Computational anatomy is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical methods for modelling and simulation of biological structures.
The field is broadly defined and includes foundations in anatomy, applied mathematics and pure mathematics, machine learning, computational mechanics, computational science, biological imaging, neuroscience, physics, probability, and statistics; it also has strong connections with fluid mechanics and geometric mechanics. Additionally, it complements newer, interdisciplinary fields like bioinformatics and neuroinformatics in the sense that its interpretation uses metadata derived from the original sensor imaging modalities (of which magnetic resonance imaging is one example). It focuses on the anatomical structures being imaged, rather than the medical imaging devices. It is similar in spirit to the history of computational linguistics, a discipline that focuses on the linguistic structures rather than the sensor acting as the transmission and communication media.
In computational anatomy, the diffeomorphism group is used to study different coordinate systems via coordinate transformations as generated via the Lagrangian and Eulerian velocities of flow in . The flows between coordinates in computational anatomy are constrained to be geodesic flows satisfying the principle of least action for the Kinetic energy of the flow. The kinetic energy is defined through a Sobolev smoothness norm with strictly more than two generalized, square-integrable derivatives for each component of the flow velocity, which guarantees that the flows in are diffeomorphisms. It also implies that the diffeomorphic shape momentum taken pointwise satisfying the Euler–Lagrange equation for geodesics is determined by its neighbors through spatial derivatives on the velocity field. This separates the discipline from the case of incompressible fluids for which momentum is a pointwise function of velocity. Computational anatomy intersects the study of Riemannian manifolds and nonlinear global analysis, where groups of diffeomorphisms are the central focus. Emerging high-dimensional theories of shape are central to many studies in computational anatomy, as are questions emerging from the fledgling field of shape statistics. The metric structures in computational anatomy are related in spirit to morphometrics, with the distinction that Computational anatomy focuses on an infinite-dimensional space of coordinate systems transformed by a diffeomorphism, hence the central use of the terminology diffeomorphometry, the metric space study of coordinate systems via diffeomorphisms.
Genesis At computational anatomy's heart is the comparison of shape by recognizing in one shape the other. This connects it to D'Arcy Wentworth Thompson's developments On Growth and Form which has led to scientific explanations of morphogenesis, the process by which patterns are formed in biology. Albrecht Durer's Four Books on Human Proportion were arguably the earliest works on computational anatomy. The efforts of Noam Chomsky in his pioneering of computational linguistics inspired the original formulation of computational anatomy as a generative model of shape and form from exemplars acted upon via transformations.
Due to the availability of dense 3D measurements via technologies such as magnetic resonance imaging (MRI), computational anatomy has emerged as a subfield of medical imaging and bioengineering for extracting anatomical coordinate systems at the morphome scale in 3D. The spirit of this discipline shares strong overlap with areas such as computer vision and kinematics of rigid bodies, where objects are studied by analysing the groups responsible for the movement in question. Computational anatomy departs from computer vision with its focus on rigid motions, as the infinite-dimensional diffeomorphism group is central to the analysis of Biological shapes. It is a branch of the image analysis and pattern theory school at Brown University pioneered by Ulf Grenander. In Grenander's general metric pattern theory, making spaces of patterns into a metric space is one of the fundamental operations since being able to cluster and recognize anatomical configurations often requires a metric of close and far between shapes. The diffeomorphometry metric of computational anatomy measures how far two diffeomorphic changes of coordinates are from each other, which in turn induces a metric on the shapes and images indexed to them. The models of metric pattern theory, in particular group action on the orbit of shapes and forms is a central tool to the formal definitions in computational anatomy.
History Computational anatomy is the study of shape and form at the morphome or gross anatomy millimeter, or morphology scale, focusing on the study of sub-manifolds of points, curves surfaces and subvolumes of human anatomy. An early modern computational neuro-anatomist was David Van Essen performing some of the early physical unfoldings of the human brain based on printing of a human cortex and cutting. Jean Talairach's publication of Talairach coordinates is an important milestone at the morphome scale demonstrating the fundamental basis of local coordinate systems in studying neuroanatomy and therefore the clear link to charts of differential geometry. Concurrently, virtual mapping in computational anatomy across high resolution dense image coordinates was already happening in Ruzena Bajcy's and Fred Bookstein's earliest developments based on computed axial tomography and magnetic resonance imagery. The earliest introduction of the use of flows of diffeomorphisms for transformation of coordinate systems in image analysis and medical imaging was by Christensen, Joshi, Miller, and Rabbitt.
