Computational anatomy

Computational anatomy is a discipline within medical imaging focusing on the study of anatomical shape and form at the morphome scale of morphology. It involves the development and application of computational, mathematical and data-analytical methods for modeling 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, medical 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 detectors acting as the transmission and communication medium(s).

In Computational anatomy, the diffeomorphism group for coordinate transformations is generated via the Lagrangian and Eulerian velocities of flow in ${\mathbb R}^3$. 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 defined via a Sobolev smoothness norm with more than two finite square-integrable derivatives for each component of the velocity of flow. This, in turn, guarantees that the flows in $\mathbb{R}^3$ are diffeomorphisms;^[1] it also implies that the diffeomorphic shape momentum in Computational anatomy, which satisfies the Euler-Lagrange equation for geodesics, is determined by its velocity and spatial derivatives. This separates the discipline from the case of incompressible fluids^[2] 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^[3] 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, connecting 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.^[4][5][6] The Grenander abstraction (see below) within the setting of deformable templates uses group actions as the comparison mechanism; the central group action of Computational anatomy is diffeomorphic action. 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.^[7]

Due to the focus on medical imaging 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^[8] 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^[9] 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,^[10][11] 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 ${\mathbb R}^3,$ points, curves surfaces and subvolumes of human anatomy. An early modern computational neuro-anatomist was David Van Essen ^[12] 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 Tailarach 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^[13] and Fred Bookstein's^[14] 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.^[15][16][17]

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,^[18] and subsequent publication.^[7] 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, $(\mathcal{G}, \mathcal{M}, \mathcal{P}) \ ,$ the group $g \in \mathcal{G}$, the orbit of shapes and forms $m \in \mathcal{M}$, and the probability laws $P$ which encode the variations of the objects in the orbit. The template or collection of templates are elements in the orbit $m_\mathrm{temp} \in \mathcal {M}$ 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 ^[19] organized by the Azencott^[20] 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 Sobelev smoothness conditions on the diffeomorphometry metric to insure existence of solutions of variational problems in the space of diffeomorphisms,^[21][22] the derivation of the Euler-Lagrange equations characterizing geodesics through the group and associated conservation laws,^[23][24][25] the demonstration of the metric properties of the right invariant metric,^[26] the demonstration that the Euler-Lagrange equations have a well-posed initial value problem with unique solutions for all time,^[27] and with the first results on sectional curvatures for the diffeomorphometry metric in landmarked spaces.^[28] Following the Los Alamos meeting in 2002,^[29] Joshi's^[30] original large deformation singular Landmark solutions in Computational anatomy were connected to peaked Solitons or Peakons^[31] 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^[2] characterization of the Euler equation for incompressible flows as describing geodesics in the group of volume preserving diffeomorphisms.^[32][33] The first algorithms, generally termed LDDMM for large deformation diffeomorphic mapping for computing connections between landmarks in volumes^[30][34][35] and spherical manifolds,^[36] curves,^[37] currents and surfaces,^[38][39][40] volumes,^[41] tensors,^[42] varifolds,^[43] and time-series^[44][45][46] 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 ^[47][48]); the key modern book laying the foundations of such ideas in global analysis are from Michor.^[49]

The applications within Medical Imaging of Computational Anatomy continued to flourish after two organized meetings at the Institute for Pure and Applied Mathematics conferences^[50][51] 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,^[52] as well as in modeling biological systems.^[53] 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^[54][55] and in neuroimaging research at the morphome scale.^[56][57] Atlas based methods and virtual textbooks^[58] which accommodate variations as in deformable templates are at the center of many neuro-image analysis platforms including Freesurfer,^[59] FSL,^[60] MRIStudio,^[61] SPM.^[62] Diffeomorphic registration,^[16] introduced in the 90's, is now an important player with existing codes bases organized around ANTS,^[63] DARTEL,^[64] DEMONS,^[65] LDDMM,^[66] StationaryLDDMM^[67] are examples of actively used computational codes for constructing correspondences between coordinate systems based on sparse features and dense images. Voxel-based morphometry(VBM) 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 $m \in \mathcal {M}$, with the group $(\mathcal{ G}, \circ )$ with law of composition $\circ$; the action of the group on shapes is denoted $g \cdot m$, where the action of the group $g \cdot m \in \mathcal{M}, m \in \mathcal {M}$ is defined to satisfy

$(g \circ g^\prime) \cdot m = g \cdot (g^\prime \cdot m)\in \mathcal { M} .$

The orbit $\mathcal{M}$ of the template becomes the space of all shapes, $\mathcal{M} \doteq \{ m = g \cdot m_\mathrm{temp}, g \in \mathcal{G} \}$.

Figure depicting three medial temporal lobe structures amgydala, entorhinal cortex and hippocampus with fiducial landmarks depicted as well embedded in the MRI background.

