Large deformation diffeomorphic metric mapping
Diffeomorphic mapping is a broad term that actually refers to a number of different algorithms, processes, and methods.
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.
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[1][2] for volumes and Joshi's landmark matching for point sets with correspondence,[3][4] with LDDMM algorithms now available for computing diffeomorphic metric maps between non-corresponding landmarks[5] and landmark matching intrinsic to spherical manifolds,[6] curves,[7] currents and surfaces,[8][9][10] tensors,[11] varifolds,[12] and time-series.
[13][14][15] The term LDDMM was first established as part of the National Institutes of Health supported Biomedical Informatics Research Network.
There are now many codes organized around diffeomorphic registration[17] including ANTS,[18] DARTEL,[19] DEMONS,[20] StationaryLDDMM,[21] FastLDDMM,[22][23] 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.
Diffeomorphic mapping 3-dimensional information across coordinate systems is central to high-resolution Medical imaging and the area of Neuroinformatics within the newly emerging field of bioinformatics.
Diffeomorphic mapping 3-dimensional coordinate systems as measured via high resolution dense imagery has a long history in 3-D beginning with Computed Axial Tomography (CAT scanning) in the early 80's by the University of Pennsylvania group led by Ruzena Bajcsy,[24] and subsequently the Ulf Grenander school at Brown University with the HAND experiments.
[27][28][29][30][31] The central focus of the sub-field of Computational anatomy (CA) within medical imaging is mapping information across anatomical coordinate systems at the 1 millimeter morphome scale.
Methods based on linear or non-linear elasticity energetics which grows with distance from the identity mapping of the template, is not appropriate for cross-sectional study.
Rather, in models based on Lagrangian and Eulerian flows of diffeomorphisms, the constraint is associated to topological properties, such as open sets being preserved, coordinates not crossing implying uniqueness and existence of the inverse mapping, and connected sets remaining connected.
[19][34] Such methods are powerful in that they introduce notions of regularity of the solutions so that they can be differentiated and local inverses can be calculated.
The disadvantages of these methods is that there was no associated global least-action property which could score the flows of minimum energy.
This contrasts the geodesic motions which are central to the study of Rigid body kinematics and the many problems solved in Physics via Hamilton's principle of least action.
In 1998, Dupuis, Grenander and Miller[35] established the conditions for guaranteeing the existence of solutions for dense image matching in the space of flows of diffeomorphisms.
These conditions require an action penalizing kinetic energy measured via the Sobolev norm on spatial derivatives of the flow of vector fields.
The large deformation diffeomorphic metric mapping (LDDMM) code that Faisal Beg derived and implemented for his PhD at Johns Hopkins University[36] developed the earliest algorithmic code which solved for flows with fixed points satisfying the necessary conditions for the dense image matching problem subject to least-action.
Computational anatomy now has many existing codes organized around diffeomorphic registration[17] including ANTS,[18] DARTEL,[19] DEMONS,[37] LDDMM,[2] StationaryLDDMM[21] as examples of actively used computational codes for constructing correspondences between coordinate systems based on dense images.
These large deformation methods have been extended to landmarks without registration via measure matching,[38] curves,[39] surfaces,[40] dense vector[41] and tensor [42] imagery, and varifolds removing orientation.
[43] Deformable shape in computational anatomy (CA)[44][45][46][47]is studied via the use of diffeomorphic mapping for establishing correspondences between anatomical coordinates in Medical Imaging.
In this setting, three dimensional medical images are modelled as a random deformation of some exemplar, termed the template
, satisfying the Lagrangian and Eulerian specification of the flow field associated to the ordinary differential equation, with
are modelled as a reproducing Kernel Hilbert space (RKHS) defined by a 1-1, differential operator
The original large deformation diffeomorphic metric mapping (LDDMM) algorithms of Beg, Miller, Trouve, Younes[51] was derived taking variations with respect to the vector field parameterization of the group, since
to gives the first variation LDDMM matching based on the principal eigenvector of the diffusion tensor matrix takes the image
uniformly distributed directions on the sphere and can characterize more complex fiber geometries by reconstructing an orientation distribution function (ODF) that characterizes the angular profile of the diffusion probability density function of water molecules.
Define the variational problem assuming that two ODF volumes can be generated from one to another via flows of diffeomorphisms
