our dual cell-based blade (Mercury Computer Systems) allows to 2D backproject 330 images/s and one can complete the 3D cone-beam backprojection in 6.8 s.

Approximate and exact cone-beam reconstruction with standard and non-standard spiral scanning

The long object problem is practically important and theoretically challenging. To solve the long object problem, spiral cone-beam CT was first proposed in 1991, and has been extensively studied since then. As a main feature of the next generation medical CT, spiral cone-beam CT has been greatly improved over the past several years, especially in terms of exact image reconstruction methods. Now, it is well established that volumetric images can be exactly and efficiently reconstructed from longitudinally truncated data collected along a rather general scanning trajectory. Here we present an overview of some key results in this area.

Volume fusion for two-circular-orbit cone-beam tomography

By using the Feldkamp-Davis-Kress (FDK) algorithm, we can efficiently produce a digital volume, called the FDK volume, from cone-beam data acquired along a circular scan orbit. Due to the insufficiency of the cone-beam data set, the FDK volume suffers from nonuniform reproduction exactness. Specifically, the midplane (on the scan-orbit plane) can be exactly reproduced, and the reproduction exactness of off-midplanes decreases as the distance from the midplane increases. We describe the longitudinal falling-off degradation by a hatlike function and the spatial distribution over the object domain by an exactness volume. With two orthogonal circular scan orbits, we can reconstruct two FDK volumes and generate two exactness volumes. We propose a volume fusion scheme to combine the two FDK volumes into a single volume. Let Va and Vb denote the two FDK volumes, let Ea and Eb denote the exactness volumes for orbits Gamma(a) and Gamma(b), respectively, then the volume fusion is defined by Vab=VaWa+VbWb, with Wa=Ea/(Ea+Eb) and Wb=1-Wa. In the result, the overall reproduction exactness of Vab is expected to outperform that of Va, or Vb, or (Va+Vb)/2. In principle, this volume-fusion scheme is applicable for general cone-beam tomography with multiple nonorthogonal and noncircular orbits.

Feldkamp-type VOI reconstruction from super-short-scan cone-beam data

Based on the fan-beam reconstruction formula recently developed by Noo et al. [Phys. Med. Biol. 47, 2525-2546 (2002)] we develop a Feldkamp-type algorithm for the reconstruction of a volume of interest (VOI) from super-short-scan data. With either a circular or spiral scanning locus in our VOI reconstruction scheme, we first estimate fan-beam data from cone-beam data using the popular "cosine correction" scheme, and perform reconstruction based on Noo's FBP-type fan-beam reconstruction. Our proposed algorithm is tested using the three-dimensional (3-D) Shepp-Logan phantom. The experimental results show that the new algorithm can be applied to multi-source 4-D CT with significantly superior temporal resolution and temporal consistency relative to the Katsevich algorithm, which is the state of the art for exact helical cone-beam reconstruction.

An alternative derivation of Katsevich's cone-beam reconstruction formula

In this paper an alternative derivation of Katsevich's cone-beam image reconstruction algorithm is presented. The starting point is the classical Tuy's inversion formula. After (i) using the hidden symmetries of the intermediate functions, (ii) handling the redundant data by weighting them, (iii) changing the weighted average into an integral over the source trajectory parameter, and (iv) imposing an additional constraint on the weighting function, a filtered backprojection reconstruction formula from cone beam projections is derived. The following features are emphasized in the present paper: First, the nontangential condition in Tuy's original data sufficiency conditions has been relaxed. Second, a practical regularization scheme to handle the singularity is proposed. Third, the derivation in the cone beam case is in the same fashion as that in the fan-beam case. Our final cone-beam reconstruction formula is the same as the one discovered by Katsevich in his most recent paper. However, the data sufficiency conditions and the regularization scheme of singularities are different. A detailed comparison between these two methods is presented.

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.

O(N3 log N) backprojection algorithm for the 3-D radon transform

We present a novel backprojection algorithm for three-dimensional (3-D) radon transform data that requires O(N3 log2 N) operations for reconstruction of an N x N x N volume from O(N2) plane-integral projections. Our algorithm uses a hierarchical decomposition of the 3-D radon transform to recursively decompose the backprojection operation. Simulations are presented demonstrating reconstruction quality comparable to the standard filtered backprojection, which requires O(N5) computations under the same circumstances.

