LDDMM y GANs: Redes Generativas Antagónicas para Registro Difeomorfico.

El Registro Difeomorfico de imágenes es un problema clave para muchas aplicaciones de la Anatomía Computacional. Tradicionalmente, el registro deformable de imagen ha sido formulado como un problema variacional, resoluble mediante costosos métodos de optimización numérica. En la última década, contribuciones en la forma de nuevos métodos basados en formulaciones tradicionales están decreciendo, mientras que más modelos basados en Aprendizaje profundo están siendo desarrollados para aprender registros deformables de imágenes. En este trabajo contribuimos a esta nueva corriente proponiendo un novedoso método LDDMM para registro difeomorfico de imágenes 3D, basado en redes generativas antagónicas. Combinamos las arquitecturas de generadores y discriminadores con mejores prestaciones en registro deformable con el paradigma LDDMM. Hemos implementado con éxito tres modelos para distintas parametrizaciones de difeomorfismos, los cuales demuestran resultados competitivos en comparación con métodos del estado del arte tanto tradicionales como basados en aprendizaje profundo.<br /

Comparing algorithms for diffeomorphic registration: Stationary LDDMM and Diffeomorphic Demons

International audienceThe stationary parameterization of diffeomorphisms is be- ing increasingly used in computational anatomy. In certain applications it provides similar results to the non-stationary parameterization alle- viating the computational charge. With this characterization for diffeo- morphisms, two different registration algorithms have been recently pro- posed: stationary LDDMM and diffeomorphic Demons. To our knowl- edge, their theoretical and practical differences have not been analyzed yet. In this article we provide a comparison between both algorithms in a common framework. To this end, we have studied the differences in the elements of both registration scenarios. We have analyzed the sen- sitivity of the regularization parameters in the smoothness of the final transformations and compared the performance of the registration re- sults. Moreover, we have studied the potential of both algorithms for the computation of essential operations for further statistical analysis. We have found that both methods have comparable performance in terms of image matching although the transformations are qualitatively different in some cases. Diffeomorphic Demons shows a slight advantage in terms of computational time. However, it does not provide as stationary LD- DMM the vector field in the tangent space needed to compute statistics or exact inverse transformations

Efficient Parallel Transport of Deformations in Time Series of Images: from Schild's to Pole Ladder

International audienceGroup-wise analysis of time series of images requires to compare longitudinal evolutions of images observed on different subjects. In medical imaging, longitudinal anatomical changes can be modeled thanks to non-rigid registration of follow-up images. The comparison of longitudinal trajectories requires the transport (or "normalization") of longitudinal deformations in a common reference frame. We previously proposed an effective computational scheme based on the Schild's ladder for the parallel transport of diffeomorphic deformations parameterized by tangent velocity fields, based on the construction of a geodesic parallelogram on a manifold. Schild's ladder may be however inefficient for transporting longitudinal deformations from image time series of multiple time points, in which the computation of the geodesic diagonals is required several times. We propose here a new algorithm, the pole ladder, in which one diagonal of the parallelogram is the baseline-to-reference frame geodesic. This drastically reduces the number of geodesics to compute. Moreover, differently from the Schild's ladder, the pole ladder is symmetric with respect to the baseline-to-reference frame geodesic. From the theoretical point of view, we show that the pole ladder is rigorously equivalent to the Schild's ladder when transporting along geodesics. From the practical point of view, we establish the computational advantages and demonstrate the effectiveness of this very simple method by comparing with standard methods of transport on simulated images with progressing brain atrophy. Finally, we illustrate its application to a clinical problem: the measurement of the longitudinal progression in Alzheimer's disease. Results suggest that an important gain in sensitivity could be expected in group-wise comparisons

Hierarchical Geodesic Models in Diffeomorphisms

Hierarchical linear models (HLMs) are a standard approach for analyzing data where individuals are measured repeatedly over time. However, such models are only applicable to longitudinal studies of Euclidean data. This paper develops the theory of hierarchical geodesic models (HGMs), which generalize HLMs to the manifold setting. Our proposed model quantifies longitudinal trends in shapes as a hierarchy of geodesics in the group of diffeomorphisms. First, individual-level geodesics represent the trajectory of shape changes within individuals. Second, a group-level geodesic represents the average trajectory of shape changes for the population. Our proposed HGM is applicable to longitudinal data from unbalanced designs, i.e., varying numbers of timepoints for individuals, which is typical in medical studies. We derive the solution of HGMs on diffeomorphisms to estimate individual-level geodesics, the group geodesic, and the residual diffeomorphisms. We also propose an efficient parallel algorithm that easily scales to solve HGMs on a large collection of 3D images of several individuals. Finally, we present an effective model selection procedure based on cross validation. We demonstrate the effectiveness of HGMs for longitudinal analysis of synthetically generated shapes and 3D MRI brain scans

PDE-constrained LDDMM via geodesic shooting and inexact Gauss-Newton-Krylov optimization using the incremental adjoint Jacobi equations

The class of non-rigid registration methods proposed in the framework of PDE-constrained Large Deformation Diffeomorphic Metric Mapping is a particularly interesting family of physically meaningful diffeomorphic registration methods. Inexact Newton-Krylov optimization has shown an excellent numerical accuracy and an extraordinarily fast convergence rate in this framework. However, the Galerkin representation of the non-stationary velocity fields does not provide proper geodesic paths. In this work, we propose a method for PDE-constrained LDDMM parameterized in the space of initial velocity fields under the EPDiff equation. The derivation of the gradient and the Hessian-vector products are performed on the final velocity field and transported backward using the adjoint and the incremental adjoint Jacobi equations. This way, we avoid the complex dependence on the initial velocity field in the derivations and the computation of the adjoint equation and its incremental counterpart. The proposed method provides geodesics in the framework of PDE-constrained LDDMM, and it shows performance competitive to benchmark PDE-constrained LDDMM and EPDiff-LDDMM methods

Doctor of Philosophy

dissertationAn important aspect of medical research is the understanding of anatomy and its relation to function in the human body. For instance, identifying changes in the brain associated with cognitive decline helps in understanding the process of aging and age-related neurological disorders. The field of computational anatomy provides a rich mathematical setting for statistical analysis of complex geometrical structures seen in 3D medical images. At its core, computational anatomy is based on the representation of anatomical shape and its variability as elements of nonflat manifold of diffeomorphisms with an associated Riemannian structure. Although such manifolds effectively represent natural biological variability, intrinsic methods of statistical analysis within these spaces remain deficient at large. This dissertation contributes two critical missing pieces for statistics in diffeomorphisms: (1) multivariate regression models for cross-sectional study of shapes, and (2) generalization of classical Euclidean, mixed-effects models to manifolds for longitudinal studies. These models are based on the principle that statistics on manifold-valued information must respect the intrinsic geometry of that space. The multivariate regression methods provide statistical descriptors of the relationships of anatomy with clinical indicators. The novel theory of hierarchical geodesic models (HGMs) is developed as a natural generalization of hierarchical linear models (HLMs) to describe longitudinal data on curved manifolds. Using a hierarchy of geodesics, the HGMs address the challenge of modeling the shape-data with unbalanced designs typically arising as a result of follow-up medical studies. More generally, this research establishes a mathematical foundation to study dynamics of changes in anatomy and the associated clinical progression with time. This dissertation also provides efficient algorithms that utilize state-of-the-art high performance computing architectures to solve models on large-scale, longitudinal imaging data. These manifold-based methods are applied to predictive modeling of neurological disorders such as Alzheimer's disease. Overall, this dissertation enables clinicians and researchers to better utilize the structural information available in medical images