Partial Differential Equation-Constrained Diffeomorphic Registration from Sum of Squared Differences to Normalized Cross-Correlation, Normalized Gradient Fields, and Mutual Information: A Unifying Framework; 35632143

This work proposes a unifying framework for extending PDE-constrained Large Deformation Diffeomorphic Metric Mapping (PDE-LDDMM) with the sum of squared differences (SSD) to PDE-LDDMM with different image similarity metrics. We focused on the two best-performing variants of PDE-LDDMM with the spatial and band-limited parameterizations of diffeomorphisms. We derived the equations for gradient-descent and Gauss-Newton-Krylov (GNK) optimization with Normalized Cross-Correlation (NCC), its local version (lNCC), Normalized Gradient Fields (NGFs), and Mutual Information (MI). PDE-LDDMM with GNK was successfully implemented for NCC and lNCC, substantially improving the registration results of SSD. For these metrics, GNK optimization outperformed gradient-descent. However, for NGFs, GNK optimization was not able to overpass the performance of gradient-descent. For MI, GNK optimization involved the product of huge dense matrices, requesting an unaffordable memory load. The extensive evaluation reported the band-limited version of PDE-LDDMM based on the deformation state equation with NCC and lNCC image similarities among the best performing PDE-LDDMM methods. In comparison with benchmark deep learning-based methods, our proposal reached or surpassed the accuracy of the best-performing models. In NIREP16, several configurations of PDE-LDDMM outperformed ANTS-lNCC, the best benchmark method. Although NGFs and MI usually underperformed the other metrics in our evaluation, these metrics showed potentially competitive results in a multimodal deformable experiment. We believe that our proposed image similarity extension over PDE-LDDMM will promote the use of physically meaningful diffeomorphisms in a wide variety of clinical applications depending on deformable image registration

Doctor of Philosophy in Computing

dissertationAn important area of medical imaging research is studying anatomical diffeomorphic shape changes and detecting their relationship to disease processes. For example, neurodegenerative disorders change the shape of the brain, thus identifying differences between the healthy control subjects and patients affected by these diseases can help with understanding the disease processes. Previous research proposed a variety of mathematical approaches for statistical analysis of geometrical brain structure in three-dimensional (3D) medical imaging, including atlas building, brain variability quantification, regression, etc. The critical component in these statistical models is that the geometrical structure is represented by transformations rather than the actual image data. Despite the fact that such statistical models effectively provide a way for analyzing shape variation, none of them have a truly probabilistic interpretation. This dissertation contributes a novel Bayesian framework of statistical shape analysis for generic manifold data and its application to shape variability and brain magnetic resonance imaging (MRI). After we carefully define the distributions on manifolds, we then build Bayesian models for analyzing the intrinsic variability of manifold data, involving the mean point, principal modes, and parameter estimation. Because there is no closed-form solution for Bayesian inference of these models on manifolds, we develop a Markov Chain Monte Carlo method to sample the hidden variables from the distribution. The main advantages of these Bayesian approaches are that they provide parameter estimation and automatic dimensionality reduction for analyzing generic manifold-valued data, such as diffeomorphisms. Modeling the mean point of a group of images in a Bayesian manner allows for learning the regularity parameter from data directly rather than having to set it manually, which eliminates the effort of cross validation for parameter selection. In population studies, our Bayesian model of principal modes analysis (1) automatically extracts a low-dimensional, second-order statistics of manifold data variability and (2) gives a better geometric data fit than nonprobabilistic models. To make this Bayesian framework computationally more efficient for high-dimensional diffeomorphisms, this dissertation presents an algorithm, FLASH (finite-dimensional Lie algebras for shooting), that hugely speeds up the diffeomorphic image registration. Instead of formulating diffeomorphisms in a continuous variational problem, Flash defines a completely new discrete reparameterization of diffeomorphisms in a low-dimensional bandlimited velocity space, which results in the Bayesian inference via sampling on the space of diffeomorphisms being more feasible in time. Our entire Bayesian framework in this dissertation is used for statistical analysis of shape data and brain MRIs. It has the potential to improve hypothesis testing, classification, and mixture models