Truncation reduction in fan-beam transmission scanning using the Radon transform consistency conditions

Erlandsson, Kjell; Visvikis, Dimitris; Waddington, W; Jarritt, P. H.

doi:10.1109/23.856536

Cited by 3 publications

(1 citation statement)

References 18 publications

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“…In medical imaging, there are numerous examples of applications where data consistency conditions (also known as range conditions) play a central role in the image reconstruction step. The idea is that certain mathematical redundancies in the projection measurements can be exploited to identify the parameters of some systematic effect, such as abrupt patient movement (Welch et al 1998, Yu and Wang 2007, Leng et al 2007, beam-hardening scaling factors (Tang et al 2011), attenuation factors in emission tomography (Natterer 1983, Hertle 1988, Welch et al 2003, Defrise et al 2012, unknown patient outline (Natterer 1993, Laurette et al 1999, Mennessier et al 1999, or to synthesize missing projections (Defrise 1995, Patch 2002b or missing rays of truncated projections (Erlandsson et al 2000).…”

Section: Introductionmentioning

confidence: 99%

Full data consistency conditions for cone-beam projections with sources on a plane

Clackdoyle

Desbat

2013

Phys. Med. Biol.

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Cone-beam consistency conditions (also known as range conditions) are mathematical relationships between different cone-beam projections, and they therefore describe the redundancy or overlap of information between projections. These redundancies have often been exploited for applications in image reconstruction. In this work we describe new consistency conditions for cone-beam projections whose source positions lie on a plane. A further restriction is that the target object must not intersect this plane. The conditions require that moments of the cone-beam projections be polynomial functions of the source positions, with some additional constraints on the coefficients of the polynomials. A precise description of the consistency conditions is that the four parameters of the cone-beam projections (two for the detector, two for the source position) can be expressed with just three variables, using a certain formulation involving homogeneous polynomials. The main contribution of this work is our demonstration that these conditions are not only necessary, but also sufficient. Thus the consistency conditions completely characterize all redundancies, so no other independent conditions are possible and in this sense the conditions are full. The idea of the proof is to use the known consistency conditions for 3D parallel projections, and to then apply a 1996 theorem of Edholm and Danielsson that links parallel to cone-beam projections. The consistency conditions are illustrated with a simulation example.

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Section: Introductionmentioning

confidence: 99%

Full data consistency conditions for cone-beam projections with sources on a plane

Clackdoyle

Desbat

2013

Phys. Med. Biol.

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Necessary and Sufficient Consistency Conditions for Fanbeam Projections Along a Line

Clackdoyle

2013

IEEE Trans. Nucl. Sci.

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The mathematical equivalence of consistency conditions in the divergent-beam computed tomography

Tang

Mou

et al. 2012

Journal of X-Ray Science and Technology: Clinical Applications

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In this paper, we discuss the mathematical equivalence among four consistency conditions in the divergent-beam computed tomography (CT). The first is the consistency condition derived by Levine et al. by degenerating the John's equation; the second is the integral invariant derived by Wei et al. using the symmetric group theory; the third is the so-called parallel-fanbeam Hilbert projection equality derived by Hamaker et al.; and the fourth is the fan-beam data consistency condition (FDCC) derived by Chen et al. using the complex analysis theory. Historically, most of these consistency conditions were derived by their corresponding authors using complicated mathematical strategies, which are usually not easy to be precisely understood by researchers with only a general engineering mathematical background. In this paper, we symmetrically re-derive all these consistency conditions using a friendly mathematical language. Based on theoretical derivation, it has been found that all these consistency conditions can be viewed as a necessary condition for the specific solution to John's equation. From the physical point of view, all these consistency conditions have been essentially expressed as a similar constraint on the projection data acquired with arbitrary two x-ray source points. Numerical simulations have been carried out to experimentally evaluate and verify their merits.

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Truncation reduction in fan-beam transmission scanning using the Radon transform consistency conditions

Cited by 3 publications

References 18 publications

Full data consistency conditions for cone-beam projections with sources on a plane

Full data consistency conditions for cone-beam projections with sources on a plane

Necessary and Sufficient Consistency Conditions for Fanbeam Projections Along a Line

The mathematical equivalence of consistency conditions in the divergent-beam computed tomography

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