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GCC and FBCC for Linear Tomosynthesis Jer´ Omeˆ Lesaint, Simon Rit, Rolf Clackdoyle, Laurent Desbat

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GCC and FBCC for Linear Tomosynthesis Jer´ Omeˆ Lesaint, Simon Rit, Rolf Clackdoyle, Laurent Desbat GCC and FBCC for linear tomosynthesis Jer´ omeˆ Lesaint, Simon Rit, Rolf Clackdoyle, Laurent Desbat Abstract—Grangeat-based Consistency Conditions (GCC) and correction in x-ray CT [9]. For FBCC, a circular cone-beam Fan-beam Consistency Conditions (FBCC) are two ways to de- micro CT application appeared recently [10]. scribe consistency (or redundancy) between cone-beam projections. In this work, we focus on GCC and FBCC and propose a Here we consider cone-beam projections that are collected in the linear tomosynthesis geometry. We propose a theoretical theoretical comparison of these two sets of DCC. We carry comparison of these two sets of consistency conditions and illustrate this work in the specific context of tomosynthesis, with an X- the comparison with numerical simulations of a thorax Forbild ray source moving along a line. It is proved that the FBCC phantom. are stronger than the GCC, in the sense that if the FBCC are Index Terms—Cone-beam computed tomography (CBCT), data satisfied, then so are the GCC, but not conversely. We also show consistency conditions (DCCs), tomosynthesis. that if all the projections are complete (non-truncated) then the FBCC and GCC are equivalent. Furthermore the hypothesis of complete projections is essential; we show that under particular I. INTRODUCTION circumstances (with truncated projections), the FBCC are more In Computed Tomography (CT), the 3D density map of a restrictive, i.e., the FBCC can fail even when the GCC are patient (or an object) is reconstructed from a set of 2D radio- satisfied. We finally prove that neither of these two sets of graphs. Should the acquisition be realized in perfect conditions conditions are sufficient. The theory is detailed in Section II. and neglecting physical side-effects (like scattering or beam- Numerical aspects are presented in Section III and finally, hardening), these radiographs (after their log-transform) follow Section IV contains discussion and conclusion. the forward line-integral model. Unfortunately, these conditions are never fullfilled and systematic effects always degrade the II. THEORY projection data. One way to detect such effects is to quantify A. The forward X-ray model how inconsistent the data are, using the concept of data consis- We consider X-ray projection data g of an object function tency conditions (DCC). DCC are equations that characterize the f, acquired along a 1D trajectory of the source parametrized image of the forward operator. A large amount of research has by a scalar λ 2 Λ ⊂ R. The source position is denoted ~sλ. been published on DCC. In parallel geometries, the well-known 2 A projection gλ associates to each unit vector ~α 2 S the Helgason-Ludwig DCC [1], [2] provide necessary and sufficient corresponding line integral: conditions on the Radon transform. In divergent geometries, Z +1 [3] and [4] also give necessary and sufficient conditions for gλ(~α) = f(~sλ + t~α) dt: (1) a source moving along a linear trajectory (in 2D) or planar 0 trajectory (in 3D) respectively. When a set of DCC is known to be complete (necessary and sufficient conditions), no additional B. The tomosynthesis geometry information can be expected from another set of DCC. On the In the following, we consider a tomosynthesis geometry, other hand, if two sets of DCC are known to be necessary where an X-ray source moves along a line parallel to the plane of but no information on the sufficiency is available, one may the detector. Let (O; x; y; z) be a 3D coordinate system. Without wonder which to use. Some works have introduced necessary- loss of generality, the detector, denoted D, is assumed to lie in only conditions, like Grangeat-based DCC (GCC) [5], [6] and the z = 0 plane. It is equipped with 2D coordinates (O; u; v) fan-beam DCC (FBCC) [3] extended to 3D projection data as where the detector origin O and the u- and v- axes coincide with suggested by [7]. As explained below, FBCC refers to what the 3D origin and the x- and y- axes respectively. Note that the would be called zeroth-order conditions in [3]. coordinates (u; v) are independent of the source position, as if Various applications of GCC and FBCC have been published. the detector was large enough to capture every projection of the Geometric calibration in circular cone-beam CB [5] and in x-ray scan. (In the numerical simulations though, the detector was