The orbit model of computational anatomy is an abstract algebra - to be compared to 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 ${\mathbb R}^n$ vectors are replaced by the finite-dimensional anatomical submanifolds (points, curves, surfaces and volumes) and images of them, and the $n \times n$ 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

The central objects are shapes or forms in Computational anatomy, one set of examples being the 0,1,2,3-dimensional submanifolds of ${\mathbb R}^3$, a second set of examples being images generated via medical imaging such as via magnetic resonance imaging (MRI) and functional magnetic resonance imaging.

Triangulated mesh surfaces depicting subcortical structures amygdala, hippocampus, thalamus, caudate, putamen, ventricles.The shapes are denoted $m(u), u \in U\subset {\mathbb R}^1 \rightarrow {\mathbb R}^2$ represented as triangulated meshes.

The 0-dimensional manifolds are landmarks or fiducial points; 1-dimensional manifolds are curves such as sulcul and gyral curves in the brain; 2-dimensional manifolds correspond to boundaries of substructures in anatomy such as the subcortical structures of the midbrain or the gyral surface of the neocortex; subvolumes correspond to subregions of the human body, the heart, the thalamus, the kidney.

The landmarks, denoted as$X \doteq \{ x_1,\dots,x_n \} \subset {\mathbb R}^3 \in \mathcal{M}$ are a collections of points with no other structure, delineating important fiducials within human shape and form (see associated landmarked image).

The sub-manifold shapes such as surfaces are denoted as$X \subset {\mathbb R}^3 \in \mathcal{M}$, collections of points modeled as parametrized by a local chart or immersion $m:U \subset {\mathbb R}^{1,2} \rightarrow {\mathbb R}^3$, $m(u), u \in U$ (see Figure showing shapes as mesh surfaces).

The images in Computational anatomy such as MR images or DTI images are denoted $I \in \mathcal{M}$, and are dense functions $I(x), x \in X \subset {\mathbb R}^{1,2,3}$ are scalars, vectors, and matrices (see Figure showing scalar image).

Groups and group actions

Showing an MRI section through a 3D brain representing a scalar image $I(x), x \in {\mathbb R}^2$ based on T1-weighting.

Groups and group actions are familiar to the Engineering community with the universal popularization and standardization of linear algebra as a basic model for analyzing signals and systems in mechanical engineering, electrical engineering and applied mathematics. In linear algebra the matrix groups (matrices with inverses) are the central structure, with group action defined by the usual definition of $A$ as an $n\times n$ matrix, acting on $x \in {\mathbb R}^n$ as $n \times 1$ vectors; the orbit in linear-algebra is the set of $n$-vectors given by $y = A \cdot x \in {\mathbb R}^n$, which is a group action of the matrices through the orbit of ${\mathbb R}^n$.

The central group in Computational anatomy defined on volumes in ${\mathbb R}^3$ are the diffeomorphisms $\mathcal{G} \doteq Diff$ which are mappings with 3-components $\phi(\cdot) = (\phi_1(\cdot),\phi_2 (\cdot),\phi_3 (\cdot))$, law of composition of functions $\phi \circ \phi^\prime (\cdot)\doteq \phi (\phi^\prime(\cdot))$, with inverse $\phi \circ \phi^{-1}(\cdot) =\phi ( \phi^{-1}(\cdot))= id$.

Most popular are scalar images, $I(x),x \in {\mathbb R}^3$, with action on the right via the inverse.

$\phi \cdot I(x) = I \circ \phi^{-1} (x), x \in {\mathbb R}^3$.

For sub-manifolds $X \subset {\mathbb R}^3 \in \mathcal{M}$, parametrized by a chart or immersion $m(u), u \in U$, the diffeomorphic action the flow of the position

$\phi \cdot m(u) \doteq \phi\circ m(u), u \in U$.

Several group actions in computational anatomy have been defined.

Lagrangian and Eulerian flows for generating diffeomorphisms

For the study of rigid body kinematics, the low-dimensional matrix 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 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.

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 differeomorphism groups used in Computational Anatomy are generated via smooth flows $\phi_t, t \in [0,1]$ which satisfy the Lagrangian and Eulerian specification of the flow fieldssas first introduced in.,^[15][17][68] satisfying the ordinary differential equation:

Showing the Lagrangian flow of coordinates $x \in X$ with associated vector fields $v_t, t \in [0,1]$ satisfying ordinary differential equation $\dot \phi_t = v_t(\phi_t), \phi_0=id$.

