Computational Anatomy

Computational anatomy is an interdisciplinary field focused on quantitative investigation and modelling of anatomical shape variability, integrating anatomy, applied and pure mathematics, machine learning, computational mechanics, computational science, biological imaging, neuroscience, physics, probability, and statistics. The field uses the diffeomorphism group to study coordinate systems via coordinate transformations, with flows constrained to be geodesic flows satisfying the principle of least action for kinetic energy, defined through a Sobolev smoothness norm with more than two square-integrable derivatives, ensuring diffeomorphic flows.

Computational anatomy Computational anatomy is an interdisciplinary field of biology https://en.wikipedia.org/wiki/Biology focused on quantitative investigation and modelling of anatomical shapes variability. 1 cite note-1 2 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 https://en.wikipedia.org/wiki/Anatomy , applied mathematics https://en.wikipedia.org/wiki/Applied mathematics and pure mathematics https://en.wikipedia.org/wiki/Pure mathematics , machine learning https://en.wikipedia.org/wiki/Machine learning , computational mechanics https://en.wikipedia.org/wiki/Computational mechanics , computational science https://en.wikipedia.org/wiki/Computational science , biological imaging https://en.wikipedia.org/wiki/Biological imaging , neuroscience https://en.wikipedia.org/wiki/Neuroscience , physics https://en.wikipedia.org/wiki/Physics , probability https://en.wikipedia.org/wiki/Probability , and statistics https://en.wikipedia.org/wiki/Statistics ; it also has strong connections with fluid mechanics https://en.wikipedia.org/wiki/Fluid mechanics and geometric mechanics https://en.wikipedia.org/wiki/Geometric mechanics . Additionally, it complements newer, interdisciplinary fields like bioinformatics https://en.wikipedia.org/wiki/Bioinformatics and neuroinformatics https://en.wikipedia.org/wiki/Neuroinformatics in the sense that its interpretation uses metadata derived from the original sensor imaging modalities of which magnetic resonance imaging https://en.wikipedia.org/wiki/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 https://en.wikipedia.org/wiki/Computational linguistics , a discipline that focuses on the linguistic structures rather than the sensor https://en.wikipedia.org/wiki/Sensor acting as the transmission https://en.wikipedia.org/wiki/Transmission medium and communication media. In computational anatomy, the diffeomorphism https://en.wikipedia.org/wiki/Diffeomorphism group is used to study different coordinate systems via coordinate transformations https://en.wikipedia.org/wiki/Change of basis as generated via the Lagrangian and Eulerian velocities of flow https://en.wikipedia.org/wiki/Lagrangian and Eulerian specification of the flow field in . The flows between coordinates in computational anatomy Lagrangian and Eulerian flows for generating diffeomorphisms are constrained to be geodesic flows The metric on geodesic flows of landmarks, surfaces, and volumes within the orbit satisfying the principle of least action for the Kinetic energy of the flow The action integral for Hamilton's principle on diffeomorphic flows . The kinetic energy is defined through a Sobolev smoothness The Sobolev smoothness condition on vector fields as modeled in a reproducing kernel Hilbert space norm with strictly more than two generalized, square-integrable https://en.wikipedia.org/wiki/Square-integrable function derivatives for each component of the flow velocity https://en.wikipedia.org/wiki/Flow velocity , which guarantees that the flows in are diffeomorphisms. 3 It also implies that the diffeomorphic shape momentum The Sobolev smoothness condition on vector fields as modeled in a reproducing kernel Hilbert space taken pointwise satisfying the Euler–Lagrange equation for geodesics The Euler–Lagrange equation on shape momentum for geodesics on the group of diffeomorphisms is determined by its neighbors through spatial derivatives on the velocity field. This separates the discipline from the case of incompressible fluids https://en.wikipedia.org/wiki/Incompressible flow for which momentum is a pointwise function of velocity. Computational anatomy intersects the study of 4 cite note-MR202082-4 Riemannian manifolds https://en.wikipedia.org/wiki/Riemannian manifolds and nonlinear global analysis https://en.wikipedia.org/wiki/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 5 cite note-5 shape statistics https://en.wikipedia.org/wiki/Shape statistics . The metric structures in computational anatomy are related in spirit to morphometrics https://en.wikipedia.org/wiki/Morphometrics , with the distinction that Computational anatomy focuses on an infinite-dimensional space of coordinate systems https://en.wikipedia.org/wiki/Coordinate system transformed by a diffeomorphism https://en.wikipedia.org/wiki/Diffeomorphism , hence the central use of the terminology , the metric space study of coordinate systems via diffeomorphisms. Diffeomorphometry: The metric space of shapes and forms diffeomorphometry Genesis edit /w/index.php?title=Computational anatomy&action=edit§ion=1 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 https://en.wikipedia.org/wiki/D'Arcy Wentworth Thompson 's developments On Growth and Form which has led to scientific explanations of morphogenesis https://en.wikipedia.org/wiki/Morphogenesis , the process by which patterns https://en.wikipedia.org/wiki/Patterns are formed in biology https://en.wikipedia.org/wiki/Biology . Albrecht Durer https://en.wikipedia.org/wiki/Albrecht Dürer 's Four Books on Human Proportion were arguably the earliest works on computational anatomy. 6 cite note-6 7 cite note-7 The efforts of 8 cite note-8 Noam Chomsky https://en.wikipedia.org/wiki/Noam Chomsky in his pioneering of computational linguistics https://en.wikipedia.org/wiki/Computational Linguistics inspired the original formulation of computational anatomy as a generative model of shape and form from exemplars acted upon via transformations. 9 cite note-:20-9 Due to the availability of dense 3D measurements via technologies such as magnetic resonance imaging https://en.wikipedia.org/wiki/Magnetic resonance imaging MRI , computational anatomy has emerged as a subfield of medical imaging https://en.wikipedia.org/wiki/Medical imaging and bioengineering https://en.wikipedia.org/wiki/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 https://en.wikipedia.org/wiki/Computer vision and kinematics https://en.wikipedia.org/wiki/Kinematics of rigid bodies https://en.wikipedia.org/wiki/Rigid bodies , where objects are studied by analysing the groups https://en.wikipedia.org/wiki/Group mathematics 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 10 pioneered by Ulf Grenander https://en.wikipedia.org/wiki/Ulf Grenander . In Grenander's general metric pattern theory https://en.wikipedia.org/wiki/Pattern Theory , making spaces of patterns into a metric space https://en.wikipedia.org/wiki/Complete 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 The Right Invariant Metric on Diffeomorphism of computational anatomy measures how far two diffeomorphic changes of coordinates are from each other, which in turn induces a 11 cite note-Miller 36-11 metric on the shapes and images The metric on shapes and forms indexed to them. The models of metric pattern theory, 12 cite note-12 in particular group action on the orbit of shapes and forms is a central tool to the formal definitions in computational anatomy. 