Computational anatomy | Geometry
Large deformation diffeomorphic metric mapping
Large deformation diffeomorphic metric mapping (LDDMM) is a specific suite of algorithms used for diffeomorphic mapping and manipulating dense imagery based on diffeomorphic metric mapping within the academic discipline of computational anatomy, to be distinguished from its precursor based on diffeomorphic mapping. The distinction between the two is that diffeomorphic metric maps satisfy the property that the length associated to their flow away from the identity induces a metric on the group of diffeomorphisms, which in turn induces a metric on the orbit of shapes and forms within the field of Computational Anatomy. The study of shapes and forms with the metric of diffeomorphic metric mapping is called diffeomorphometry. A diffeomorphic mapping system is a system designed to map, manipulate, and transfer information which is stored in many types of spatially distributed medical imagery. Diffeomorphic mapping is the underlying technology for mapping and analyzing information measured in human anatomical coordinate systems which have been measured via Medical imaging. Diffeomorphic mapping is a broad term that actually refers to a number of different algorithms, processes, and methods. It is attached to many operations and has many applications for analysis and visualization. Diffeomorphic mapping can be used to relate various sources of information which are indexed as a function of spatial position as the key index variable. Diffeomorphisms are by their Latin root structure preserving transformations, which are in turn differentiable and therefore smooth, allowing for the calculation of metric based quantities such as arc length and surface areas. Spatial location and extents in human anatomical coordinate systems can be recorded via a variety of Medical imaging modalities, generally termed multi-modal medical imagery, providing either scalar and or vector quantities at each spatial location. Examples are scalar T1 or T2 magnetic resonance imagery, or as 3x3 diffusion tensor matrices diffusion MRI and diffusion-weighted imaging, to scalar densities associated to computed tomography (CT), or functional imagery such as temporal data of functional magnetic resonance imaging and scalar densities such as Positron emission tomography (PET). Computational anatomy is a subdiscipline within the broader field of neuroinformatics within bioinformatics and medical imaging. The first algorithm for dense image mapping via diffeomorphic metric mapping was Beg's LDDMM for volumes and Joshi's landmark matching for point sets with correspondence, with LDDMM algorithms now available for computing diffeomorphic metric maps between non-corresponding landmarks and landmark matching intrinsic to spherical manifolds, curves, currents and surfaces, tensors, varifolds, and time-series. The term LDDMM was first established as part of the National Institutes of Health supported Biomedical Informatics Research Network. In a more general sense, diffeomorphic mapping is any solution that registers or builds correspondences between dense coordinate systems in medical imaging by ensuring the solutions are diffeomorphic. There are now many codes organized around diffeomorphic registration including ANTS, DARTEL, DEMONS, StationaryLDDMM, FastLDDMM, as examples of actively used computational codes for constructing correspondences between coordinate systems based on dense images. The distinction between diffeomorphic metric mapping forming the basis for LDDMM and the earliest methods of diffeomorphic mapping is the introduction of a Hamilton principle of least-action in which large deformations are selected of shortest length corresponding to geodesic flows. This important distinction arises from the original formulation of the Riemannian metric corresponding to the right-invariance. The lengths of these geodesics give the metric in the metric space structure of human anatomy. Non-geodesic formulations of diffeomorphic mapping in general does not correspond to any metric formulation. (Wikipedia).