$\frac{d}{dt} \phi_t = v_t \circ \phi_t , \ \phi_0 = id \ ;$

(Lagrangian flow)

with $v \doteq (v_1,v_2,v_3)$ the vector fields on ${\mathbb R}^3$ termed the Eulerian velocity of the particles at position $\phi$ of the flow. The vector fields are functions in a function space, modelled as a smooth Hilbert space of high-dimension, with the Jacobian of the flow $\ D\phi \doteq (\frac{\partial \phi_i}{\partial x_j})$ 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^[69][70] for large deformations in image matching; $\dot \phi_t(x)$ is the instantaneous velocity of particle $x$ at time $t$ .

The inverse $\phi_t^{-1}, t \in [0,1]$ required for the group is defined on the Eulerian vector-field with advective inverse flow

$\frac{d}{dt} \phi_t^{-1} = - (D \phi_t^{-1}) v_t , \ \phi_0^{-1} =id \ .$

(Inverse Transport flow)

The diffeomorphism group of computational anatomy

The group of diffeomorphisms is very big. To ensure smooth flows of diffeomorphisms avoiding shock-like solutions for the inverse, the vector fields must be at least 1-time continuously differentiable in space.^[71][72] For diffeomorphisms on ${\mathbb R}^3$, vector fields are modelled as elements of the Hilbert space $(V, \| \cdot \|_V )$ using the Sobolev embedding theorems so that each element $v_i \in H_0^3, i=1,2,3,$ has 3-square-integrable derivatives, thusly embedding in 1-time continuously differentiable functions.^[71][72]

The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:

$Diff_V \doteq \{\varphi=\phi_1: \dot \phi_t = v_t \circ \phi_t , \phi_0 = id, \int_0^1 \|v_t \|_V dt < \infty \} \ .$

(Diffeomorphism Group)

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 $(V, \| \cdot \|_V )$ as a reproducing kernel Hilbert space (RKHS), with the norm defined by a 1-1, differential operator$A: V \rightarrow V^*$, Green's inverse $K = A^{-1}$. The norm of the Hilbert space is induced by the differential operator. For $\sigma(v) \doteq Av \in V^*$ a generalized function or distribution, define the linear form as $(\sigma|w)\doteq \int_{{\mathbb R}^3} \sum_i w_i(x) \sigma_i(dx)$. This determines the norm on $(V, \| \cdot \|_V )$ according to

$\langle v , w \rangle_V \doteq (A v | w), \ \| v\|_V^2 \doteq (Av|v), \ v,w \in V \ .$

Since $A$ is a differential operator, finiteness of the norm-square $(Av|v) < \infty$ includes derivatives from the differential operator implying smoothness of the vector fields.The Sobolev embedding theorem arguments were made in^[71][72] demonstrating that 1-continuous derivative is required for smooth flows.

For proper choice of $A$ then $(V,\| \cdot \|_V)$ is an RKHS with the operator $K = A^{-1}: V^* \rightarrow V$ termed the Green's operator generated from the Green's function (scalar case) for the vector field case. The Green's kernels associated to the differential operator smooths since the kernel $k(\cdot,\cdot)$ is continuously differentiable in both variables implying

$K\sigma (x)_i \doteq \sum_j \int_{{\mathbb R}^3} k_{ij}(x,y) \sigma_j(dy)$

When $\sigma \doteq \mu dx$, a vector density, $(\sigma|v) \doteq \int v \cdot \mu dx$.

Diffeomorphometry: The metric space of shapes and forms

The study of metrics on groups of diffeomorphisms and the study of metrics between manifolds and surfaces has been an area of significant investigation.^{[26][73][74][75][76][77]} In Computational anatomy, the diffeomorphometry metric measures how close and far two shapes or images are from each other. Informally, 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 don't form a vector space. In the Riemannian orbit model of Computational anatomy, diffeomorphisms acting on the forms $\phi \cdot m \in \mathcal {M}, \phi \in Diff_V, m \in \mathcal{M}$ don't act linearly. There are many ways to define metrics, and for the sets associated to shapes the Hausdorff metric is another. The method we use to induce the 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

Define the distance on the group of diffeomorphisms

:$d_{Diff_V}(\psi, \varphi) = \inf_{v_t} \left(\int_0^1 (Av_t|v_t)dt: \phi_0 = \psi, \phi_1 = \varphi, \dot \phi_t = v_t \circ \phi_t \right)^{1/2} \ ;$

(metric-diffeomorphisms)

this is the right-invariant metric of diffeomorphometry,^[26][78] invariant to reparameterization of space since for all $\phi \in Diff_V \, \, \,$,

$d_{Diff_V}(\psi, \varphi) = d_{Diff_V}(\psi \circ \phi, \varphi \circ \phi)$.

