Spatiotemporal PET Reconstruction Using ML-EM with Learned Diffeomorphic Deformation

Abstract

Patient movement in emission tomography deteriorates reconstruction quality because of motion blur. Gating the data improves the situation somewhat: each gate contains a movement phase which is approximately stationary. A standard method is to use only the data from a few gates, with little movement between them. However, the corresponding loss of data entails an increase of noise. Motion correction algorithms have been implemented to take into account all the gated data, but they do not scale well, especially not in 3D. We propose a novel motion correction algorithm which addresses the scalability issue. Our approach is to combine an enhanced ML-EM algorithm with deep learning based movement registration. The training is unsupervised, and with artificial data. We expect this approach to scale very well to higher resolutions and to 3D, as the overall cost of our algorithm is only marginally greater than that of a standard ML-EM algorithm. We show that we can significantly decrease the noise corresponding to a limited number of gates.

The ML–EM algorithm in continuum: sparse measure solutions

“…In that case, movement can be modelled by deformations of images which do not easily carry over to discretised images (simply because interesting deformations do not preserve the grid). It is then desirable to express the problem in a continuous setting, in order to both analyse the algorithms proposed in the literature, and to derive new ones [14,15].…”

Deep Learning Based Joint PET Image Reconstruction and Motion Estimation

“…Similar to our method, Öktem et al proposed a joint estimation method that incorporated a DL based on probabilistic diffeomorphic deformation model which is differentiable and invertible. However, their algorithm does not find the maximum of a joint objective function and cannot guarantee the convergence [39].…”

ML-EM algorithm with known continuous movement model

“…In other words, we recover the algorithm for gated data, proposed in [9] for the intensity-preserving action, and generalised in [13,15]. The ensuing analysis is up to our knowledge the first rigorous justification for these informally-derived algorithms, under the assumption of piecewise-constant movement.…”

“…We also emphasise that in the case of gated data, i.e., when a physical device allows to group counts per phase in which the movement can be assumed to be stationary, an ML-EM algorithm has been derived informally in the literature [9, 13,15]. We recover that algorithm as well when the W t are assumed to be piecewise constant in time.…”

“…Our approach can also serve as a building block for joint motion estimation and reconstruction, when the movement model is unknown. A typical strategy to solve the resulting optimisation problem is to intertwine estimating the image and the transformation (typically a diffeomorphism) [9, 13,15]. Thus, given a current Outline of the paper.…”