Automatic alignment for three-dimensional tomographic reconstruction

In tomographic reconstruction, the goal is to reconstruct an unknown object from a collection of line integrals. Given a complete sampling of such line integrals for various angles and directions, explicit inverse formulas exist to reconstruct the object. Given noisy and incomplete measurements, the inverse problem is typically solved through a regularized least-squares approach. A challenge for both approaches is that in practice the exact directions and offsets of the x-rays are only known approximately due to, e.g. calibration errors. Such errors lead to artifacts in the reconstructed image. In the case of sufficient sampling and geometrically simple misalignment, the measurements can be corrected by exploiting so-called consistency conditions. In other cases, such conditions may not apply and we have to solve an additional inverse problem to retrieve the angles and shifts. In this paper we propose a general algorithmic framework for retrieving these parameters in conjunction with an algebraic reconstruction technique. The proposed approach is illustrated by numerical examples for both simulated data and an electron tomography dataset

Stability estimates for the regularized inversion of the truncated Hilbert transform

In limited data computerized tomography, the 2D or 3D problem can be reduced to a family of 1D problems using the differentiated backprojection (DBP) method. Each 1D problem consists of recovering a compactly supported function $f \in L^2(\mathcal F)$, where $\mathcal F$ is a finite interval, from its partial Hilbert transform data. When the Hilbert transform is measured on a finite interval $\mathcal G$ that only overlaps but does not cover $\mathcal F$ this inversion problem is known to be severely ill-posed [1]. In this paper, we study the reconstruction of $f$ restricted to the overlap region $\mathcal F \cap \mathcal G$. We show that with this restriction and by assuming prior knowledge on the $L^2$ norm or on the variation of $f$, better stability with H\"older continuity (typical for mildly ill-posed problems) can be obtained.Comment: added one remark, larger fonts for axis labels in figure

Regularized 4D-CT reconstruction from a single dataset with a spatio-temporal prior

X-ray Computerized Tomography (CT) reconstructions can be severely impaired by the patient’s respiratory motion and cardiac beating. Motion must thus be recovered in addition to the 3D reconstruction problem. The approach generally followed to reconstruct dynamic volumes consists of largely increasing the number of projections so that independent reconstructions be possible using only subsets of projections from the same phase of the cyclic movement. Apart from this major trend, motion compensation (MC) aims at recovering the object of interest and its motion by accurately modeling its deformation over time, allowing to use the whole dataset for 4D reconstruction in a coherent way.We consider a different approach for dynamic reconstruction based on inverse problems, without any additional measurements, nor explicit knowledge of the motion. The dynamic sequence is reconstructed out of a single data set, only assuming the motion’s continuity and periodicity. This inverse problem is solved by the minimization of the sum of a data-fidelity term, consistent with the dynamic nature of the data, and a regularization term which implements an efficient spatio-temporal version of the total variation (TV). We demonstrate the potential of this approach and its practical feasibility on 2D and 3D+t reconstructions of a mechanical phantom and patient data