The metric on shapes and forms

The distance on shapes and forms,^[79]$d_{\mathcal{M}}:\mathcal{M} \times \mathcal{M}\rightarrow \R^+$,

:$d_{\mathcal{M}}(m,n)=\inf_{\phi \in \operatorname{Diff}_V: \phi \cdot m = n } d_{\operatorname{Diff}_V}(id,\phi) \ ;$

(metric-shapes-forms)

the images^[80] are denoted with the orbit as $I \in \mathcal{I}$ and metric $, d_{\mathcal{I}}$.

The action integral for Hamilton's principle on diffeomorphic flows

In classical mechanics the evolution of physical systems is described by solutions to the Euler–Lagrange equations associated to the Least-action principle of Hamilton. This is a standard way, for example of obtaining Newton's laws of motion of free particles. More generally, the Euler-Lagrange equations can be derived for systems of 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 $\phi, \dot{\phi}$, the two related via the Eulerian velocity $v \doteq \dot{\phi} \circ \phi^{-1}$. Hamilton's principle for generating the Euler-Lagrange equation requires the action integral on the Lagrangian given by

$J(\phi) \doteq \int_0^1 L(\phi_t,\dot \phi_t) dt \ ;$

(Hamiltonian-Integrated-Lagrangian)

the Lagrangian is given by the kinetic energy:

$L(\phi_t, \dot \phi_t ) \doteq \frac{1}{2} (A (\dot \phi_t \circ \phi_t^{-1}) | \dot \phi_t \circ \phi_t^{-1} ) = \frac{1}{2} (Av_t | v_t ) \ .$

(Lagrangian-Kinetic-Energy)

Diffeomorphic or Eulerian shape momentum

In computational anatomy, $Av$ was first called the Eulerian or fiffeomorphic dhape momentum ^[81] since when integrated against Eulerian velocity $v$ gives energy density, and since there is a conservation of diffeomorphic shape momentum which holds. The operator $A$ is the generalized moment of inertia or inertial operator.

The Euler–Lagrange equation on shape momentum for geodesics on the group of diffeomorphisms

Classical calculation of the Euler-Lagrange equation from Hamilton's 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, given by operator $ad_v: w \in V \mapsto V$ which involves the Jacobian given by

$ad_v[w]\doteq [v,w]\doteq (Dv)w-(Dw)v \in V$.

Defining the adjoint $ad_v^*: V^* \rightarrow V^*,$ then the first order variation gives the Eulerian shape momentum $Av \in V^*$ satisfying the generalized equation:

$\frac{d}{dt} Av_t + ad_{v_t}^* (Av_t)=0 \ , \ t \in [0,1] \ ;$

(EL-General)

meaning for all smooth $w \in V ,$

$\left( \frac{d}{dt} Av_t + ad_{v_t}^* (Av_t) \mid w \right) = (\frac{d}{dt} Av_t\mid w) + (Av_t\mid(Dv_t)w-(Dw)v_t) =0 .$

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 ${\mathbb R}^3$ which are dimension $\leq 2$ in Lebesgue measure. In such cases, the energy is still well defined $(Av_t\mid v_t)$ since although $Av_t$ is a generalized function, the vector fields are smooth and the Eulerian momentum is understood via it's 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 $Av \in V^*$ was derived in.^[82] In Riemannian Metric and Lie-Bracket Interpretation of the Euler-Lagrange 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 $A=identity$ for incompressible, divergence free, fluids.^[33][83]

Diffeomorphic shape momentum: a classical vector function

For the momentum density case $(Av_t \mid w) =\int_X \mu_t \cdot w \, dx$, then Euler–Lagrange equation has a classical solution:

$\frac{d}{dt} \mu_t + (Dv_t)^T \mu_t +(D\mu_t)v_t + ( \nabla \cdot v) \mu_t =0 \ , t \in [0,1].$

(EL-Classic)

The Euler-Lagrange equation on diffeomorphisms, classically defined for momentum densities first appeared in^[84] for medical image analysis.

Riemannian exponential (geodesic positioning) and Riemannian logarithm (geodesic coordinates)

Global positioning systems based on systems of satellites provides a spatial navigation sytstem on the globe allowing electronic receivers to determine their location in the 3-dimensional coordinate system of longitude, latitude, and altitude to within meter scale.

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.^[9] Solving the geodesic from the initial condition $v_0$ is termed the Riemannian-exponential, a mapping $Exp_{\rm id}(\cdot): V \to Diff_V$ at identity to the group.