13 cite note-13 History edit /w/index.php?title=Computational anatomy&action=edit§ion=2 Computational anatomy is the study of shape and form at the morphome https://en.wikipedia.org/wiki/Morphome or gross anatomy https://en.wikipedia.org/wiki/Gross anatomy millimeter, or morphology https://en.wikipedia.org/wiki/Morphology biology scale, focusing on the study of sub- manifolds https://en.wikipedia.org/wiki/Manifolds of points, curves surfaces and subvolumes of human anatomy. An early modern computational neuro-anatomist was David Van Essen 14 performing some of the early physical unfoldings of the human brain based on printing of a human cortex and cutting. Jean Talairach's https://en.wikipedia.org/wiki/Jean Talairach publication of Talairach coordinates https://en.wikipedia.org/wiki/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 https://en.wikipedia.org/wiki/Differential geometry . Concurrently, virtual mapping in computational anatomy across high resolution dense image coordinates was already happening in Ruzena Bajcy's https://en.wikipedia.org/wiki/Ruzena Bajcsy and Fred Bookstein's 15 cite note-15 earliest developments based on 16 cite note-16 computed axial tomography https://en.wikipedia.org/wiki/Computed Tomography and magnetic resonance imagery https://en.wikipedia.org/wiki/Magnetic resonance imaging . 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. 17 cite note-Christensen-17 18 cite note-Christensen 1435–1447-18 19 cite note-:14-19 The first formalization of computational anatomy as an orbit of exemplar templates under diffeomorphism https://en.wikipedia.org/wiki/Diffeomorphism group action https://en.wikipedia.org/wiki/Group action mathematics 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, 20 and subsequent publication. This was the basis for the strong departure from much of the previous work on advanced methods for 9 cite note-:20-9 spatial normalization https://en.wikipedia.org/wiki/Spatial normalization and image registration https://en.wikipedia.org/wiki/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 https://en.wikipedia.org/wiki/Homeomorphism and diffeomorphisms https://en.wikipedia.org/wiki/Diffeomorphism carry smooth submanifolds smoothly. They are generated via Lagrangian and Eulerian flows https://en.wikipedia.org/wiki/Lagrangian and Eulerian specification of the flow field 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 21 organized by the Azencott school at 22 cite note-22 Ecole-Normale Cachan https://en.wikipedia.org/wiki/École Normale Supérieure on the "Mathematics of Shape Recognition" and the 1998 Trimestre at Institute Henri Poincaré https://en.wikipedia.org/wiki/Institut Henri Poincaré organized by David Mumford https://en.wikipedia.org/wiki/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 https://en.wikipedia.org/wiki/Calculus of variations problems in the space of diffeomorphisms, 23 cite note-23 24 the derivation of the Euler–Lagrange equations characterizing geodesics through the group and associated conservation laws, 25 cite note-25 26 cite note-Miller 209–228-26 the demonstration of the metric properties of the right invariant metric, 27 cite note-Miller 447–509-27 the demonstration that the Euler–Lagrange equations have a well-posed 28 cite note-:8-28 initial value problem https://en.wikipedia.org/wiki/Initial value problem with unique solutions for all time, and with the first results on sectional curvatures for the diffeomorphometry metric in landmarked spaces. 29 cite note-29 Following the Los Alamos meeting in 2002, 30 cite note-30 Joshi's 31 cite note-31 original large deformation singular 32 cite note-Joshi 1357–1370-32 Landmark solutions in computational anatomy were connected to peaked or https://en.wikipedia.org/wiki/Soliton solitons peakons as solutions for the 33 cite note-33 Camassa–Holm https://en.wikipedia.org/wiki/Camassa–Holm equation 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 https://en.wikipedia.org/wiki/Vladimir Arnold characterization of the 4 cite note-MR202082-4 Euler equation https://en.wikipedia.org/wiki/Euler equations fluid dynamics for incompressible flows as describing geodesics in the group of volume preserving diffeomorphisms. 34 cite note-34 The first algorithms, generally termed LDDMM for large deformation diffeomorphic mapping for computing connections between landmarks in volumes 35 cite note-Mumford-35 32 cite note-Joshi 1357–1370-32 36 cite note-36 and spherical manifolds, 37 cite note-37 curves, 38 cite note-38 currents and surfaces, 39 cite note-39 40 cite note-:18-40 41 cite note-Vaillant 1149–1159-41 volumes, 42 cite note-42 tensors, 43 cite note-43 varifolds, 44 cite note-44 and time-series 45 cite note-Charon 2547–2580-45 46 cite note-:15-46 47 cite note-47 have followed. 48 cite note-48 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 https://en.wikipedia.org/wiki/Bernhard Riemann 's Habilitation https://en.wikipedia.org/wiki/Habilitation Ueber die Hypothesen, welche der Geometrie zu Grunde liegen 49 cite note-49 50 ; the key modern book laying the foundations of such ideas in global analysis are from Michor. 51 cite note-51 The applications within medical imaging of computational anatomy continued to flourish after two organized meetings at the Institute for Pure and Applied Mathematics https://en.wikipedia.org/wiki/Institute for Pure and Applied Mathematics conferences 52 cite note-52 53 at University of California, Los Angeles https://en.wikipedia.org/wiki/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. 54 cite note-54 Since the late 1990s, computational anatomy has become an important part of developing 55 cite note-55 emerging technologies https://en.wikipedia.org/wiki/Emerging technologies for the field of medical imaging. Digital atlases are a fundamental part of modern Medical-school education 56 cite note-56 and in neuroimaging research at the morphome scale. 57 cite note-57 58 cite note-58 Atlas based methods and virtual textbooks 59 cite note-59 which accommodate variations as in deformable templates are at the center of many neuro-image analysis platforms including Freesurfer, 60 cite note-60 FSL, 61 cite note-61 MRIStudio, 62 cite note-62 SPM. 63 cite note-63 Diffeomorphic registration, 64 cite note-64 introduced in the 1990s, is now an important player with existing codes bases organized around ANTS, 18 cite note-Christensen 1435–1447-18 DARTEL, 65 cite note-stnava/ANTs-65 DEMONS, 66 cite note-Ashburner 95–113-66 LDDMM, 67 cite note-Software - Tom Vercauteren-67 StationaryLDDMM, 68 cite note-NITRC: LDDMM: Tool/Resource Info-68 FastLDDMM, 69 cite note-openaire.eu-69 are examples of actively used computational codes for constructing correspondences between coordinate systems based on sparse features and dense images. 70 cite note-70 Voxel-based morphometry https://en.wikipedia.org/wiki/Voxel-based morphometry is an important technology built on many of these principles. The deformable template orbit model of computational anatomy edit /w/index.php?title=Computational anatomy&action=edit§ion=3 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 https://en.wikipedia.org/wiki/Group action mathematics 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 . The orbit model of computational anatomy is an abstract algebra https://en.wikipedia.org/wiki/Abstract algebra – to be compared to linear algebra https://en.wikipedia.org/wiki/Linear algebra – since the groups act nonlinearly on the shapes. This is a generalization of the classical models of linear algebra, in which the set of finite dimensional vectors are replaced by the finite-dimensional anatomical submanifolds points, curves, surfaces and volumes and images of them, and the matrices of linear algebra are replaced by coordinate transformations based on linear and affine groups and the more general high-dimensional diffeomorphism groups. Shapes and forms edit /w/index.php?title=Computational anatomy&action=edit§ion=4 The central objects are shapes or forms in computational anatomy, one set of examples being the 0,1,2,3-dimensional submanifolds of , a second set of examples being images generated via medical imaging https://en.wikipedia.org/wiki/Medical imaging such as via magnetic resonance imaging https://en.wikipedia.org/wiki/Magnetic resonance imaging MRI and functional magnetic resonance imaging https://en.wikipedia.org/wiki/Functional magnetic resonance imaging . The 0-dimensional manifolds are landmarks or fiducial points; 