# Diffeomorphism Frameworks

CA [79] aims to develop models to understand the anatomical variability of organs and tissues, including *(a) automated construction of anatomical manifolds, points, curves, surfaces, and subvolumes; (b) comparison of these manifolds; and (c) the statistical codification of the variability of anatomy via probability measures allowing for inference and hypothesis testing of disease states* [80]. An applicable framework to achieve these goals should provide: (a) a mathematical model to describe the space and variability of anatomy, i.e., the shape space and the transformations between shapes; (b) a computable distance metric to measure the difference between shapes; and (c) statistical analysis tools for the shape space.

In the last decade, a diffeomorphism-based CA framework developed by A. Trouve [81], M. Miller [82], L. Younes [83], X. Pennec [84], D. Holm [85], S. Durrleman [86], and their collaborators has undergone tremendous progress to reach these goals. Existing computational frameworks developed along this road map fall into two categories: the Riemannian manifold solution and the Lie group solution with their corresponding terms large deformation diffeomorphic metric mapping (LDDMM) well-known representatives [81-83, 85, 87-89] and stationary vector fields (SVF) [90-93], respectively. Of the two, the LDDMM framework is more mathematically fundamental and general. The SVF can be regarded as a simpler alternative which provides better computational efficiency at the cost of more theoretical limitations.

Taking LDDMM as a general framework of diffeomorphism-based CA, its key components to solve the abovementioned tasks include:

- • Accommodation of variation in the shape space by introducing groups of diffeomorphic transformations carrying individual elements from one to another.
- • Defining a Riemannian metric defined on the shape space to measure continuous deformations of shapes, that is, paths in shape space. The distance between shapes can be defined as the length of the shortest path, in other words, the geodesic that connects two shapes.
- • Providing a Riemannian exponential map that generates the geodesic allowing linearization of shape space. When shapes are represented as initial velocity fields of geodesics connecting them with a fixed reference shape, one effectively works in the linear tangent space over the reference shape. The exponential map allows calculation of statistics in shape space.

In practice, finding the distance and the corresponding *optimal* curve connecting shapes in the shape space to realize this distance is formalized as a registration problem. Once the optimal deformation curves are found, their linearized representations can be used for statistical analysis on the shape manifold.

Sections 2.3.4.1 and 2.3.4.2 explain the LDDMM- and SVF-based image registration and their underlying geometry. Section 2.3.4.3 sketches the basic ideas of statistical analysis on shape space. Typical applications and future directions of diffeomorphism-based CA are addressed in Sect. 2.3.4.4.