The Riemannian exponential satisfies $Exp_{id} (v_0)= \phi_1$ for initial condition $\dot \phi_0 = v_0$, vector field dynamics $\dot \phi_t = v_t \circ \phi_t, t \in [0,1]$,

• for classical equation diffeomorphic shape momentum $(Av_t \mid w) =\int_X Av_t \cdot w \, dx$, $Av \in V$, then

$\ \ \ \frac{d}{dt} Av_t + (Dv_t)^T Av_t +(DAv_t)v_t + ( \nabla \cdot v) Av_t =0 \ ;$

• for generalized equation, then $Av \in V^*$,$w \in V$,

$\ \ \ (\frac{d}{dt} Av_t\mid w) + (Av_t\mid(Dv_t)w-(Dw)v_t) =0 .$

Computing the flow $v_0$ onto coordinates Riemannian logarithm,^[79][85] mapping $Log_{\rm id}(\cdot): Diff_V \to V$ at identity from $\varphi$ to vector field $v_0 \in V$;

$Log_{ id}( \varphi) = v_0 \ \text{initial condition of EL geodesic } \dot \phi_0 = v_0, \phi_0=id, \phi_1=\varphi \ .$

Extended to the entire group they become

$\phi= Exp_\varphi(v_0\circ \varphi) \doteq Exp_{id} (v_0) \circ \varphi$ ; $Log_\varphi(\phi)\doteq Log_{id}( \phi \circ\varphi^{-1}) \circ \varphi$ .

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 for the finite dimensional version).The geodesic metric is a local flattening of the Riemannian coordinate system (see figure).

Showing metric local flattening of coordinatized manifolds of shapes and forms. The local metric is given by the norm of the vector field $\|v_0\|_V$ of the geodesic mapping $Exp_{id}(v_0) \cdot m$

Hamiltonian formulation of computational anatomy

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 $t \mapsto \phi_t \in \operatorname{Diff}_V$ and the control the vector field $t \mapsto v_t \in V$ related via $\dot \phi_t = v_t \cdot \phi_t,\phi_0=id .$ The Hamiltonian view ^{[79] [86] [87] [88][89]} reparameterizes the momentum distribution $Av \in V^*$ in terms of the conjugate momentum or canonical momentum, introduced as a Lagrange multiplier $p: \dot \phi \mapsto (p\mid\dot \phi)$ constraining the Lagrangian velocity $\dot \phi_t = v_t \circ \phi_t$.accordingly:

$H(\phi_t,p_t,v_t)=(p_t\mid v_t \circ \phi_t)-\frac{1}{2}(Av_t\mid v_t) .$

This function is the extended Hamiltonian. The Pontryagin maximum principle^[79] gives the optimizing vector field which determines the geodesic flow satisfying $\dot \phi_t = v_t \circ \phi_t , \phi_0=id,$ as well as the reduced Hamiltonian

$H(\phi_t,p_t) \doteq \max_v H( \phi_t, p_t,v) \ .$

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 $dx$ on ${\mathbb R}^3$. 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

$v_t \doteq \arg max_{v} H(\phi_t,p_t,v)$

with dynamics of canonical momentum reparameterizing the vector field along the geodesic

$\begin{cases} \dot \phi_t = \frac{\partial H( \phi_t, p_t)}{\partial p} \\ \dot p_t = -\frac{\partial H(\phi_t,p_t)}{\partial \phi} \\ \end{cases}$

(Hamiltonian-Dynamics)

Stationarity of the Hamiltonian and kinetic energy along Euler–Lagrange

Whereas the vector fields are extended across the entire background space of ${\mathbb R}^3$, the geodesic flows associated to the submanifolds has Eulerian shape momentum which evolves as a generalized function $Av_t \in V^*$ concentrated to the submanifolds. For landmarks^[90][91][92] the geodesics have Eulerian shape momentum which are a superposition of delta distributions 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 travelling with the surface.^[9]

The geodesics connecting coordinate systems satisfying EL-General have stationarity of the Lagrangian. The Hamiltonian is given by the extremum along the path $t \in [0,1]$, $H(\phi,p) = \max_v H(\phi,p,v)$, equalling the Lagrangian-Kinetic-Energy and is stationary along EL-General. Defining the geodesic velocity at the identity $v_0 = \arg \max_v H(\phi_0,p_0,v)$, then along the geodesic

$H(\phi_t,p_t) = H(\phi_0,p_0)=\frac{1}{2} (p_0\mid v_0) = \frac{1}{2} (Av_0\mid v_0)=\frac{1}{2} (Av_t\mid v_t)$

(Hamiltonian-Geodesics)

The stationarity of the Hamiltonian demonstrates the interpretation of the Lagrange multiplier as momentum; integrated against velocity $\dot \phi$ gives energy density. The canonical momentum has many names. In optimal control, the flows $\phi$ is interpreted as the state, and $p$ is interpreted as conjugate state, or conjugate momentum.^[93] The geodesi of EL implies specification of the vector fields $v_0$ or Eulerian momentum $Av_0$ at $t=0$, or specification of canonical momentum $p_0$ determines the flow.

