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

Clackdoyle, Rolf; Desbat, Laurent

doi:10.1088/0031-9155/58/23/8437

Cited by 17 publications

(17 citation statements)

References 42 publications

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“…Helgason-Ludwig [21], [22] and John's equation [23] have yet to be established. Possibly, some such formulations of consistency are special cases of ECC.…”

Section: Discussionmentioning

confidence: 99%

Epipolar Consistency in Transmission Imaging

Aichert

Berger

Wang

et al. 2015

IEEE Trans. Med. Imaging

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This paper presents the derivation of the Epipolar Consistency Conditions (ECC) between two X-ray images from the Beer-Lambert law of X-ray attenuation and the Epipolar Geometry of two pinhole cameras, using Grangeat's theorem. We motivate the use of Oriented Projective Geometry to express redundant line integrals in projection images and define a consistency metric, which can be used, for instance, to estimate patient motion directly from a set of X-ray images. We describe in detail the mathematical tools to implement an algorithm to compute the Epipolar Consistency Metric and investigate its properties with detailed random studies on both artificial and real FD-CT data. A set of six reference projections of the CT scan of a fish were used to evaluate accuracy and precision of compensating for random disturbances of the ground truth projection matrix using an optimization of the consistency metric. In addition, we use three X-ray images of a pumpkin to prove applicability to real data. We conclude, that the metric might have potential in applications related to the estimation of projection geometry. By expression of redundancy between two arbitrary projection views, we in fact support any device or acquisition trajectory which uses a cone-beam geometry. We discuss certain geometric situations, where the ECC provide the ability to correct 3D motion, without the need for 3D reconstruction.

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“…Helgason-Ludwig [21], [22] and John's equation [23] have yet to be established. Possibly, some such formulations of consistency are special cases of ECC.…”

Section: Discussionmentioning

confidence: 99%

Epipolar Consistency in Transmission Imaging

Aichert

Berger

Wang

et al. 2015

IEEE Trans. Med. Imaging

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“…These consistency conditions include the Helgason-Ludwig moment conditions [19], [20], which have been extensively studied for a set of parallel-beam projections and extended to divergent fan-beams using rebinning strategies. Several consistency conditions were developed directly for divergent-beam geometry: the use of John's equation for cone-beam projections [21], [22]; the extension of John's equation for straight line source trajectory [23]; the integral invariant condition [24]; the parallel-fanbeam Hilbert projection equality [25] and the associated data consistency condition [26]; and the new range conditions for cone beam projections [27]. These consistency conditions have also been proposed as guiding principles to either complete the missing line integrals or to compensate for the object movement [28].…”

Section: Introductionmentioning

confidence: 99%

An Empirical Data Inconsistency Metric (DIM) Driven CT Image Reconstruction Method

Chen

2019

IEEE Trans. Med. Imaging

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Current CT image reconstruction methods generally assume that a complete and consistent data set was acquired during data acquisition. In practice, however, the acquired data are often not consistent from one view angle to another. In this case, the application of a well developed image reconstruction algorithm to an inconsistent data set still generates artifacts in the reconstructed images. Therefore, it is highly desirable to develop a simple method to classify the acquired data set into several consistency classes and to incorporate the data consistency information into an image reconstruction framework to simultaneously reconstruct several sub-images of the same image object according to the data consistency level of the acquired data set. In this work, an empirical data inconsistency metric (DIM) was introduced to characterize the inconsistency level of the acquired cone beam CT projection data at each view angle caused by the polychromatic x-ray spectrum and the use of tube potential modulation. The entire acquired data set is then sorted into several subsets of view angles based upon the value of the DIM at each view angle. After data classification, the previously published algorithm, synchronized multiartifact reduction with tomographic reconstruction (SMART-RECON), was applied to simultaneously reconstruct images for these subimages in different spectral consistency classes. Each sub-image is consistent with the subset of the projection view angles for a given range of DIM values. To validate the method, numerical simulation studies from an anthropomorphic numerical phantom, a hybrid phantom of known truth with human anatomy, and in vivo human subject data were conducted to demonstrate the practical utility of the proposed method.

