An inversion method for the cone-beam transform (original) (raw)
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A unified framework for exact cone-beam reconstruction formulas
Medical Physics, 2005
In this paper, we present concise proofs of several recently developed exact cone-beam reconstruction methods in the Tuy inversion framework, including both filtered-backprojection and backprojection-filtration formulas in the cases of standard spiral, nonstandard spiral, and more general scanning loci. While a similar proof of the Katsevich formula was previously reported, we present a new proof of the Zou and Pan backprojection-filtration formula. Our proof combines both odd and even data extensions so that only the cone-beam transform itself is utilized in the backprojection-filtration inversion. More importantly, our formulation is valid for general smooth scanning curves, in agreement with an earlier paper from our group ͓Ye, Zhao, Yu, and Wang, Proc. SPIE 5535, 293-300 ͑Aug. 6 2004͔͒. As a consequence of that proof, we obtain a new inversion formula, which is in a two-dimensional filtering backprojection format. A possibility for generalization of the Katsevich filtered-backprojection reconstruction method is also discussed from the viewpoint of this framework.
Implementation of Tuy's cone-beam inversion formula
Physics in Medicine and Biology, 1994
Tuy's cone-beam inversion formula was modified to develop a cone-beam reconswction algorithm. The algorithm was implemented far a cone-beun vertex orbit consisting of a circle and WO Orthogonal lines. This orbit geometry satisfies the cone-beam data sufficiency condition and is easy to implement on commercial single photon emission computed tomography (SPECT) systems. The algorithm, which consists of W O derivative steps, one rebinning step, and one three-dimensional backprojection step, was verified by computer simulations and by reconstructing physical phantom data collected on a clinical SPECT system. The proposed algorithm gives equivalent results and is as efficient as other analytical cone-beam recanstmclion algorithms.
Inverse Problems, 2004
The inversion problem for the 3D parallel-beam exponential ray transform is solved through inversion of a set of the 2D exponential Radon transforms with complex-valued angle-dependent attenuation. An inversion formula for the latter 2D transform is derived; it generalizes the known Kuchment-Shneiberg formula valid for real angle-dependent attenuation. We derive an explicit theoretically exact solution of the 3D problem which is valid for arbitrary closed trajectory that does not intersect itself. A simple reconstruction algorithm is described, applicable for certain sets of trajectories satisfying Orlov's condition. In the latter case, our inversion technique is as stable as the Tretiak-Metz inversion formula. Possibilities of further reduction of noise sensitivity are briefly discussed in the paper. The work of our algorithm is illustrated by an example of image reconstruction from two circular orbits.
A simplified approach for the generation of projection data for cone beam geometry
Pramana, 2011
To test a developed reconstruction algorithm for cone beam geometry, whether it is transmission or emission tomography, one needs projection data. Generally, mathematical phantoms are generated in three dimensions and the projection for all rotation angles is calculated. For non-symmetric objects, the process is cumbersome and computation intensive. This paper describes a simple methodology for the generation of projection data for cone beam geometry for both transmission and emission tomographies by knowing the object's attenuation and/or source spatial distribution details as input. The object details such as internal geometrical distribution are nowhere involved in the projection data calculation. This simple approach uses the pixilated object matrix values in terms of the matrix indices and spatial geometrical coordinates. The projection data of some typical phantoms (generated using this approach) are reconstructed using standard FDK algorithm and Novikov's inversion formula. Correlation between the original and reconstructed images has been calculated to compare the image quality.
Exact inversion of the conical Radon transform with a fixed opening angle
2013
We study a new class of Radon transforms defined on circular cones called the conical Radon transform. In R^3 it maps a function to its surface integrals over circular cones, and in R^2 it maps a function to its integrals along two rays with a common vertex. Such transforms appear in various mathematical models arising in medical imaging, nuclear industry and homeland security. This paper contains new results about inversion of conical Radon transform with fixed opening angle and vertical central axis in R^2 and R^3. New simple explicit inversion formulae are presented in these cases. Numerical simulations were performed to demonstrate the efficiency of the suggested algorithm in 2D.