1-dimensional manifolds are curves such as sulcal and gyral curves in the brain; 2-dimensional manifolds correspond to boundaries of substructures in anatomy such as the subcortical structures of the midbrain https://en.wikipedia.org/wiki/Midbrain or the gyral surface of the neocortex https://en.wikipedia.org/wiki/Neocortex ; subvolumes correspond to subregions of the human body, the heart https://en.wikipedia.org/wiki/Heart , the thalamus https://en.wikipedia.org/wiki/Thalamus , the kidney. The landmarks are a collections of points with no other structure, delineating important fiducials within human shape and form see associated landmarked image . The sub- manifold https://en.wikipedia.org/wiki/Manifold shapes such as surfaces are collections of points modeled as parametrized by a local chart or immersion https://en.wikipedia.org/wiki/Immersion mathematics , see Figure showing shapes as mesh surfaces . The images such as MR images or DTI images , and are dense functions are scalars, vectors, and matrices see Figure showing scalar image . Groups and group actions edit /w/index.php?title=Computational anatomy&action=edit§ion=5 Groups https://en.wikipedia.org/wiki/Group mathematics and group actions https://en.wikipedia.org/wiki/Group action mathematics are familiar to the Engineering community with the universal popularization and standardization of linear algebra https://en.wikipedia.org/wiki/Linear algebra as a basic model for analyzing signals and systems https://en.wikipedia.org/wiki/Signals and systems in mechanical engineering https://en.wikipedia.org/wiki/Mechanical engineering , electrical engineering https://en.wikipedia.org/wiki/Electrical engineering and applied mathematics https://en.wikipedia.org/wiki/Applied mathematics . In linear algebra the matrix groups matrices with inverses are the central structure, with group action defined by the usual definition of as an matrix, acting on as vectors; the orbit in linear-algebra is the set of -vectors given by , which is a group action of the matrices through the orbit of . The central group in computational anatomy defined on volumes in are the diffeomorphisms https://en.wikipedia.org/wiki/Diffeomorphisms which are mappings with 3-components , law of composition of functions , with inverse . Most popular are scalar images, , with action on the right via the inverse. For sub- manifolds https://en.wikipedia.org/wiki/Manifold , parametrized by a chart or immersion https://en.wikipedia.org/wiki/Immersion mathematics , the diffeomorphic action the flow of the position Several group actions in computational anatomy https://en.wikipedia.org/wiki/Group actions in computational anatomy have been defined. citation needed https://en.wikipedia.org/wiki/Wikipedia:Citation needed Lagrangian and Eulerian flows for generating diffeomorphisms edit /w/index.php?title=Computational anatomy&action=edit§ion=6 For the study of rigid body https://en.wikipedia.org/wiki/Rigid body kinematics https://en.wikipedia.org/wiki/Kinematics , the low-dimensional matrix Lie groups https://en.wikipedia.org/wiki/Lie groups have been the central focus. The matrix groups are low-dimensional mappings, which are diffeomorphisms that provide one-to-one correspondences between coordinate systems, with a smooth inverse. The matrix group https://en.wikipedia.org/wiki/Matrix group of rotations and scales can be generated via a closed form finite-dimensional matrices which are solution of simple ordinary differential equations with solutions given by the matrix exponential https://en.wikipedia.org/wiki/Matrix exponential . For the study of deformable shape in computational anatomy, a more general diffeomorphism group has been the group of choice, which is the infinite dimensional analogue. The high-dimensional diffeomorphism groups used in Computational Anatomy are generated via smooth flows which satisfy the Lagrangian and Eulerian specification of the flow fields https://en.wikipedia.org/wiki/Lagrangian and Eulerian specification of the flow field as first introduced in, 17 cite note-Christensen-17 19 cite note-:14-19 71 satisfying the ordinary differential equation https://en.wikipedia.org/wiki/Ordinary differential equation : | Lagrangian flow | with the vector fields on termed the Eulerian https://en.wikipedia.org/wiki/Lagrangian and Eulerian specification of the flow field velocity of the particles at position of the flow. The vector fields are functions in a function space https://en.wikipedia.org/wiki/Function space , modelled as a smooth Hilbert https://en.wikipedia.org/wiki/Hilbert space space of high-dimension, with the Jacobian of the flow a high-dimensional field in a function space as well, rather than a low-dimensional matrix as in the matrix groups. Flows were first introduced 72 cite note-72 73 for large deformations in image matching; is the instantaneous velocity of particle at time . The inverse required for the group is defined on the Eulerian vector-field with advective https://en.wikipedia.org/wiki/Advection inverse flow | inverse transport flow | The diffeomorphism group of computational anatomy edit /w/index.php?title=Computational anatomy&action=edit§ion=7 The group of diffeomorphisms is very big. To ensure smooth flows of diffeomorphisms avoiding shock-like solutions https://en.wikipedia.org/wiki/Advection Solving the equation for the inverse, the vector fields must be at least 1-time continuously differentiable in space. 74 cite note-:2-74 75 For diffeomorphisms on , vector fields are modelled as elements of the Hilbert space using the Sobolev https://en.wikipedia.org/wiki/Sobolev space embedding theorems so that each element has strictly greater than 2 generalized square-integrable spatial derivatives thus is sufficient , yielding 1-time continuously differentiable functions. 74 cite note-:2-74 75 cite note-:4-75 The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm: | diffeomorphism group | where with the linear operator mapping to the dual space https://en.wikipedia.org/wiki/Dual space , with the integral calculated by integration by parts https://en.wikipedia.org/wiki/Integration by parts when is a generalized function https://en.wikipedia.org/wiki/Generalized function in the dual space. The Sobolev smoothness condition on vector fields as modeled in a reproducing kernel Hilbert space The modelling approach used in computational anatomy enforces a continuous differentiability condition on the vector fields by modelling the space of vector fields as a reproducing kernel Hilbert space https://en.wikipedia.org/wiki/Reproducing kernel Hilbert space RKHS , with the norm defined by a 1-1, differential operator, Green's inverse . The norm of the Hilbert space is induced by the differential operator. For a generalized function or distribution, define the linear form as . This determines the norm on according to Since is a differential operator, finiteness of the norm-square includes derivatives from the differential operator implying smoothness of the vector fields.The Sobolev embedding https://en.wikipedia.org/wiki/Sobolev embedding theorem arguments were made in 74 cite note-:2-74 75 demonstrating that 1-continuous derivative is required for smooth flows. For proper choice of then is an RKHS with the operator termed the Green's https://en.wikipedia.org/wiki/Green's function for the three-variable Laplace equation operator generated from the Green's function https://en.wikipedia.org/wiki/Green's function scalar case for the vector field case. The Green's kernels associated to the differential operator smooths since the kernel is continuously differentiable in both variables implying When , a vector density, Diffeomorphometry: The metric space of shapes and forms edit /w/index.php?title=Computational anatomy&action=edit§ion=9 The study of metrics on groups of diffeomorphisms and the study of metrics between manifolds and surfaces has been an area of significant investigation. 