The first formalization of computational anatomy as an orbit of exemplar templates under diffeomorphism group action was in the original lecture given by Grenander and Miller with that title in May 1997 at the 50th Anniversary of the Division of Applied Mathematics at Brown University, and subsequent publication. This was the basis for the strong departure from much of the previous work on advanced methods for spatial normalization and image registration which were historically built on notions of addition and basis expansion. The structure preserving transformations central to the modern field of Computational Anatomy, homeomorphisms and diffeomorphisms carry smooth submanifolds smoothly. They are generated via Lagrangian and Eulerian flows which satisfy a law of composition of functions forming the group property, but are not additive.
The original model of computational anatomy was as the triple, the group , the orbit of shapes and forms , and the probability laws which encode the variations of the objects in the orbit. The template or collection of templates are elements in the orbit of shapes.
The Lagrangian and Hamiltonian formulations of the equations of motion of computational anatomy took off post 1997 with several pivotal meetings including the 1997 Luminy meeting organized by the Azencott school at Ecole-Normale Cachan on the "Mathematics of Shape Recognition" and the 1998 Trimestre at Institute Henri Poincaré organized by David Mumford "Questions Mathématiques en Traitement du Signal et de l'Image" which catalyzed the Hopkins-Brown-ENS Cachan groups and subsequent developments and connections of computational anatomy to developments in global analysis.
The developments in computational anatomy included the establishment of the Sobolev smoothness conditions on the diffeomorphometry metric to insure existence of solutions of variational problems in the space of diffeomorphisms, the derivation of the Euler–Lagrange equations characterizing geodesics through the group and associated conservation laws, the demonstration of the metric properties of the right invariant metric, the demonstration that the Euler–Lagrange equations have a well-posed initial value problem with unique solutions for all time, and with the first results on sectional curvatures for the diffeomorphometry metric in landmarked spaces. Following the Los Alamos meeting in 2002, Joshi's original large deformation singular Landmark solutions in computational anatomy were connected to peaked solitons or peakons as solutions for the Camassa–Holm equation. Subsequently, connections were made between computational anatomy's Euler–Lagrange equations for momentum densities for the right-invariant metric satisfying Sobolev smoothness to Vladimir Arnold's characterization of the Euler equation for incompressible flows as describing geodesics in the group of volume preserving diffeomorphisms. The first algorithms, generally termed LDDMM for large deformation diffeomorphic mapping for computing connections between landmarks in volumes and spherical manifolds, curves, currents and surfaces, volumes, tensors, varifolds, and time-series have followed.
These contributions of computational anatomy to the global analysis associated to the infinite dimensional manifolds of subgroups of the diffeomorphism group is far from trivial. The original idea of doing differential geometry, curvature and geodesics on infinite dimensional manifolds goes back to Bernhard Riemann's Habilitation (Ueber die Hypothesen, welche der Geometrie zu Grunde liegen); the key modern book laying the foundations of such ideas in global analysis are from Michor.
The applications within medical imaging of computational anatomy continued to flourish after two organized meetings at the Institute for Pure and Applied Mathematics conferences at University of California, Los Angeles. Computational anatomy has been useful in creating accurate models of the atrophy of the human brain at the morphome scale, as well as Cardiac templates, as well as in modeling biological systems. Since the late 1990s, computational anatomy has become an important part of developing emerging technologies for the field of medical imaging. Digital atlases are a fundamental part of modern Medical-school education and in neuroimaging research at the morphome scale. Atlas based methods and virtual textbooks which accommodate variations as in deformable templates are at the center of many neuro-image analysis platforms including Freesurfer, FSL, MRIStudio, SPM. Diffeomorphic registration, introduced in the 1990s, is now an important player with existing codes bases organized around ANTS, DARTEL, DEMONS, LDDMM, StationaryLDDMM, FastLDDMM, are examples of actively used computational codes for constructing correspondences between coordinate systems based on sparse features and dense images. Voxel-based morphometry is an important technology built on many of these principles.
The deformable template orbit model of computational anatomy The model of human anatomy is a deformable template, an orbit of exemplars under group action. Deformable template models have been central to Grenander's metric pattern theory, accounting for typicality via templates, and accounting for variability via transformation of the template. An orbit under group action as the representation of the deformable template is a classic formulation from differential geometry. The space of shapes are denoted , with the group with law of composition ; the action of the group on shapes is denoted , where the action of the group is defined to satisfy
The orbit of the template becomes the space of all shapes, , being homogenous under the action of the elements of .