## Final Report Summary - FCCA (Five Challenges in Computational Anatomy)

The goal of computational anatomy is to model and study the variability of anatomical shape. The idea of computational anatomy originates in the seminal book “Growth and Form” by D'Arcy Thompson (Thompson [1917]).

Comparison [of shapes] finds its solution in the elementary use of a certain method of the mathematician. This method is the Method of Coordinates, on which is based the Theory of Transformations. (Thompson [1917) page 1032).

So it was, that shape comparison became a branch of mathematics, and computational anatomy became a science. To have a specific example in mind, let us consider computational neuro-anatomy; i.e., the study of the form and shape of the brain (Grenander [1998]). The observations in this case are the diagnostic tools accessible to the clinician. Of particular interest to us are noninvasive imaging techniques, such as computed tomography (CT), magnetic resonance imaging (MRI), functional MRI and diffusion tensor imaging (DTI). The observations ignore all the processes taking place at cellular, organ and environmental level, which together influence and form the anatomical shape of the brain. The observations simply provide images, and the shape of each image is encoded in its data structure. Given two images, the problem of finding the transformation between them is called the problem of image registration. One compares these transformations of the various data structures in order to infer information about anatomical shape and its variability.

The FCCA project was based on the mathematical concept of the momentum map, which for a given data structure collectivises the information in the images into manageable coherent geometric properties, e.g., landmarks, rather than individual voxels. We measured the deformation of shape required in registering one image to another as the cost for the evolution of the momentum map for a given data structure to pass along an optimal path. The optimal path minimised a certain cost associated with the given data structure, and was obtained from a variational principal defined on the space of smooth invertible transformations known as diffeos. This variational principle implies a fundamental equation known as EPDiff, which is short for the Euler-Poincaré equation on the diffeos. Thus, EPDiff governs the optimal motion of the momentum map that determines the process of registering two given images. (Bruveris [2015], Holm [2004,2005,2009]) This registration quantifies the comparison of shapes by measuring the minimum distance between them along the optimal path (minimum cost) for the momentum map evolution. The momentum map concept thereby unified our approach for meeting the goals of the FCCA project in applying modern transformation theory to computational anatomy.

The momentum map concept had been pursued vigorously in the PI’s earlier work in fluid dynamics of various types, particularly in using variational principles from geometric mechanics to formulate the dynamics of momentum maps in fluid dynamics (Holm [1998]).

The FCCA project began by establishing three new capabilities for computational image registration by using the momentum map approach. These were the capabilities to: (1) register images of arbitrary data structures; (2) register combined data structures (data fusion); (3) register images at different resolutions. We then enhanced these three new capabilities by extending them into two additional challenge domains: Longitudinal investigations of sequences of images; and registration of images with different topologies (metamorphosis). In the latter stages of the project, we recognised that all of these new deterministic capabilities based on momentum map dynamics could be made stochastic and then could be used as a basis for quantifying uncertainty and variability in computational anatomy. This is a new direction for computational anatomy, in which activity is now burgeoning. The momentum map approach for quantifying uncertainty will be a new direction for our endeavours.

Comparison [of shapes] finds its solution in the elementary use of a certain method of the mathematician. This method is the Method of Coordinates, on which is based the Theory of Transformations. (Thompson [1917) page 1032).

So it was, that shape comparison became a branch of mathematics, and computational anatomy became a science. To have a specific example in mind, let us consider computational neuro-anatomy; i.e., the study of the form and shape of the brain (Grenander [1998]). The observations in this case are the diagnostic tools accessible to the clinician. Of particular interest to us are noninvasive imaging techniques, such as computed tomography (CT), magnetic resonance imaging (MRI), functional MRI and diffusion tensor imaging (DTI). The observations ignore all the processes taking place at cellular, organ and environmental level, which together influence and form the anatomical shape of the brain. The observations simply provide images, and the shape of each image is encoded in its data structure. Given two images, the problem of finding the transformation between them is called the problem of image registration. One compares these transformations of the various data structures in order to infer information about anatomical shape and its variability.

The FCCA project was based on the mathematical concept of the momentum map, which for a given data structure collectivises the information in the images into manageable coherent geometric properties, e.g., landmarks, rather than individual voxels. We measured the deformation of shape required in registering one image to another as the cost for the evolution of the momentum map for a given data structure to pass along an optimal path. The optimal path minimised a certain cost associated with the given data structure, and was obtained from a variational principal defined on the space of smooth invertible transformations known as diffeos. This variational principle implies a fundamental equation known as EPDiff, which is short for the Euler-Poincaré equation on the diffeos. Thus, EPDiff governs the optimal motion of the momentum map that determines the process of registering two given images. (Bruveris [2015], Holm [2004,2005,2009]) This registration quantifies the comparison of shapes by measuring the minimum distance between them along the optimal path (minimum cost) for the momentum map evolution. The momentum map concept thereby unified our approach for meeting the goals of the FCCA project in applying modern transformation theory to computational anatomy.

The momentum map concept had been pursued vigorously in the PI’s earlier work in fluid dynamics of various types, particularly in using variational principles from geometric mechanics to formulate the dynamics of momentum maps in fluid dynamics (Holm [1998]).

The FCCA project began by establishing three new capabilities for computational image registration by using the momentum map approach. These were the capabilities to: (1) register images of arbitrary data structures; (2) register combined data structures (data fusion); (3) register images at different resolutions. We then enhanced these three new capabilities by extending them into two additional challenge domains: Longitudinal investigations of sequences of images; and registration of images with different topologies (metamorphosis). In the latter stages of the project, we recognised that all of these new deterministic capabilities based on momentum map dynamics could be made stochastic and then could be used as a basis for quantifying uncertainty and variability in computational anatomy. This is a new direction for computational anatomy, in which activity is now burgeoning. The momentum map approach for quantifying uncertainty will be a new direction for our endeavours.