28 cite note-:8-28 76 cite note-76 77 cite note-77 78 cite note-78 79 cite note-79 80 The diffeomorphometry metric measures how close and far two shapes or images are from each other; the metric length is the shortest length of the flow which carries one coordinate system into the other. Oftentimes, the familiar Euclidean metric is not directly applicable because the patterns of shapes and images do not form a vector space. In the Riemannian orbit model of computational anatomy https://en.wikipedia.org/wiki/Riemannian Metric and Lie-Bracket Interpretation of the Euler Equation on Geodesics , diffeomorphisms acting on the forms do not act linearly. There are many ways to define metrics, and for the sets associated to shapes the Hausdorff metric https://en.wikipedia.org/wiki/Hausdorff metric is another. The method we use to induce the Riemannian metric https://en.wikipedia.org/wiki/Riemannian metric is used to induce the metric on the orbit of shapes by defining it in terms of the metric length between diffeomorphic coordinate system transformations of the flows. Measuring the lengths of the geodesic flow between coordinates systems in the orbit of shapes is called diffeomorphometry . The right-invariant metric on diffeomorphisms edit /w/index.php?title=Computational anatomy&action=edit§ion=10 Define the distance on the group of diffeomorphisms | metric-diffeomorphisms | this is the right-invariant metric of diffeomorphometry, 11 cite note-Miller 36-11 28 invariant to reparameterization of space since for all , - . The metric on shapes and forms edit /w/index.php?title=Computational anatomy&action=edit§ion=11 The distance on shapes and forms, 81 , | metric-shapes-forms | the images 28 are denoted with the orbit as and metric . The action integral for Hamilton's principle on diffeomorphic flows edit /w/index.php?title=Computational anatomy&action=edit§ion=12 In classical mechanics the evolution of physical systems is described by solutions to the Euler–Lagrange equations associated to the Least-action principle https://en.wikipedia.org/wiki/Least-action principle of Hamilton https://en.wikipedia.org/wiki/Hamilton's principle . This is a standard way, for example of obtaining Newton's laws of motion https://en.wikipedia.org/wiki/Newton's laws of motion of free particles. More generally, the Euler–Lagrange equations can be derived for systems of generalized coordinates https://en.wikipedia.org/wiki/Generalized coordinates . The Euler–Lagrange equation in computational anatomy describes the geodesic shortest path flows between coordinate systems of the diffeomorphism metric. In computational anatomy the generalized coordinates are the flow of the diffeomorphism and its Lagrangian velocity , the two related via the Eulerian velocity . Hamilton's principle https://en.wikipedia.org/wiki/Hamilton principle for generating the Euler–Lagrange equation requires the action integral on the Lagrangian given by | Hamiltonian-integrated-Lagrangian | the Lagrangian is given by the kinetic energy: | Lagrangian-kinetic-energy | Diffeomorphic or Eulerian shape momentum edit /w/index.php?title=Computational anatomy&action=edit§ion=13 In computational anatomy, was first called the Eulerian or diffeomorphic shape momentum 82 since when integrated against Eulerian velocity gives energy density, and since there is a conservation of diffeomorphic shape momentum Conservation laws on diffeomorphic shape momentum for computational anatomy which holds. The operator is the generalized moment of inertia https://en.wikipedia.org/wiki/Moment of inertia or inertial operator. The Euler–Lagrange equation on shape momentum for geodesics on the group of diffeomorphisms edit /w/index.php?title=Computational anatomy&action=edit§ion=14 Classical calculation of the Euler–Lagrange equation from Hamilton's principle https://en.wikipedia.org/wiki/Hamilton principle requires the perturbation of the Lagrangian on the vector field in the kinetic energy with respect to first order perturbation of the flow. This requires adjustment by the Lie bracket of vector field https://en.wikipedia.org/wiki/Lie bracket of vector fields , given by operator which involves the Jacobian given by - . Defining the adjoint then the first order variation gives the Eulerian shape momentum satisfying the generalized equation: | EL-general | meaning for all smooth Computational anatomy is the study of the motions of submanifolds, points, curves, surfaces and volumes. Momentum associated to points, curves and surfaces are all singular, implying the momentum is concentrated on subsets of which are dimension in Lebesgue measure https://en.wikipedia.org/wiki/Lebesgue measure . In such cases, the energy is still well defined since although is a generalized function, the vector fields are smooth and the Eulerian momentum is understood via its action on smooth functions. The perfect illustration of this is even when it is a superposition of delta-diracs, the velocity of the coordinates in the entire volume move smoothly. The Euler–Lagrange equation EL-general on diffeomorphisms for generalized functions was derived in. In 83 cite note-:0-83 Riemannian Metric and Lie-Bracket Interpretation of the Euler–Lagrange Equation on Geodesics https://en.wikipedia.org/wiki/Riemannian Metric and Lie-Bracket Interpretation of the Euler Equation on Geodesics derivations are provided in terms of the adjoint operator and the Lie bracket for the group of diffeomorphisms. It has come to be called EPDiff equation for diffeomorphisms connecting to the Euler-Poincare method having been studied in the context of the inertial operator for incompressible, divergence free, fluids. 35 cite note-Mumford-35 84 cite note-:3-84 Diffeomorphic shape momentum: a classical vector function edit /w/index.php?title=Computational anatomy&action=edit§ion=15 For the momentum density case , then Euler–Lagrange equation has a classical solution: | | EL-Classic | The Euler–Lagrange equation on diffeomorphisms, classically defined for momentum densities first appeared in 85 for medical image analysis. Riemannian exponential geodesic positioning and Riemannian logarithm geodesic coordinates edit /w/index.php?title=Computational anatomy&action=edit§ion=16 In medical imaging and computational anatomy, positioning and coordinatizing shapes are fundamental operations; the system for positioning anatomical coordinates and shapes built on the metric and the Euler–Lagrange equation a geodesic positioning system as first explicated in Miller Trouve and Younes. 11 Solving the geodesic from the initial condition is termed the Riemannian-exponential, a mapping at identity to the group. The Riemannian exponential satisfies for initial condition , vector field dynamics , - for classical equation diffeomorphic shape momentum , , then - for generalized equation, then , , Computing the flow onto coordinates Riemannian logarithm , 11 cite note-Miller 36-11 81 mapping at identity from to vector field ; Extended to the entire group they become - ; . These are inverses of each other for unique solutions of Logarithm; the first is called geodesic positioning , the latter geodesic coordinates see exponential map, Riemannian geometry https://en.wikipedia.org/wiki/Exponential map Riemannian geometry for the finite dimensional version . The geodesic metric is a local flattening of the Riemannian coordinate system see figure . Hamiltonian formulation of computational anatomy edit /w/index.php?title=Computational anatomy&action=edit§ion=17 In computational anatomy the diffeomorphisms are used to push the coordinate systems, and the vector fields are used as the control within the anatomical orbit or morphological space. The model is that of a dynamical system, the flow of coordinates and the control the vector field related via The Hamiltonian view 81 cite note-Miller null2-81 86 cite note-86 87 cite note-87 88 cite note-88 89 reparameterizes the momentum distribution in terms of the conjugate momentum or ntroduced as a Lagrange multiplier constraining the Lagrangian velocity .accordingly: canonical momentum , iThis function is the extended