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“…The latter conditions can also be regarded as countably many algebraic conditions characterizing the range of I 0 over functions in the so-called parallel geometry. Translating these conditions into fan-beam coordinates has been and continues to be an object of active study [2,4], in particular for their many applications to the imaging modalities of Computerized tomography and Positron Emission Tomography: motion artifact reduction as in, e.g., [49,51]; consistent extrapolation of truncated data as in, e.g., [48,50]; monitoring of CT systems for faulty detector channels [29]; see also the detailed introduction in [3] on the matter. In the present article, we show that another range characterization provided in [32] is equivalent 2 to the moment conditions for functions with compact support.…”

Section: Introductionmentioning

confidence: 99%

Efficient tensor tomography in fan-beam coordinates

Monard¹

2016

IPI

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We propose a thorough analysis of the tensor tomography problem on the Euclidean unit disk parameterized in fan-beam coordinates. This includes, for the inversion of the Radon transform over functions, using another range characterization first appearing in [32] to enforce in a fast way classical moment conditions at all orders. When considering directiondependent integrands (e.g., tensors), a problem where injectivity no longer holds, we propose a suitable representative (other than the traditionally sought-after solenoidal candidate) to be reconstructed, as well as an efficient procedure to do so. Numerical examples illustrating the method are provided at the end. 1 A non-trapping Riemannian manifold (dim ≥ 2) with boundary is called simple if it has no conjugate points and its boundary is strictly convex.On studying question (ii), the solenoidal-potential decomposition of an m tensor f = dh+f s (with h an m − 1 tensor vanishing on ∂M ), true for any L 2 tensor f and where each summand is continuous in terms of f , suggests that, since If = If s , the "solenoidal" tensor f s is a good candidate as a representative to be reconstructed from data If . In this direction, Sharafutdinov provided a reconstruction of f s in Euclidean free space in any dimension in [39, Theorem 2.12.2], Kazantsev and Bukhgeim proposed in [22] a reconstruction algorithm for a tensor of arbitrary order in the case of the Euclidean unit disk, see also [21,46,8,6] for further works on the topic and the recent work [7]. Another approach is to make use of the theory of A-analytic functionsà la Bukhgeim, which has successfully led to range characterizations and inversion formulas, most recently in [36,38,37] for the ray transform in Euclidean convex domains over functions, vector fields and two-tensors, including attenuation coefficients.The solenoidal representative is arguably not an easy quantity to work with: Sharafutdinov's formulas involve iterated use of the non-local operator ∆ −1 d 2 , where ∆ denotes componentwise Laplacian, and the expression of d 2 depends on the tensor order. A first salient feature of this paper is to exploit another decomposition, arising naturally for instance when performing Fourier analysis on tangent fibers: doing this comes with a different inner product structure than the one traditionally used on tensor fields, and in particular, this motivates another tensor decomposition than the solenoidal-potential one. This decomposition has proved to be more natural in earlier contexts such as the search for conformal Killing tensor fields (see for instance [41, 42] or [5, Theorem 1.5]). Here we exploit this other decomposition to provide, for any tensor field, an element to be reconstructed with some advantages: its Fourier components are made up of a function to be inverted for via "traditional" inverse X-ray transform, plus additional terms which can be easily expressed in terms of analytic functions, the reconstruction of which via ad hoc Cauchy integrals is straighforward and efficient; in addition, the X-ray ...

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

Cited by 17 publications

References 42 publications

Epipolar Consistency in Transmission Imaging

Epipolar Consistency in Transmission Imaging

An Empirical Data Inconsistency Metric (DIM) Driven CT Image Reconstruction Method

Efficient tensor tomography in fan-beam coordinates

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