tomosynthesis [8] were described using GCC, and also jitter- displaced horizontally.) The X-ray source moves along a line L such that ~sλ has coordinates (λ, 0; d) for λ 2 Λ, where d is the J. Lesaint, R. Clackdoyle and L. Desbat are with the TIMC-IMAG source-to-detector distance and Λ is an interval. The object of laboratory, CNRS UMR 5525 and Universite´ Grenoble Alpes (e-mail : [email protected]). interest is assumed to be entirely contained between D and the S. Rit is with Univ. Lyon, INSA-Lyon, UCB Lyon 1, UJM-Saint Etienne, plane parallel to D and containing L. See Figure 1. CNRS, Inserm, CREATIS UMR5220, U1206, Centre Leon´ Berard,´ F-69373, LYON, France. This work is partially supported by the Agence Nationale de la Recherche C. The family of planes containing the source trajectory (France), Labex CAMI, number ANR-11-LABX-0004-01, Labex PRIMES, number ANR-11-LABX-0063 and project “DROITE”, number ANR-12-BS01- We now focus on the family of planes which contain the 0018. source line. They will play an important role both in the 114 The fifth international conference on image formation in X-ray computed tomography O x; u gλ(u; v) = gλ(~α(θ; ')). We also use g~λ to denote the projection D gλ weighted by the cosine of the incidence angle of the ray. Given the coordinates of the source position (λ, 0; d), we have: d z g~λ(u; v) = gλ(u; v) (2) p(u − λ)2 + v2 + d2 d D. Grangeat-based consistency condition Consistency conditions are equations derived from the for- ward model that ideal projection data must satisfy. Consistency conditions can be used to detect - and possibly correct for - inconsistencies introduced in the data by systematic effect such as mis-calibration, motion, scattering or beam-hardening. Let Rf(Pθ) denote the 3D Radon transform of f over the plane Pθ. We recall the well-known result of Grangeat [11] L (expressed with our notation): Z +1 Source motion 1 @ @ g~λ(u; v) du = Rf(Pθ) (3) cos2 θ @v @s −∞ s=~sλ·~nθ Fig. 1. Top-view of the tomosynthesis acquisition geometry. The X-ray source where the derivative of Rf is taken in the direction ~nθ. Again, moves at constant distance d from the detector plane, along a horizontal line. v and θ are related through v = d tan θ. For fixed Pθ (equivalently, for fixed v = d tan θ), we note that ~sλ · ~nθ = d sin θ is constant throughout the trajectory, (u; v) so the right-hand-side of Equation 3 does not depend on the Pθ source position (it only depends on v, not on λ). We let y; v @ R G(λ, v) = @v g~λ(u; v) du, so the left-hand-side of Equation 3, (1= cos2 θ)G(λ, v), must be independent of λ. We thus obtain x; u O the following necessary DCC for this particular acquisition ' ~α z geometry: θ The projection data (gλ)λ2Λ are consistent only if, for all v, ~n θ L G(λ, v) is a constant function of λ. ~s0 This is what will be called in the sequel, the Grangeat-based ~sλ Consistency Conditions, GCC for short. D Grangeat’s result, Equation 3, is also the basis of Epipolar consistency conditions described in [6]. Fig. 2. The family of planes Pθ. Each plane is defined by a normal vector ~nθ which makes an angle θ with the central plane. To each detector pixel (u; v) E. Fan-beam consistency conditions is associated a unit vector ~α together with its spherical coordinates (θ; '). Another way to quantify the redundancy/consistency of a set of projections is to fix the v coordinate and consider the corresponding plane Pθ and the fan-beam projections therein. Grangeat case and in the fan-beam case. For θ 2 − π ; π , 2 2 We then consider fan-beam consistency conditions (see [3] for a we let Pθ denote the plane which contains the line L and complete proof) extended to 3D cone-beam projections acquired makes an angle θ with the central plane (y = 0). We let in our particular tomosynthesis geometry: the set of projections ~nθ = (0; cos θ; sin θ) denote the unit vector orthogonal to (gλ)λ2Λ is consistent if and only if, for each n = 0; 1; 2::: and this plane. There is a one-to-one correspondence between the π π each θ 2 − 2 ; 2 , the function: detector’s horizontal rows (with offset v) and the planes Pθ via π Z 2 the relation d tan θ = v. See Figure 2. gλ(α(θ; ')) n Jn(λ, θ) = tan ' d'; (4) π cos ' − 2 is a polynomial of order n in λ. We assume that the detector is fixed so from de- In particular, for n = 0, the quantity J0 does not depend on tector coordinates (u; v), the unit vector ~α from Equa- the source position λ. It only depends on θ. And since there is a tion 1 is given by ~α = ((u; v; 0) − ~sλ) =k(u; v; 0) − one-to-one correspondence between θ and v, we will write (with p 2 2 2 ~sλk = (u − λ, v; −d)= (u − λ) + v + d .
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