Grangeat's Formula Research Articles

The Formula of Grangeat for Tensor Fields of Arbitrary Order in n Dimensions

The cone beam transform of a tensor field of order m in n ≥ 2 dimensions is considered. We prove that the image of a tensor field under this transform is related to a derivative of the n-dimensional Radon transform applied to a projection of the tensor field. Actually the relation we show reduces for m = 0 and n = 3 to the well-known formula of Grangeat. In that sense, the paper contains a generalization of Grangeat's formula to arbitrary tensor fields in any dimension. We further briefly explain the importance of that formula for the problem of tensor field tomography. Unfortunately, for m > 0, an inversion method cannot be derived immediately. Thus, we point out the possibility to calculate reconstruction kernels for the cone beam transform using Grangeat's formula.

Projection-based Bolus Detection for Computed Tomographic Angiography

Computed tomographic (CT) angiography is important for imaging studies on cardiovascular structures, peripheral vessels, and solid organs. In practice, a CT angiography scan is triggered by the bolus arrival at a prespecified anatomical location, which is determined using CT fluoroscopy. In this article, we propose a projection-based method adapted from the Grangeat formula to detect the bolus arrival. Then, we evaluate our new method in numerical and animal studies. Our results indicate that this method allows significantly better temporal resolution and is computationally more efficient, as compared with the image-based methods.

Analysis of Cone‐Beam Artifacts in off‐Centered Circular CT for Four Reconstruction Methods

Cone-beam (CB) acquisition is increasingly used for truly three-dimensional X-ray computerized tomography (CT). However, tomographic reconstruction from data collected along a circular trajectory with the popular Feldkamp algorithm is known to produce the so-called CB artifacts. These artifacts result from the incompleteness of the source trajectory and the resulting missing data in the Radon space increasing with the distance to the plane containing the source orbit. In the context of the development of integrated PET/CT microscanners, we introduced a novel off-centered circular CT cone-beam geometry. We proposed a generalized Feldkamp formula (α-FDK) adapted to this geometry, but reconstructions suffer from increased CB artifacts. In this paper, we evaluate and compare four different reconstruction methods for correcting CB artifacts in off-centered geometry. We consider the α-FDK algorithm, the shift-variant FBP method derived from the T-FDK, an FBP method based on the Grangeat formula, and an iterative algebraic method (SART). The results show that the low contrast artifacts can be efficiently corrected by the shift-variant method and the SART method to achieve good quality images at the expense of increased computation time, but the geometrical deformations are still not compensated for by these techniques.

Grangeat-type helical half-scan computerized tomography algorithm for reconstruction of a short object.

Currently, cone-beam computerized tomography (CT) and micro-CT scanners are under rapid development for major biomedical applications. Half-scan cone-beam image reconstruction algorithms assume only part of a scanning turn, and are advantageous in terms of temporal resolution and image artifacts. While the existing half-scan cone-beam algorithms are in the Feldkamp framework, we have published a half-scan algorithm in the Grangeat framework for a circular trajectory [Med. Phys. 30, 689-700 (2003)]. In this paper, we extend our previous work to a helical case without data truncation. We modify the Grangeat's formula for utilization and estimation of Radon data. Specifically, we categorize each characteristic point in the Radon space into singly, doubly, triply sampled, and shadow regions, respectively. A smooth weighting strategy is designed to compensate for data redundancy and inconsistency. In the helical half-scan case, the concepts of projected trajectories and transition points on meridian planes are introduced to guide the design of weighting functions. Then, the shadow region is recovered via linear interpolation after smooth weighting. The Shepp-Logan phantom is used to verify the correctness of the formulation, and demonstrate the merits of the Grangeat-type half-scan algorithm. Our Grangeat-type helical half-scan algorithm is not only valuable for quantitative and/or dynamic biomedical applications of CT and micro-CT, but also serves as an intermediate step towards solving the long object problem.

Artifacts associated with implementation of the Grangeat formula.

To compensate for image artifacts introduced in approximate cone-beam reconstruction, exact cone-beam reconstruction algorithms are being developed for medical x-ray CT. Although the exact cone-beam approach is theoretically error-free, it is subject to image artifacts due to the discrete nature of numerical implementation. We report a study on image artifacts associated with the Grangeat algorithm as applied to a circular scanning locus. Three types of artifacts are found, which are thorn, wrinkle, and V-shaped artifacts. The thorn pattern is created by inappropriate extrapolation into the shadow zone in the radon domain. If the shadow zone is filled in with continuous data, the thorn artifacts along the boundary of the shadow zone can be removed. The wrinkle appearance arises if interpolated first derivatives of the radon data are not smooth between adjacent detector planes. In particular, the nearest-neighbor interpolation method should not be used. If the number of projections is not small, the bilinear interpolation method is effective to suppress the wrinkle artifacts. The V-shaped artifacts on the meridian plane come from the line integrations through the transition zones where derivative data change abruptly. Two remedies are to increase the sampling rate and suppress data noise.