A cone beam filtered backprojection (CB-FBP) reconstruction algorithm for a circle-plus-two-arc orbit

The circle-plus-arc orbit possesses advantages over other "circle-plus" orbits for the application of x-ray cone beam (CB) volume CT in image-guided interventional procedures requiring intraoperative imaging, in which movement of the patient table is to be avoided. A CB circle-plus-two-arc orbit satisfying the data sufficiency condition and a filtered backprojection (FBP) algorithm to reconstruct longitudinally unbounded objects is presented here. In the circle suborbit, the algorithm employs Feldkamp's formula and another FBP implementation. In the arc suborbits, an FBP solution is obtained originating from Grangeat's formula, and the reconstruction computation is significantly reduced using a window function to exclude redundancy in Radon domain. The performance of the algorithm has been thoroughly evaluated through computer-simulated phantoms and preliminarily evaluated through experimental data, revealing that the algorithm can regionally reconstruct longitudinally unbounded objects exactly and efficiently, is insensitive to the variation of the angle sampling interval along the arc suborbits, and is robust over practical x-ray quantum noise. The algorithm's merits include: only 1D filtering is implemented even in a 3D reconstruction, only separable 2D interpolation is required to accomplish the CB backprojection, and the algorithm structure is appropriate for parallel computation.

Cone-beam image reconstruction using spherical harmonics

Image reconstruction from cone-beam projections is required for both x-ray computed tomography (CT) and single photon emission computed tomography (SPECT). Grangeat's algorithm accurately performs cone-beam reconstruction provided that Tuy's data sufficiency condition is satisfied and projections are complete. The algorithm consists of three stages: (a) Forming weighted plane integrals by calculating the line integrals on the cone-beam detector, and obtaining the first derivative of the plane integrals (3D Radon transform) by taking the derivative of the weighted plane integrals. (b) Rebinning the data and calculating the second derivative with respect to the normal to the plane. (c) Reconstructing the image using the 3D Radon backprojection. A new method for implementing the first stage of Grangeat's algorithm was developed using spherical harmonics. The method assumes that the detector is large enough to image the whole object without truncation. Computer simulations show that if the trajectory of the cone vertex satisfies Tuy's data sufficiency condition, the proposed algorithm provides an exact reconstruction.

Cone-beam reconstruction by backprojection and filtering

A new analytical method for tomographic image reconstruction from cone-beam projections acquired on the source orbits lying on a cylinder is presented. By application of a weighted cone-beam backprojection, the reconstruction problem is reduced to an image-restoration problem characterized by a shift-variant point-spread function that is given analytically. Assuming that the source is relatively far from the imaged object, a formula for an approximate shift-invariant inverse filter is derived; the filter is presented in the Fourier domain. Results of numerical experiments with circular and helical orbits are considered.

Feldkamp-type cone-beam tomography in the wavelet framework

X-ray computed tomography (CT) is in transition from fan-beam to cone-beam geometry. For cone-beam volumetric imaging, reduction of radiation exposure remains an important issue. Because the wavelet approach was shown to be effective and flexible for two-dimensional (2-D) local region reconstruction, we are motivated to perform wavelet local CT in cone-beam geometry. In this paper, we formulate the Feldkamp cone-beam reconstruction from the wavelet perspective, derive both full-scan and half-scan Feldkamp-type formulas for either global or local reconstruction, and demonstrate the feasibility and utility in synthetic and real data. It is found that using the wavelet Feldkamp approach, a three-dimensional (3-D) region of interest (ROI) can be reconstructed with neither severe image artifacts nor any significant constant bias in our simulation and experiments.

A cone-beam reconstruction algorithm for circle-plus-arc data-acquisition geometry

In cone-beam computerized tomography (CT), projections acquired with the focal spot constrained on a planar orbit cannot provide a complete set of data to reconstruct the object function exactly. There are severe distortions in the reconstructed noncentral transverse planes when the cone angle is large. In this work, a new method is proposed which can obtain a complete set of data by acquiring cone-beam projections along a circle-plus-arc orbit. A reconstruction algorithm using this circle-plus-arc orbit is developed, based on the Radon transform and Grangeat's formula. This algorithm first transforms the cone-beam projection data of an object to the first derivative of the three-dimensional (3-D) Radon transform, using Grangeat's formula, and then reconstructs the object using the inverse Radon transform. In order to reduce interpolation errors, new rebinning equations have been derived accurately, which allows one-dimensional (1-D) interpolation to be used in the rebinning process instead of 3-D interpolation. A noise-free Defrise phantom and a Poisson noise-added Shepp-Logan phantom were simulated and reconstructed for algorithm validation. The results from the computer simulation indicate that the new cone-beam data-acquisition scheme can provide a complete set of projection data and the image reconstruction algorithm can achieve exact reconstruction. Potentially, the algorithm can be applied in practice for both a standard CT gantry-based volume tomographic imaging system and a C-arm-based cone-beam tomographic imaging system, with little mechanical modification required.

Imaging modalities in x-ray computerized tomography and in selected volume tomography

This review of different principles used in x-ray computerized tomography (CT) starts with attenuation (transmission) CT. The pros and cons of different geometrical solutions, single-ray, fan-beam and cone-beam, are discussed. Attenuation CT measures the spatial distribution of the linear attenuation coefficient, mu. The contributions of different interaction processes to mu have also been used for CT. Fluorescence CT is based on measurements of the contribution, cZtauZ/rho, from an element Z with concentration cZ, to the linear attenuation coefficient. Diffraction CT measures the differential coherent cross section d sigma (theta)(coh)/d omega, Compton CT the incoherent scatter cross section sigma. The usefulness of these modalities is illustrated. CT methods based on secondary photons have a competitor in selected volume tomography. These two tomography methods are compared. A proposal to perform Compton profile tomography is also discussed, as is the promising method of phase-contrast x-ray CT.