The metric on geodesic flows of landmarks, surfaces, and volumes within the orbit

Illustration of geodesic flow for one landmark, demonstrating diffeomorphic motion of background space. Red arrow shows $p_0(1)$, blue curve shows $\varphi_t(x_1)$, black grid shows $\varphi_t$

In Computational anatomy the submanifolds are pointsets, curves, surfaces and subvolumes which are the basic primitive forming the index sets or background space of medically imaged human anatomy. The geodesic flows of the submanifolds such as the landmarks, surface and subvolumes and the distance as measured by the geodesic flows of such coordinates, form the basic measuring and transporting tools of diffeomorphometry.

What is so important about the RKHS norm defining the kinetic energy in the action principle is that the vector fields of the geodesic motions of the submanifolds are superpositions of Green's Kernel's. For landmarks the superposition is a sum of weight kernels weighted by the canonical momentum which determines the inner product, for surfaces it is a surface integral, and for dense volumes it is a volume integral.

At $t=0$ the geodesic has vector field $v_0 = K p_0$ determined by the conjugate momentum and the Green's kernel of the inertial operator defining the Eulerian momentum $K=A^{-1}$. The metric distance between coordinate systems connected via the geodesic determined by the induced distance between identity and group element:

$d_{Diff_V}(id,\varphi) = \|Log_{id}(\varphi)\|_V=\| v_0 \|_V = \sqrt{2H(id,p_0)}$

Landmark and surface submanifolds have Lagrange multiplier associated to a sum and surface integral, respectively; dense volumes an integral with respect to Lebesgue measure.

Landmark or pointset geodesics

For Landmarks, the Hamiltonian momentum is defined on the indices, $p(i), i=1,\dots,n$ with the inner product given by $(p_t|v_t \circ \phi_t) \doteq \textstyle \sum_{i} \displaystyle p_t(i)\cdot v_t \circ \phi_t( x_i)$ and Hamiltonian $H(\phi_t,p_t) =\frac{1}{2}\textstyle \sum_j \sum_i \displaystyle p_t(i)\cdot K(\phi_t (x_i),\phi_t (x_j)) p_t(j)$. The dynamics take the forms

$\begin{cases} v_t = \textstyle \sum_{i} \displaystyle K(\cdot, \phi_t (x_i)) p_t(i) , \ \\ \dot p_t (i) = - (Dv_t)^T_{|_{\phi_t(x_i)}} p_t(i), i=1,2,\dot, n \\ \end{cases}$

with the metric between landmarks $d^2 =(p_0|v_0) =\textstyle \sum_i p_0(i)\cdot \sum_j \displaystyle K(x_i,x_j) p_0(j).$

Surface geodesics

For surfaces, the Hamiltonian momentum is defined across the surface with the inner product $(p_t\mid v_t \circ \phi_t) \doteq \textstyle \int_U \displaystyle p_t(u)\cdot v_t \circ \phi_t ( m(u)) du$, with $H(\phi_t,p_t) =\frac{1}{2} \int_U \int_U p_t(u)\cdot K(\phi_t (m(u)), \phi_t (m(v))) p_t(v) \, du \, dv$. The dynamics

$\begin{cases} v_t= \textstyle \int_U \displaystyle K(\cdot, \phi_t ( m(u)))p_t(u)\,du \ , \\ \dot p_t(u) = - (Dv_t)^T_{|_{\phi_t(m(u))} } p_t(u), u \in U \end{cases}$

with the metric between surface coordinates $d^2 = (p_0 \mid v_0) =\int_U p_0(u)\cdot \int_U K(m(u), m(u^\prime)) p_0(u^\prime) \, du \, du^\prime$

Volume geodesics

For volumes the Hamiltonian momentum is $(p_t\mid v_t \circ \phi_t ) \doteq \int_{{\mathbb R}^3} p_t(x)\cdot v_t \circ \phi_t (x) \, dx$ with $H(\phi_t,p_t) = \frac{1}{2}\int_{{\mathbb R}^3} \int_{{\mathbb R}^3} p_t(x)\cdot K(\phi_t(x),\phi_t(y)) p_t(y) \, dx \, dy \displaystyle$. The dynamics $\begin{cases} v_t=\textstyle \int_X \displaystyle K(\cdot, \phi_t(x))p_t(x)\,dx \ , \\ \dot p_t(x) = - (Dv_t)^T_{|_{\phi_t(x)} } p_t(x), x \in {\mathbb R}^3 \end{cases}$

with the metric between volumes $\displaystyle d^2 =(p_0\mid v_0) = \int_{{\mathbb R}^3} p_0(x)\cdot \int_{{\mathbb R}^3} K(x,y) p_0(y)dy \ dx.$

Conservation laws on diffeomorphic shape momentum for computational anatomy

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 and hand with the EL-General. In Computational anatomy, $Av$ is the Eulerian Momentum since when integrated against Eulerian velocity $v$ gives energy density; operator $A$ the generalized moment of inertia or inertial operator which acting on the Eulerian velocity gives momentum which is conserved along the geodesic:

