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Margaret Duff, Ivor J. A. Simpson, Matthias J. Ehrhardt, Neill D. F. Campbell

Learned regularization for MRI reconstruction can provide complex data-driven priors to inverse problems while still retaining the control and insight of a variational regularization method. Moreover, unsupervised learning, without paired training data, allows the learned regularizer to remain flexible to changes in the forward problem such as noise level, sampling pattern or coil sensitivities. One such approach uses generative models, trained on ground-truth images, as priors for inverse problems, penalizing reconstructions far from images the generator can produce. In this work, we utilize variational autoencoders (VAEs) that generate not only an image but also a covariance uncertainty matrix for each image. The covariance can model changing uncertainty dependencies caused by structure in the image, such as edges or objects, and provides a new distance metric from the manifold of learned images. We demonstrate these novel generative regularizers on radially sub-sampled MRI knee measurements from the fastMRI dataset and compare them to other unlearned, unsupervised and supervised methods. Our results show that the proposed method is competitive with other state-of-the-art methods and behaves consistently with changing sampling patterns and noise levels.

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Ivor J. A. Simpson, Ashley McManamon, Balázs Örzsik, Alan J. Stone, Nicholas P. Blockley, Iris Asllani, Alessandro Colasanti, Mara Cercignani

Streamlined qBOLD acquisitions enable experimentally straightforward observations of brain oxygen metabolism. $R_2^\prime$ maps are easily inferred; however, the Oxygen extraction fraction (OEF) and deoxygenated blood volume (DBV) are more ambiguously determined from the data. As such, existing inference methods tend to yield very noisy and underestimated OEF maps, while overestimating DBV. This work describes a novel probabilistic machine learning approach that can infer plausible distributions of OEF and DBV. Initially, we create a model that produces informative voxelwise prior distribution based on synthetic training data. Contrary to prior work, we model the joint distribution of OEF and DBV through a scaled multivariate logit-Normal distribution, which enables the values to be constrained within a plausible range. The prior distribution model is used to train an efficient amortized variational Bayesian inference model. This model learns to infer OEF and DBV by predicting real image data, with few training data required, using the signal equations as a forward model. We demonstrate that our approach enables the inference of smooth OEF and DBV maps, with a physiologically plausible distribution that can be adapted through specification of an informative prior distribution. Other benefits include model comparison (via the evidence lower bound) and uncertainty quantification for identifying image artefacts. Results are demonstrated on a small study comparing subjects undergoing hyperventilation and at rest. We illustrate that the proposed approach allows measurement of gray matter differences in OEF and DBV and enables voxelwise comparison between conditions, where we observe significant increases in OEF and $R_2^\prime$ during hyperventilation.

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Ivor J. A. Simpson, Sara Vicente, Neill D. F. Campbell

This paper proposes using a sparse-structured multivariate Gaussian to provide a closed-form approximator for the output of probabilistic ensemble models used for dense image prediction tasks. This is achieved through a convolutional neural network that predicts the mean and covariance of the distribution, where the inverse covariance is parameterised by a sparsely structured Cholesky matrix. Similarly to distillation approaches, our single network is trained to maximise the probability of samples from pre-trained probabilistic models, in this work we use a fixed ensemble of networks. Once trained, our compact representation can be used to efficiently draw spatially correlated samples from the approximated output distribution. Importantly, this approach captures the uncertainty and structured correlations in the predictions explicitly in a formal distribution, rather than implicitly through sampling alone. This allows direct introspection of the model, enabling visualisation of the learned structure. Moreover, this formulation provides two further benefits: estimation of a sample probability, and the introduction of arbitrary spatial conditioning at test time. We demonstrate the merits of our approach on monocular depth estimation and show that the advantages of our approach are obtained with comparable quantitative performance.

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Jan Rühaak, Thomas Polzin, Stefan Heldmann, Ivor J. A. Simpson, Heinz Handels, Jan Modersitzki, Mattias P. Heinrich

We present a novel algorithm for the registration of pulmonary CT scans. Our method is designed for large respiratory motion by integrating sparse keypoint correspondences into a dense continuous optimization framework. The detection of keypoint correspondences enables robustness against large deformations by jointly optimizing over a large number of potential discrete displacements, whereas the dense continuous registration achieves subvoxel alignment with smooth transformations. Both steps are driven by the same normalized gradient fields data term. We employ curvature regularization and a volume change control mechanism to prevent foldings of the deformation grid and restrict the determinant of the Jacobian to physiologically meaningful values. Keypoint correspondences are integrated into the dense registration by a quadratic penalty with adaptively determined weight. Using a parallel matrix-free derivative calculation scheme, a runtime of about 5 min was realized on a standard PC. The proposed algorithm ranks first in the EMPIRE10 challenge on pulmonary image registration. Moreover, it achieves an average landmark distance of 0.82 mm on the DIR-Lab COPD database, thereby improving upon the state of the art in accuracy by 15%. Our algorithm is the first to reach the inter-observer variability in landmark annotation on this dataset.

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