Exact reconstruction in 2D dynamic CT: compensation of time-dependent affine deformations

Abstract

This work is dedicated to the reduction of reconstruction artefacts due to motion occurring during the acquisition of computerized tomographic projections. This problem has to be solved when imaging moving organs such as the lungs or the heart. The proposed method belongs to the class of motion compensation algorithms, where the model of motion is included in the reconstruction formula. We address two fundamental questions. First what conditions on the deformation are required for the reconstruction of the object from projections acquired sequentially during the deformation, and second how do we reconstruct the object from those projections. Here we answer these questions in the particular case of 2D general time-dependent affine deformations, assuming the motion parameters are known. We treat the problem of admissibility conditions on the deformation in the parallel-beam and fan-beam cases. Then we propose exact reconstruction methods based on rebinning or sequential FBP formulae for each of these geometries and present reconstructed images obtained with the fan-beam algorithm on simulated data.

Algebraic and analytic reconstruction methods for dynamic tomography

“…In dynamic tomography, the attenuation f to be reconstructed is also a function of t. In our approach, we consider a time dependent deformation of the space as in [9], [10]. We introduce a time dependent deformation model Γ t and we assume that, for t ∈ T , Γ t are known bijective appropriately smooth functions on R 3 whose inverse are smooth too.…”

“…In particular, if the patient motion is globally rigid, this amounts to using virtual source and detector positions and applying the reconstruction algorithm as in the static case [11], [12], [13] or to deform data prior to reconstruction [14], [15]. These approaches can be generalized to deformations transforming the set of acquisition lines of each cone beam projection into other sets of concurent lines [9], [10], [16]. Thus, analytic approaches essentially allow for the compensation of deformations in subclasses of those presented in section I-B.…”

“…We recall briefly in this section a class of 3D deformations which can be analytically compensated in 3D Cone Beam reconstruction, see [23] and [16], [10] for 2D fan-beam dynamic tomography. In order to stay in the framework of 3D CB reconstruction, we consider the deformations Γ t which transform the set of convergent acquisition lines at time t into an other set of convergent lines at the reference time, i.e.…”

Compensation of Some Time Dependent Deformations in Tomography

“…CT scanner rotation speed and sampling parameter adjustment to the heart period were also suggested in order to improve the reconstructed images [7]. This paper shares the same approach as Crawford et al [8] and Roux et al [9]. The idea is to introduce a time (denoted by t) dependent motion or a deformation model Γ t within the reconstruction.…”

“…In [9] the deformation compensation is incorporated within analytical reconstruction algorithms for time dependent ane deformations Γ t in 2D parallel 1 and fan beam tomography (a generalization to 3D is given in [10]). The 2D fan beam method is based on the recent reconstruction framework proposed by Noo et al [11].…”