Hamiltonian. The Pontryagin maximum principle https://en.wikipedia.org/wiki/Pontryagin maximum principle 81 gives the optimizing vector field which determines the geodesic flow satisfying as well as the reduced Hamiltonian The Lagrange multiplier in its action as a linear form has its own inner product of the canonical momentum acting on the velocity of the flow which is dependent on the shape, e.g. for landmarks a sum, for surfaces a surface integral, and. for volumes it is a volume integral with respect to on . In all cases the Greens kernels carry weights which are the canonical momentum evolving according to an ordinary differential equation which corresponds to EL but is the geodesic reparameterization in canonical momentum. The optimizing vector field is given by with dynamics of canonical momentum reparameterizing the vector field along the geodesic | Hamiltonian-dynamics | Stationarity of the Hamiltonian and kinetic energy along Euler–Lagrange edit /w/index.php?title=Computational anatomy&action=edit§ion=18 Whereas the vector fields are extended across the entire background space of , the geodesic flows associated to the submanifolds has Eulerian shape momentum which evolves as a generalized function concentrated to the submanifolds. For landmarks 90 cite note-:5-90 91 cite note-:10-91 92 the geodesics have Eulerian shape momentum which are a superposition of delta distributions Landmark or pointset geodesics travelling with the finite numbers of particles; the diffeomorphic flow of coordinates have velocities in the range of weighted Green's Kernels. For surfaces, the momentum is a surface integral of delta distributions Surface geodesics travelling with the surface. 11 cite note-Miller 36-11 The geodesics connecting coordinate systems satisfying EL-general have stationarity of the Lagrangian. The Hamiltonian is given by the extremum along the path , , equalling the and is stationary along Lagrangian-kinetic-energy math Lagrangian-kinetic-energy . Defining the geodesic velocity at the identity , then along the geodesic EL-general math EL-general | Hamiltonian-geodesics | The stationarity of the Hamiltonian demonstrates the interpretation of the Lagrange multiplier as momentum; integrated against velocity gives energy density. The canonical momentum has many names. In optimal control https://en.wikipedia.org/wiki/Optimal control , the flows is interpreted as the state, and is interpreted as conjugate state, or conjugate momentum. 93 The geodesi of EL implies specification of the vector fields or Eulerian momentum at , or specification of canonical momentum determines the flow. The metric on geodesic flows of landmarks, surfaces, and volumes within the orbit edit /w/index.php?title=Computational anatomy&action=edit§ion=19 In computational anatomy the submanifolds are pointsets, curves, surfaces and subvolumes which are the basic primitives. The geodesic flows between the submanifolds determine the distance, and form the basic measuring and transporting tools of diffeomorphometry https://en.wikipedia.org/wiki/Diffeomorphometry . At the geodesic has vector field determined by the conjugate momentum and the Green's kernel of the inertial operator defining the Eulerian momentum . The metric distance between coordinate systems connected via the geodesic determined by the induced distance between identity and group element: Conservation laws https://en.wikipedia.org/wiki/Conservation law physics on diffeomorphic shape momentum for computational anatomy edit /w/index.php?title=Computational anatomy&action=edit§ion=20 Given the least-action there is a natural definition of momentum associated to generalized coordinates; the quantity acting against velocity gives energy. The field has studied two forms, the momentum associated to the Eulerian vector field termed Eulerian diffeomorphic shape momentum , and the momentum associated to the initial coordinates or canonical coordinates termed canonical diffeomorphic shape momentum . Each has a conservation law. The conservation of momentum goes hand in hand with the EL-general . In computational anatomy, is the Eulerian momentum https://en.wikipedia.org/wiki/Momentum since when integrated against Eulerian velocity gives energy density; operator the generalized moment of inertia https://en.wikipedia.org/wiki/Moment of inertia or inertial operator which acting on the Eulerian velocity gives momentum which is conserved along the geodesic: | Euler-conservation-constant-energy | Conservation of Eulerian shape momentum was shown in 94 and follows from ; conservation of canonical momentum was shown in EL-general math EL-general 81 cite note-Miller null2-81 The proof follow from defining , implying The proof on canonical momentum is shown from : - . Geodesic interpolation of information between coordinate systems via variational problems edit /w/index.php?title=Computational anatomy&action=edit§ion=21 Construction of diffeomorphic correspondences between shapes calculates the initial vector field coordinates and associated weights on the Greens kernels . These initial coordinates are determined by matching of shapes, called large-deformation diffeomorphic metric mapping LDDMM . LDDMM has been solved for landmarks with and without correspondence 32 cite note-Joshi 1357–1370-32 95 cite note-95 96 cite note-96 97 cite note-97 and for dense image matchings. 98 cite note-98 99 cite note-:13-99 curves, 100 cite note-:12-100 surfaces, 101 cite note-101 41 cite note-Vaillant 1149–1159-41 dense vector 102 cite note-102 and tensor 103 cite note-103 imagery, and varifolds removing orientation. 104 cite note-Cao 1216–1230-104 LDDMM calculates geodesic flows of the 105 cite note-105 onto target coordinates, adding to the action integral an endpoint matching condition measuring the correspondence of elements in the orbit under coordinate system transformation. Existence of solutions were examined for image matching. EL-general math EL-general The solution of the variational problem satisfies the 24 cite note-:142-24 for with boundary condition. EL-general math EL-general Matching based on minimizing kinetic energy action with endpoint condition edit /w/index.php?title=Computational anatomy&action=edit§ion=22 Conservation from EL-general extends the B.C. at to the rest of the path . The inexact matching problem with the endpoint matching term has several alternative forms. One of the key ideas of the stationarity of the Hamiltonian along the geodesic solution is the integrated running cost reduces to initial cost at t = 0, geodesics of the are determined by their initial condition . EL-general math EL-general The running cost is reduced to the initial cost determined by of Kernel-Surf.-Land.-Geodesics . Matching based on geodesic shooting edit /w/index.php?title=Computational anatomy&action=edit§ion=23 The matching problem explicitly indexed to initial condition is called shooting, which can also be reparamerized via the conjugate momentum . Dense image matching in computational anatomy edit /w/index.php?title=Computational anatomy&action=edit§ion=24 Dense image matching has a long history now with the earliest efforts 106 cite note-106 107 exploiting a small deformation framework. Large deformations began in the early 1990s, 18 cite note-Christensen 1435–1447-18 with the first existence to solutions to the variational problem for flows of diffeomorphisms for dense image matching established in. 19 cite note-:14-19 Beg solved via one of the earliest LDDMM algorithms based on solving the variational matching with endpoint defined by the dense imagery with respect to the vector fields, taking variations with respect to the vector fields. 24 cite note-:142-24 Another solution for dense image matching reparameterizes the optimization problem in terms of the state giving the solution in terms of the infinitesimal action defined by the 99 cite note-:13-99 advection https://en.wikipedia.org/wiki/Advection equation. 