$\begin{matrix} \text{Eulerian} & \ \ \ \ \frac{d}{dt} (Av_t | ((D \phi_t) w) \circ \phi_t^{-1}) = 0 \ , \ t \in [0,1] . \\ & \\ \text{Canonical} & \ \ \ \ \ \ \ \ \ \ \ \frac{d}{dt} (p_t | (D\phi_t) w) = 0 \ , \ t \in [0,1] \ \text{ for all} \ w \in V \ . \end{matrix}$

(Euler-Conservation-Constant-Energy)

Conservation of Eulerian shape momentum was shown in^[94] and follows from EL-General; conservation of canonical momentum was shown in^[79]

The proof follow from defining $w_t =((D \phi_t) w) \circ \phi_t^{-1}$, $\frac{d}{dt} w_t =(Dv_t) w_t - (D w_t) v_t$ implying $\frac{d}{dt} (Av_t | ((D \phi_t) w) \circ \phi_t^{-1})= (\frac{d}{dt}Av_t | ((D \phi_t) w) \circ \phi_t^{-1})+(Av_t |\frac{d}{dt} ((D \phi_t) w) \circ \phi_t^{-1})=(\frac{d}{dt} Av_t | w_t )+(Av_t | (D v_t) w_t-(Dw_t)v_t)=0.$

The proof on Canonical momentum is shown from $\dot p_t =- (Dv_t)_{|_{\phi_t}}^T p_t$:

$\frac{d}{dt} (p_t | (D\phi_t) w) =(\dot p_t | (D\phi_t) w)+(p_t |\frac{d}{dt} (D\phi_t) w) = (\dot p_t | (D \phi_t) w)+(p_t |(Dv_t)_{|_{\phi_t} }(D \phi_t) w) = 0$.

Geodesic interpolation of information between coordinate systems via variational problems

Construction of diffeomorphic correspondences between shapes calculates the initial vector field coordinates $v_0 \in V$ and associated weights on the Greens kernels $p_0$. 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^{[30][95][96][97][98]} and for dense image matchings.^[99][100] curves,^[101] surfaces,^[39][102] dense vector^[103] and tensor^[104] imagery, and varifolds removing orientation.^[105] LDDMM calculates geodesic flows of the EL-General onto target coordinates, adding to the action integral $\frac{1}{2} \int_0^1 (Av_t|v_t) dt$ an endpoint matching condition $E: \phi_1 \rightarrow R^+$ measuring the correspondence of elements in the orbit under coordinate system transformation. Existence of solutions were examined for image matching.^[22] The solution of the variational problem satisfies the EL-General for $t \in [0,1)$ with boundary condition.

Matching based on minimizing kinetic energy action with endpoint condition

$\text{min}_{\phi:v= \dot \phi \circ \phi^{-1}, \phi_0=id} C(\phi) \doteq \frac{1}{2} \int_0^1 (Av_t|v_t) dt +E( \phi_1)$

$\begin{cases} \text{Euler Conservation} \ \ \ \ \ \ \ \ & \ \ \ \frac{d}{dt} Av_t + ad_{v_t}^* (Av_t)=0 , \ t \in [0,1) \ , \\ \text{Boundary Condition} & \ \ \ \phi_0=id, Av_1 =- \frac{\partial E(\phi)}{\partial \phi}|_{\phi=\phi_1} \ . \end{cases}$

Conservation from EL-General extends the B.C. at $t=1$ to the rest of the path $t \in [0,1)$ .The inexact matching problem with the endpoint matching term $E(\phi_1)$ 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 EL-General are determined by their initial condition $v_0$.

The running cost is reduced to the initial cost determined by $v_0=Kp_0$ of Kernel-Surf.-Land.-Geodesics.

Matching based on geodesic shooting

$\min_{v_0} C(v_0) \doteq \frac{1}{2} (Av_0\mid v_0) +E(\mathrm{Exp}_\mathrm{id}(v_0) \cdot I_0) \ ;$

$\min_{p_0} C(p_0) = \frac{1}{2}(p_0|Kp_0)+ E(\mathrm{Exp}_\text{id}(Kp_0) \cdot I_0)$

The matching problem explicitly indexed to initial condition $v_0$ is called shooting, which can also be reparamerized via the conjugate momentum $p_0$.

