Motion Estimation From Moments Of Projection Data For Dynamic CT
Gokul Deepak, M
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In X-ray computed tomography, motion of the object (breathing, for example) while X-ray projections are acquired for tomographic reconstruction leads to mo- tion artifacts in the reconstructed image. Object motion (such as that of breathing lungs) during acquisition of a computed tomography scan causes artifacts in the reconstructed image due to the reason that the source and detectors require a finite amount of time to rotate around the object while acquiring measurements even as the object is changing with time. With traditional reconstruction algorithms, the object is assumed to be stationary while data is acquired. However, in the case of dynamic tomography, the projection data that is acquired is not consistent, as it is data measured from an object that is deformed at each view angle of measurement. In this work, we propose a method for estimation of general (non-rigid) small motion for dynamic tomography from motion-corrupted projection data. For a static object, the Helgason-Ludwig consistency conditions impose some structure on the moments of the projections. However in the case of dynamic object (result- ing in motion-corrupted projections) this is violated. In the proposed method, we estimate motion parameters of the general motion model from the moments of the dynamic projections. The dynamic object can be modeled as f (g(x, t)) where g is a time-dependent warping function. The non-linear problem of solving a system involving composition of functions is dealt with in the Fourier transform space where it simplifies into a problem involving multiplicatively separable functions. The system is then linearized to solve for object motion. We assume a general basis function in our model. For numerical simulations, we use polynomial and B-spline basis functions as special cases of the basis functions. Simulation is performed by applying known deformations to the Shepp-Logan phantom, to a head slice of the Visible Human phantom and a thorax slice of the Zubal phantom. Simulations are performed for projections generated by parallel- beam and fan-beam geometry. Simulation for fan-beam geometry are performed by rebinning the motion corrupted fan beam projections to parallel beam projections, followed by the proposed motion estimation method. Simulation for the Visible Human phantom and the thorax slice of the Zubal phantom are performed for fan- beam geometry. Poisson noise is also added to the generated dynamic projections before motion estimation is performed. To solve the ill-posed problem of motion estimation by the proposed method, we use a Tikhonov type regularization that involves minimizing an objective function that is the sum of a data discrepancy term, a term that penalizes temporal variation of motion, and another term to penalize large magnitudes of motion. Using the estimated motion, the original image has been reconstructed from the motion corrupted projection data, with the knowledge of the underlying motion which is estimated by the proposed algorithm, by an algebraic technique similar to the dynamic SART algorithm from the literature. Here, a SART-type coefficient matrix is computed using ray tracing with rays whose paths are warped according to the estimated motion. The dynamic image at t = 0 is then reconstructed with using the computed dynamic SART matrix.