11 cite note-Miller 36-11 27 cite note-Miller 447–509-27 100 cite note-:12-100 For Beg's LDDMM, denote the Image with group action . Viewing this as an optimal control problem, the state of the system is the diffeomorphic flow of coordinates , with the dynamics relating the control to the state given by . The endpoint matching condition gives the variational problem | Dense-image-matching | Beg's iterative LDDMM algorithm https://en.wikipedia.org/wiki/Large deformation diffeomorphic metric mapping Beg's Iterative LDDMM Algorithm has fixed points which satisfy the necessary optimizer conditions. The iterative algorithm is given in Beg's LDDMM algorithm for dense image matching https://en.wikipedia.org/wiki/LDDMM Beg's LDDMM algorithm for image matching . Hamiltonian LDDMM in the reduced advected state edit /w/index.php?title=Computational anatomy&action=edit§ion=26 Denote the Image , with state and the dynamics related state and control given by the advective term https://en.wikipedia.org/wiki/Advection . The endpoint gives the variational problem | Dense-image-matching | Viallard's iterative Hamiltonian LDDMM https://en.wikipedia.org/wiki/Large deformation diffeomorphic metric mapping Hamiltonian LDDMM for Dense Image Matching has fixed points which satisfy the necessary optimizer conditions. Diffusion tensor image matching in computational anatomy edit /w/index.php?title=Computational anatomy&action=edit§ion=27 Dense LDDMM tensor matching 104 cite note-Cao 1216–1230-104 108 takes the images as 3x1 vectors and 3x3 tensors solving the variational problem matching between coordinate system based on the principle eigenvectors of the diffusion tensor MRI https://en.wikipedia.org/wiki/Diffusion MRI image DTI denoted consisting of the -tensor at every voxel. Several of the group actions defined based on the Frobenius matrix norm https://en.wikipedia.org/wiki/Matrix norm between square matrices . Shown in the accompanying figure is a DTI image illustrated via its color map depicting the eigenvector orientations of the DTI matrix at each voxel with color determined by the orientation of the directions. Denote the tensor image with eigen-elements , . Coordinate system transformation based on DTI imaging has exploited two actions one based on the principle eigen-vector or entire matrix https://en.wikipedia.org/wiki/Group actions in computational anatomy Tensor matrices . LDDMM matching based on the principal eigenvector of the diffusion tensor matrix takes the image as a unit vector field defined by the first eigenvector. The group action becomes LDDMM matching based on the entire tensor matrix has group action becomes transformed eigenvectors - . The variational problem matching onto the principal eigenvector or the matrix is described LDDMM Tensor Image Matching https://en.wikipedia.org/wiki/Large deformation diffeomorphic metric mapping LDDMM Diffusion Tensor Image Matching . High angular resolution diffusion image HARDI matching in computational anatomy edit /w/index.php?title=Computational anatomy&action=edit§ion=28 High angular resolution diffusion imaging HARDI addresses the well-known limitation of DTI, that is, DTI can only reveal one dominant fiber orientation at each location. HARDI measures diffusion along uniformly distributed directions on the sphere and can characterize more complex fiber geometries. HARDI can be used to reconstruct an orientation distribution function https://en.wikipedia.org/wiki/Orientation distribution function ODF that characterizes the angular profile of the diffusion probability density function of water molecules. The ODF is a function defined on a unit sphere, . Dense LDDMM ODF matching 109 takes the HARDI data as ODF at each voxel and solves the LDDMM variational problem in the space of ODF. In the field of information geometry https://en.wikipedia.org/wiki/Information geometry , the space of ODF forms a Riemannian manifold with the Fisher-Rao metric. For the purpose of LDDMM ODF mapping, the square-root representation is chosen because it is one of the most efficient representations found to date as the various Riemannian operations, such as geodesics, exponential maps, and logarithm maps, are available in closed form. In the following, denote square-root ODF as , where is non-negative to ensure uniqueness and . The variational problem for matching assumes that two ODF volumes can be generated from one to another via flows of diffeomorphisms , which are solutions of ordinary differential equations starting from the identity map . Denote the action of the diffeomorphism on template as , , are respectively the coordinates of the unit sphere, and the image domain, with the target indexed similarly, ,,. 110 cite note-110 The group action of the diffeomorphism on the template is given according to - , where is the Jacobian of the affine-transformed ODF and is defined as This group action of diffeomorphisms on ODF reorients the ODF and reflects changes in both the magnitude of and the sampling directions of due to affine transformation. It guarantees that the volume fraction of fibers oriented toward a small patch must remain the same after the patch is transformed. The LDDMM variational problem is defined as where the logarithm of is defined as where is the normal dot product https://en.wikipedia.org/wiki/Dot product between points in the sphere under the metric. This LDDMM-ODF mapping algorithm has been widely used to study brain white matter degeneration in aging, Alzheimer's disease, and vascular dementia. 111 The brain white matter atlas generated based on ODF is constructed via Bayesian estimation. Regression analysis on ODF is developed in the ODF manifold space in. 112 cite note-112 113 cite note-113 Metamorphosis edit /w/index.php?title=Computational anatomy&action=edit§ion=29 The principle mode of variation represented by the orbit model is change of coordinates. For setting in which pairs of images are not related by diffeomorphisms but have photometric variation or image variation not represented by the template, active appearance modelling https://en.wikipedia.org/wiki/Active appearance model has been introduced, originally by Edwards-Cootes-Taylor 114 and in 3D medical imaging in. In the context of computational anatomy in which metrics on the anatomical orbit has been studied, 115 cite note-115 metamorphosis for modelling structures such as tumors and photometric changes which are not resident in the template was introduced in for magnetic resonance image models, with many subsequent developments extending the metamorphosis framework. 28 cite note-:8-28 116 cite note-116 117 cite note-117 118 cite note-118 For image matching the image metamorphosis framework enlarges the action so that with action . In this setting metamorphosis combines both the diffeomorphic coordinate system transformation of computational anatomy as well as the early morphing https://en.wikipedia.org/wiki/Morphing technologies which only faded or modified the photometric or image intensity alone. Then the matching problem takes a form with equality boundary conditions: Matching landmarks, curves, surfaces edit /w/index.php?title=Computational anatomy&action=edit§ion=30 Transforming coordinate systems based on Landmark point https://en.wikipedia.org/wiki/Landmark point or fiducial marker https://en.wikipedia.org/wiki/Fiducial marker features dates back to Bookstein's early work on small deformation spline methods 119 for interpolating correspondences defined by fiducial points to the two-dimensional or three-dimensional background space in which the fiducials are defined. Large deformation landmark methods came on in the late 1990s. 26 cite note-Miller 209–228-26 32 cite note-Joshi 1357–1370-32 The above Figure depicts a series of landmarks associated three brain structures, the amygdala, entorhinal cortex, and hippocampus. 