The Generalized Back Projection Theorem for Cone Beam Reconstruction
IEEE Transactions on Nuclear Science, 2000
The use of cone beam scanners raises the problem of three dimensional reconstruction from divergent projections. After a survey on bidimensional analytical reconstruction methods we examine their application to the 3D problem. Finally, it is shown that the back projection theorem can be generalized to cone beam projections. This allows to state a new inversion formula suitable for both the 4 rr parallel and divergent geometries. It leads to the generalization of the "rho-f iltered back projection " algorithm which is outlined.
Physics in Medicine and Biology, 2006
In this paper, a shift-invariant filtered backprojection cone-beam image reconstruction algorithm is derived, based upon Katsevich's general inversion scheme, and validated for the source trajectory of two concentric circles. The source trajectory is complete according to Tuy's data sufficiency condition and is used as the basis for an exact image reconstruction algorithm. The algorithm proceeds according to the following steps. First, differentiate the cone-beam projection data with respect to the detector coordinates and with respect to the source trajectory parameter. The data are then separately filtered along three different orientations in the detector plane with a shift-invariant Hilbert kernel. Eight different filtration groups are obtained via linear combinations of weighted filtered data. Voxel-based backprojection is then carried out from eight sets of view angles, where separate filtered data are backprojected from each set according to the backprojection sets' associated filtration group. The algorithm is first derived for a scanning configuration consisting of two concentric and orthogonal circles. By performing an affine transformation on the image object, the developed image reconstruction algorithm has been generalized to the case where the two concentric circles are not orthogonal. Numerical simulations are presented to validate the reconstruction algorithm and demonstrate the dose advantage of the equal weighting scheme.
The cone-beam transform and spherical convolution operators
Inverse Problems, 2018
The cone-beam tomography consists of integrating a function defined on the three-dimensional space along every ray that starts on a certain scanning curve. Based on Grangeat's formula, Louis [2016, Inverse Problems 32 115005] states reconstruction formulas based on a new generalized Funk-Radon transform on the sphere. In this article, we give a singular value decomposition of this generalized Funk-Radon transform. We use this result to derive a singular value decomposition of the cone-beam transform with sources on the sphere thus generalizing a result of Kazantsev [2015, J. Inverse Ill-Posed Probl. 23(2):173-185].
A cone-beam tomography algorithm for orthogonal circle-and-line orbit
Physics in Medicine and Biology, 1992
A cone-beam algorithm which provides a practical implementation of B D Smith's cane-beam inversion formula is presented. For a cone-beam vertex orbit consisting of a circle and an orthogonal line. This geometry is easy to implement in a s p~c~s y s t e m , and it satisfies the cone-beam data sufficiency condition. The proposed algorithm is in the form of a convolution-back projection, and requires a pre-filtering procedure. Computer rimulalions show a reduction of the artifacts that are found with the Feldkamp algorithm where the cone-beam vertex orbit is a circle.
Cone-beam reconstruction from n-sin trajectories with transversely-truncated projections
HAL (Le Centre pour la Communication Scientifique Directe), 2020
In cone-beam tomography, we define the n-sin source trajectory as having n periods of a sinusoid traced on an imaginary cylinder enclosing the object. A 2-sin is commonly known as a saddle, and it is known that the convex hull of a saddle is the same as the union of all of its chords. The convex hull of a closed trajectory is the Tuy region, where cone-beam reconstruction is possible if there are no truncated projections. However, with truncated projections, the method of differentiated backprojection and Hilbert inversion can be applied along a chord if the chord is visible (not truncated) in the projections. Here, we consider a particular transaxial truncation which prevents chords from always being visible, but we establish that the more powerful method of M-lines can be applied to ensure reconstruction in the reduced field-of-view. The 3-sin, on the other hand, has a Tuy region which is not filled by its chords, and we do not have any cone-beam theory to determine if reconstruction is possible with transverse reconstruction. In our preliminary numerical experiment, the 3-sin seemed to perform equally well as the 2-sin trajectory even though there were no chords passing through the slice we examined. We tentatively suggest that there might be other, yet unknown theory that explains why 3-sin reconstruction is possible with the specified transaxial truncation. We believe that these results on transverse truncation and reconstruction from 2-sin and 3-sin trajectories are new.