An efficient Fourier method for 3-D radon inversion in exact cone-beam CT reconstruction

The radial derivative of the three-dimensional (3-D) radon transform of an object is an important intermediate result in many analytically exact cone-beam reconstruction algorithms. We briefly review Grangeat's approach for calculating radon derivative data from cone-beam projections and then present a new, efficient method for 3-D radon inversion, i.e., reconstruction of the image from the radial derivative of the 3-D radon transform, called direct Fourier inversion (DFI). The method is based directly on the 3-D Fourier slice theorem. From the 3-D radon derivative data, which is assumed to be sampled on a spherical grid, the 3-D Fourier transform of the object is calculated by performing fast Fourier transforms (FFT's) along radial lines in the radon space. Then, an interpolation is performed from the spherical to a Cartesian grid using a 3-D gridding step in the frequency domain. Finally, this 3-D Fourier transform is transformed back to the spatial domain via 3-D inverse FFT. The algorithm is computationally efficient with complexity in the order of N3 logN. We have done reconstructions of simulated 3-D radon derivative data assuming sampling conditions and image quality requirements similar to those in medical computed tomography (CT).

Three-dimensional reconstruction from cone-beam data using an efficient Fourier technique combined with a special interpolation filter

We here present LINCON(FAST) which is an exact method for 3D reconstruction from cone-beam projection data. The new method is compared to the LINCON method which is known to be fast and to give good image quality. Both methods have O(N3 log N) complexity and are based on Grangeat's result which states that the derivative of the Radon transform of the object function can be obtained from cone-beam projections. One disadvantage with LINCON is that the rather computationally intensive chirp z-transform is frequently used. In LINCON(FAST), FFT and interpolation in the Fourier domain are used instead, which are less computationally demanding. The computation tools involved in LINCON(FAST) are solely FFT, 1D eight-point interpolation, multiplicative weighting and tri-linear interpolation. We estimate that LINCON(FAST) will be 2-2.5 times faster than LINCON. The interpolation filter belongs to a special class of filters developed by us. It turns out that the filter must be very carefully designed to keep a good image quality. Visual inspection of experimental results shows that the image quality is almost the same for LINCON and the new method LINCON(FAST). However, it should be remembered that LINCON(FAST) can never give better image quality than LINCON, since LINCON(FAST) is designed to approximate LINCON as well as possible.

Implementations, comparisons, and an investigation of heuristic techniques for cone-beam tomography

A novel cone-beam reconstruction method was proposed in 1985. The first objective of the work reported here is to implement this reconstruction method. The second objective is to compare it with the method developed by Feldkamp et al. (1984). Although the resulting reconstruction was not perfect, the proposed method did eliminate the axial distortion associated with Feldkamp's method. A second cone-beam reconstruction method was proposed in 1987. Two major challenges arise when this method is implemented. One is to minimize the error that results from violating an assumed condition on the distribution of the cones. The second is to minimize the error that results from the convolution of a discontinuous function which is introduced to compensate for the redundancy in the data set. The third objective of the work reported here was to investigate several heuristic techniques to minimize these errors. Techniques were found that did mitigate these errors and using these techniques resulted in images that are more accurate than those resulting from the 1985 method.

Methodologic aspects of computed microtomography to monitor the development of osteoporosis in gastrectomized rats

RATIONALE AND OBJECTIVES

We investigated the methodologic development of computed microtomography (CMT) for monitoring the development of osteoporosis in male Sprague-Dawley rats.

METHODS

Eight rats were gastrectomized and eight rats were sham operated. Femurs, tibias, and tails were prepared, and CMT scans with spatial resolutions of 5-500 microns were made. Bone diameters, bone areas, and moments of inertia were determined from the CMT scans. Optimal slice position and the need for spatial resolution and energy optimization for future in vivo applications were investigated.

RESULTS

Gastrectomy caused dramatic changes in the bone architecture of the tibia and the femur. The main features were vacuolization of the bone and reduced amounts of compact bone. Although the outer diameters of tubular bones (femur and tibia) were largely unaffected, their inner diameters were greatly increased following gastrectomy. Relative bone area and moment of inertia were greatly reduced. The optimal photon energy was 12 keV.

CONCLUSION

It is possible to monitor gastrectomy-evoked changes in bone morphology at various sites in rats using CMT scanning. The changes are suggestive of osteoporosis. By optimizing the energy spectrum and spatial resolution, as well as choosing the proper slice position, it should be possible to keep absorbed doses low enough to avoid acute radiation injury in repeated in vivo measurements.