120 cite note-120 Matching geometrical objects like unlabelled point distributions, curves or surfaces is another common problem in computational anatomy. Even in the discrete setting where these are commonly given as vertices with meshes, there are no predetermined correspondences between points as opposed to the situation of landmarks described above. From the theoretical point of view, while any submanifold in , can be parameterized in local charts , all reparametrizations of these charts give geometrically the same manifold. Therefore, early on in computational anatomy, investigators have identified the necessity of parametrization invariant representations. One indispensable requirement is that the end-point matching term between two submanifolds is itself independent of their parametrizations. This can be achieved via concepts and methods borrowed from Geometric measure theory https://en.wikipedia.org/wiki/Geometric measure theory , in particular currents https://en.wikipedia.org/wiki/Current mathematics 40 and varifolds https://en.wikipedia.org/wiki/Varifold which have been used extensively for curve and surface matching. 45 cite note-Charon 2547–2580-45 Landmark or point matching with correspondence edit /w/index.php?title=Computational anatomy&action=edit§ion=31 Denoted the landmarked shape with endpoint , the variational problem becomes | | Landmark-Matching | The geodesic Eulerian momentum is a generalized function , supported on the landmarked set in the variational problem. The endpoint condition with conservation implies the initial momentum at the identity of the group: The iterative algorithm for large deformation diffeomorphic metric mapping for landmarks https://en.wikipedia.org/wiki/LDDMM Joshi's original LDDMM landmark matching is given. Measure matching: unregistered landmarks edit /w/index.php?title=Computational anatomy&action=edit§ion=32 Glaunes and co-workers first introduced diffeomorphic matching of pointsets in the general setting of matching distributions. 121 As opposed to landmarks, this includes in particular the situation of weighted point clouds with no predefined correspondences and possibly different cardinalities. The template and target discrete point clouds are represented as two weighted sums of Diracs and living in the space of signed measures https://en.wikipedia.org/wiki/Signed measure of . The space is equipped with a Hilbert metric obtained from a real positive kernel on , giving the following norm: The matching problem between a template and target point cloud may be then formulated using this kernel metric for the endpoint matching term: where is the distribution transported by the deformation. Curve matching edit /w/index.php?title=Computational anatomy&action=edit§ion=33 In the one dimensional case, a curve in 3D can be represented by an embedding , and the group action of Diff becomes . However, the correspondence between curves and embeddings is not one to one as the any reparametrization , for a diffeomorphism of the interval 0,1 , represents geometrically the same curve. In order to preserve this invariance in the end-point matching term, several extensions of the previous 0-dimensional measure matching approach can be considered. Curve matching with currents In the situation of oriented curves, currents give an efficient setting to construct invariant matching terms. In such representation, curves are interpreted as elements of a functional space dual to the space vector fields, and compared through kernel norms on these spaces. Matching of two curves and writes eventually as the variational problem with the endpoint term is obtained from the norm the derivative being the tangent vector to the curve and a given matrix kernel of . Such expressions are invariant to any positive reparametrizations of and , and thus still depend on the orientation of the two curves. Curve matching with varifolds Varifold is an alternative to currents when orientation becomes an issue as for instance in situations involving multiple bundles of curves for which no "consistent" orientation may be defined. Varifolds directly extend 0-dimensional measures by adding an extra tangent space direction to the position of points, leading to represent curves as measures on the product of and the Grassmannian https://en.wikipedia.org/wiki/Grassmannian of all straight lines in . The matching problem between two curves then consists in replacing the endpoint matching term by with varifold norms of the form: where is the non-oriented line directed by tangent vector and two scalar kernels respectively on and the Grassmannian. Due to the inherent non-oriented nature of the Grassmannian representation, such expressions are invariant to positive and negative reparametrizations. Surface matching edit /w/index.php?title=Computational anatomy&action=edit§ion=34 Surface matching share many similarities with the case of curves. Surfaces in are parametrized in local charts by embeddings , with all reparametrizations with a diffeomorphism of U being equivalent geometrically. Currents and varifolds can be also used to formalize surface matching. Surface matching with currents Oriented surfaces can be represented as 2-currents which are dual to differential 2-forms. In , one can further identify 2-forms with vector fields through the standard wedge product of 3D vectors. In that setting, surface matching writes again: with the endpoint term given through the norm with the normal vector to the surface parametrized by . This surface mapping algorithm has been validated for brain cortical surfaces against CARET and FreeSurfer. 122 LDDMM mapping for multiscale surfaces is discussed in. 123 cite note-123 Surface matching with varifolds For non-orientable or non-oriented surfaces, the varifold framework is often more adequate. Identifying the parametric surface with a varifold in the space of measures on the product of and the Grassmannian, one simply replaces the previous current metric by: where is the non-oriented line directed by the normal vector to the surface. Growth and atrophy from longitudinal time-series edit /w/index.php?title=Computational anatomy&action=edit§ion=35 There are many settings in which there are a series of measurements, a time-series to which the underlying coordinate systems will be matched and flowed onto. This occurs for example in the dynamic growth and atrophy models and motion tracking such as have been explored in 46 cite note-:15-46 124 cite note-124 125 cite note-:16-125 126 An observed time sequence is given and the goal is to infer the time flow of geometric change of coordinates carrying the exemplars or templars through the period of observations. The generic time-series matching problem considers the series of times is . The flow optimizes at the series of costs giving optimization problems of the form - . There have been at least three solutions offered thus far, piecewise geodesic, 46 principal geodesic and splines. 126 cite note-:17-126 127 cite note-127 The random orbit model of computational anatomy edit /w/index.php?title=Computational anatomy&action=edit§ion=36 The random orbit model of computational anatomy first appeared in 128 cite note-:1-128 129 cite note-:9-129 130 modelling the change in coordinates associated to the randomness of the group acting on the templates, which induces the randomness on the source of images in the anatomical orbit of shapes and forms and resulting observations through the medical imaging devices. Such a random orbit model in which randomness on the group induces randomness on the images was examined for the Special Euclidean Group for object recognition in. 131 cite note-131 Depicted in the figure is a depiction of the random orbits around each exemplar, , generated by randomizing the flow by generating the initial tangent space vector field at the identity , and then generating random object . The random orbit model induces the prior on shapes and images conditioned on a particular atlas . For this the generative model generates the mean field as a random change in coordinates of the template according to , where the diffeomorphic change in coordinates is generated randomly via the geodesic flows. The prior on random transformations on is induced by the flow , with constructed