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.

Quasi-exact filtered backprojection algorithm for long-object problem in helical cone-beam tomography.

Exact reconstruction from axially truncated cone-beam projections acquired with a helical vertex path is a challenging problem for which solutions are currently under investigation by some researchers. This paper deals with a difficult class of this problem called the long-object problem. Its purpose is to reconstruct a central region of interest (ROI) of a long object when the helical path extends only a little bit above and below the ROI. By extending the authors' recent approach based on the triangular decomposition of the Grangeat formula, we derive quasi-exact reconstruction algorithms whose overall structure is of filtered backprojection (FBP) style. Unlike the previous similar approaches to the long-object problem, the proposed FBP algorithms do not require additional two circular scans at the ends of the helical path. Furthermore, the algorithms require a significantly smaller detector area and achieve improved image quality even for a large pitch compared with the approximate Feldkamp algorithms. One drawback of the proposed algorithms is the computational time, which is much longer than for the Feldkamp algorithms. We show some simulation results to demonstrate the performances of the proposed algorithms.

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.

Cone-beam filtered-backprojection algorithm for truncated helical data

This paper investigates 3D image reconstruction from truncated cone-beam (CB) projections acquired with a helical vertex path. First, we show that a rigorous derivation of Grangeat's formula for truncated projections leads to a small additional term compared with previously published similar formulations. This correction term is called the boundary term. Next, this result is used to develop a CB filtered-backprojection (FBP) algorithm for truncated helical projections. This new algorithm only requires the CB projections to be measured within the region that is bounded, in the detector, by the projections of the upper and lower turns of the helix. Finally, simulations with mathematical phantoms demonstrate that: (i) the boundary term is necessary to obtain high-quality imaging of low-contrast structures and (ii) good image quality is obtained even with large values of the pitch of the helix.

Minimal residual cone-beam reconstruction with attenuation correction in SPECT

This paper presents an iterative method based on the minimal residual algorithm for tomographic attenuation compensated reconstruction from attenuated cone-beam projections given the attenuation distribution. Unlike conjugate-gradient based reconstruction techniques, the proposed minimal residual based algorithm solves directly a quasisymmetric linear system, which is a preconditioned system. Thus it avoids the use of normal equations, which improves the convergence rate. Two main contributions are introduced. First, a regularization method is derived for quasisymmetric problems, based on a Tikhonov-Phillips regularization applied to the factorization of the symmetric part of the system matrix. This regularization is made spatially adaptive to avoid smoothing the region of interest. Second, our existing reconstruction algorithm for attenuation correction in parallel-beam geometry is extended to cone-beam geometry. A circular orbit is considered. Two preconditioning operators are proposed: the first one is Grangeat's inversion formula and the second one is Feldkamp's inversion formula. Experimental results obtained on simulated data are presented and the shadow zone effect on attenuated data is illustrated.

Cone-beam reconstruction from general discrete vertex sets using Radon rebinning algorithms

Addresses image reconstruction in cone-beam tomography from an arbitrary discrete set of positions of the cone vertex. As a first step in the analysis of the problem, the authors define some measures of how close a discrete vertex set comes to satisfying Tuy's condition (1983). Next, they propose 3 rebinning algorithms which use Grangeat's formula (1991) and Marr's algorithm (1980), and are capable of accurate reconstructions. The first algorithm is designed to accurately process cone-beam data finely sampled along a vertex path satisfying Tuy's condition. The second algorithm applies to pair-complete vertex sets. The third algorithm is suited to process any discrete vertex set. The efficacy of the algorithms is illustrated with reconstructions from computer-simulated data using several vertex sets, including a set of randomly placed vertices.

Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits

B.D. Smith (ibid., vol.MI-4, p.15-25, 1985; Opt. Eng., vol.29, p.524-34, 1990) and P. Grangeat (These de doctorat, 1987; Lecture Notes in Mathematics 1497, p.66-97, 1991) derived a cone-beam inversion formula that can be applied when a nonplanar orbit satisfying the completeness condition is used. Although Grangeat's inversion formula is mathematically different from Smith's one, they have similar overall structures to each other. The contribution of the present paper is two-fold. First, based on the derivation of Smith, the authors point out that Grangeat's inversion formula and Smith's one can be conveniently described using a single formula (the Smith-Grangeat inversion formula) that is in the form of space-variant filtering followed by cone-beam back projection. Furthermore, the resulting formula is reformulated for data acquisition systems with a planar detector to obtain a new reconstruction algorithm. Second, the authors make two significant modifications to the new algorithm to reduce artifacts and numerical errors encountered in direct implementation of the new algorithm. As for exactness of the new algorithm, the following fact can be stated. The algorithm based on Grangeat's intermediate function is exact for any complete orbit, whereas that based on Smith's intermediate function should be considered as an approximate inverse excepting the special case where almost every plane in 3D space meets the orbit. The validity of the new algorithm is demonstrated by simulation studies.