as a Gaussian random field prior . The density on the random observables at the output of the sensor are given by Shown in the Figure on the right the cartoon orbit are a random spray of the subcortical manifolds generated by randomizing the vector fields supported over the submanifolds. The Bayesian model of computational anatomy edit /w/index.php?title=Computational anatomy&action=edit§ion=37 The central statistical model of computational anatomy in the context of medical imaging https://en.wikipedia.org/wiki/Medical imaging has been the source-channel model of Shannon theory https://en.wikipedia.org/wiki/Shannon theory ; 128 cite note-:1-128 129 cite note-:9-129 130 the source is the deformable template of images , the channel outputs are the imaging sensors with observables see Figure . See The Bayesian model of computational anatomy https://en.wikipedia.org/wiki/The Bayesian model of computational anatomy for discussions i MAP estimation with multiple atlases, ii MAP segmentation with multiple atlases, MAP estimation of templates from populations. Statistical shape theory in computational anatomy edit /w/index.php?title=Computational anatomy&action=edit§ion=38 Shape https://en.wikipedia.org/wiki/Shape in computational anatomy is a local theory, indexing shapes and structures to templates to which they are bijectively https://en.wikipedia.org/wiki/Bijectively?action=edit&redlink=1 mapped. Statistical shape https://en.wikipedia.org/wiki/Statistical shape analysis in computational anatomy is the empirical study of diffeomorphic correspondences between populations and common template coordinate systems. This is a strong departure from Procrustes Analyses https://en.wikipedia.org/wiki/Procrustes and shape theories pioneered by David G. Kendall https://en.wikipedia.org/wiki/David George Kendall 132 in that the central group of Kendall's theories are the finite-dimensional Lie groups, whereas the theories of shape in computational anatomy 133 cite note-133 134 cite note-134 have focused on the diffeomorphism group, which to first order via the Jacobian can be thought of as a field–thus infinite dimensional–of low-dimensional Lie groups of scale and rotations. 135 cite note-135 The random orbit model provides the natural setting to understand empirical shape and shape statistics within computational anatomy since the non-linearity of the induced probability law on anatomical shapes and forms is induced via the reduction to the vector fields at the tangent space at the identity of the diffeomorphism group. The successive flow of the Euler equation induces the random space of shapes and forms . Performing empirical statistics on this tangent space at the identity is the natural way for inducing probability laws on the statistics of shape. Since both the vector fields and the Eulerian momentum are in a Hilbert space the natural model is one of a Gaussian random field, so that given test function , then the inner-products with the test functions are Gaussian distributed with mean and covariance. This is depicted in the accompanying figure where sub-cortical brain structures are depicted in a two-dimensional coordinate system based on inner products of their initial vector fields that generate them from the template is shown in a 2-dimensional span of the Hilbert space. Template estimation from populations edit /w/index.php?title=Computational anatomy&action=edit§ion=39 The study of shape and statistics in populations are local theories, indexing shapes and structures to templates to which they are bijectively mapped. Statistical shape is then the study of diffeomorphic correspondences relative to the template. A core operation is the generation of templates from populations, estimating a shape that is matched to the population. There are several important methods for generating templates including methods based on Frechet https://en.wikipedia.org/wiki/Frechet averaging, 137 and statistical approaches based on the expectation-maximization algorithm https://en.wikipedia.org/wiki/Expectation-maximization algorithm and the Bayes Random orbit models of computational anatomy. 136 cite note-ncbi.nlm.nih.gov-136 Shown in the accompanying figure is a subcortical template reconstruction from the population of MRI subjects. 138 cite note-138 139 cite note-139 Software for diffeomorphic mapping edit /w/index.php?title=Computational anatomy&action=edit§ion=40 Software suites https://en.wikipedia.org/wiki/Software suite containing a variety of diffeomorphic mapping algorithms include the following: - ANTS 65 cite note-stnava/ANTs-65 - DARTEL 66 cite note-Ashburner 95–113-66 Voxel-based morphometry https://en.wikipedia.org/wiki/Voxel-based morphometry - DEFORMETRICA 140 cite note-140 - DEMONS 67 cite note-Software - Tom Vercauteren-67 - LDDMM 68 cite note-NITRC: LDDMM: Tool/Resource Info-68 Large deformation diffeomorphic metric mapping https://en.wikipedia.org/wiki/Large deformation diffeomorphic metric mapping - LDDMM based on frame-based kernel 141 cite note-141 - StationaryLDDMM 69 cite note-openaire.eu-69 Cloud software edit /w/index.php?title=Computational anatomy&action=edit§ion=41 - MRICloud 142 cite note-142 See also edit /w/index.php?title=Computational anatomy&action=edit§ion=42 References edit /w/index.php?title=Computational anatomy&action=edit§ion=43 ↑ cite ref-1 "Computational Anatomy – Asclepios" https://team.inria.fr/asclepios/research/computational-anatomy/ . team.inria.fr . 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Bulletin of the American Mathematical Society . 75 5 : 962–967. doi https://en.wikipedia.org/wiki/Doi identifier : 10.1090/s0002-9904-1969-12315-3 https://doi.org/10.1090%2Fs0002-9904-1969-12315-3 . 1 cite ref-Mumford 35-0 2 cite ref-Mumford 35-1 Mumford, David; Michor, Peter W. 2013 . "On Euler's equation and 'EPDiff'". Journal of Geometric Mechanics . 5 3 : 319–344. arXiv https://en.wikipedia.org/wiki/ArXiv identifier : 1209.6576 https://arxiv.org/abs/1209.6576 . Bibcode https://en.wikipedia.org/wiki/Bibcode identifier : 2012arXiv1209.6576M https://ui.adsabs.harvard.edu/abs/2012arXiv1209.6576M . doi https://en.wikipedia.org/wiki/Doi identifier : 10.3934/jgm.2013.5.319 https://doi.org/10.3934%2Fjgm.2013.5.319 . ↑ cite ref-36 Scherzer, Otmar 2010-11-23 .. Springer Science & Business Media. 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PMC https://en.wikipedia.org/wiki/PMC identifier 3140704 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3140704 . PMID https://en.wikipedia.org/wiki/PMID identifier 17185000 https://pubmed.ncbi.nlm.nih.gov/17185000 . ↑ cite ref-42 Durrleman, Stanley; Pennec, Xavier; Trouvé, Alain; Ayache, Nicholas 2009-10-01 . "Statistical models of sets of curves and surfaces based on currents". Medical Image Analysis . 13 5 : 793–808. CiteSeerX https://en.wikipedia.org/wiki/CiteSeerX identifier 10.1.1.221.5224 https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.221.5224 . doi https://en.wikipedia.org/wiki/Doi identifier : 10.1016/j.media.2009.07.007 https://doi.org/10.1016%2Fj.media.2009.07.007 . PMID https://en.wikipedia.org/wiki/PMID identifier 19679507 https://pubmed.ncbi.nlm.nih.gov/19679507 . {{ : Cite uses deprecated parameter cite journal https://en.wikipedia.org/wiki/Template:Cite journal }} |citeseerx= help https://en.wikipedia.org/wiki/Help:CS1 errors deprecated params ↑ cite ref-43 M.F. Beg and M. I. Miller and A. Trouve and L. Younes 2005 . "Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms" https://www.researchgate.net/publication/220660081 . International Journal of Computer Vision . 61 2 : 139–157. Bibcode https://en.wikipedia.org/wiki/Bibcode identifier : 2005IJCV...61..139B https://ui.adsabs.harvard.edu/abs/2005IJCV...61..139B . doi https://en.wikipedia.org/wiki/Doi identifier : 10.1023/B:VISI.0000043755.93987.aa https://doi.org/10.1023%2FB%3AVISI.0000043755.93987.aa . S2CID https://en.wikipedia.org/wiki/S2CID identifier 17772076 https://api.semanticscholar.org/CorpusID:17772076 . Retrieved 2016-01-27 – via ResearchGate.