Dense image matching in computational anatomy

Dense image matching has a long history now with the earliest efforts^[106][107] exploiting a small deformation framework. Large deformations began in the early 90's,^[16][17] with the first existence to solutions to the variational problem for flows of diffeomorphisms for dense image matching established in.^[22] 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.^[99] Another solution for dense image matching reparameterizes the optimization problem in terms of the state $q_t \doteq I \circ \phi_t^{-1}, q_0 = I$ giving the solution in terms of the infinitesimal action defined by the advection equation.^[9][25][100]

LDDMM dense image matching

For Beg's LDDMM, denote the Image $I(x), x \in X$ with group action $\phi \cdot I \doteq I \circ \phi^{-1}$. Viewing this as an optimal control problem, the state of the system is the diffeomorphic flow of coordinates $\phi_t, t \in [0,1]$, with the dynamics relating the control $v_t, t \in [0,1]$ to the state given by $\dot \phi = v \circ \phi$. The endpoint matching condition $E(\phi_1) \doteq \| I\circ \phi_1^{-1} -I^\prime\|^2$

gives the variational problem

$\begin{matrix} & \ \ \ \ \ \min_{v: \dot \phi = v \circ \phi} C(v) \doteq \frac{1}{2} \int_0^1 (Av_t|v_t)dt +\frac{1}{2} \int_{{\mathbb R}^3} | I\circ \phi_1^{-1}(x)-I^\prime (x)|^2dx \end{matrix}$

(Dense-Image-Matching)

$\begin{cases} & \text{Endpoint Condition:} \ \ \ \ \ \ Av_1 =\mu_1 dx, \mu_1 =(I\circ \phi_1^{-1}-I^\prime ) \nabla (I \circ \phi_1^{-1}) \ ,\\ &\text{Conservation:} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Av_t =\mu_t \, dx, \ \mu_t = (D \phi_t^{-1})^T \mu_0 \circ \phi_t^{-1}|D \phi_t^{-1}| \ . \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mu_0=(I - I^\prime \circ \phi_1) \nabla I |D \phi_1| \ . \\ \end{cases}$

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.

Hamiltonian LDDMM in the reduced advected state

Denote the Image $I(x), x \in X$, with state $q_t \doteq I \circ \phi_t^{-1}$ and the dynamics related state and control given by the advective term $\dot q_t =- \nabla q_t \cdot v_t$. The endpoint $E(q_1) \doteq \| q_1-I^\prime\|^2$ gives the variational problem

$\begin{matrix} & \ \ \ \ \ \min_{q: \dot q =v \circ q} C(v) \doteq \frac{1}{2} \int_0^1 (Av_t|v_t)dt +\frac{1}{2} \int_{{\mathbb R}^3} | q_1(x)-I^\prime(x)|^2dx \end{matrix}$

(Dense-Image-Matching)

Viallard's iterative Hamiltonian LDDMM has fixed points which satisfy the necessary optimizer conditions.

Diffusion tensor image matching in computational anatomy

Image showing a diffusion tensor image with three color levels depicting the orientations of the three eigenvectors of the matrix image $I(x), x \in {\mathbb R}^2$, matrix valued image; each of three colors represents a direction.

Dense LDDMM tensor matching^[104][108] solves the variational problem matching between coordinate system based on the principle eigenvectors of the diffusion tensor MRI image (DTI) denoted $M(x), x \in {\mathbb R}^3$ consisting of the $3 \times 3$-tensor at every voxel. Several of the group actions defined based on the Frobenius matrix norm between square matrices $\| A \|_F^2 \doteq trace A^T A$ . 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.

Coordinate system transformation based on DTI imaging has exploited two actions, one based on the principle eigen-vector or entire matrix.

• LDDMM matching based on diffusion tensor matrix

Denote the $3 \times 3$ tensor image $M(x), x \in {\mathbb R}^3$ with eigen-elements $\{\lambda_i(x), e_i(x), i=1,2,3\}$, eigenvalues $\lambda_1 \geq \lambda_2 \geq \lambda_3$, and $e_1$, $e_2$, $e_3$ eigenvectors. The group action becomes $\varphi \cdot M = (\lambda_1 \hat{e}_1 \hat{e}_1^{T}+\lambda_2 \hat{e}_2 \hat{e}_2^{T} +\lambda_3 \hat{e}_3 \hat{e}_3^{T}) \circ \varphi^{-1} , \$ transformed eigenvectors

$\begin{align} \hat{e}_1 & = \frac{D \varphi e_1}{\|D \varphi e_1\|} \ , \ \ \ \hat{e}_2 = \frac{D \varphi e_2-\langle \hat{e}_1,D \varphi e_2\rangle \hat{e}_1}{\sqrt{\|D \varphi e_2\|^2-\langle \hat{e}_1,D \varphi e_2\rangle^2}}\ , \ \ \ \hat{e}_3 = \hat{e}_1 \times \hat{e}_2 \end{align}$,

with endpoint $E(\phi_1) \doteq \int_{{\mathbb R}^3} \| \phi_1 \cdot M(x) - M^\prime(x) \|_F^2 dx$. The variational problem matching onto $M^\prime (x), x \in {\mathbb R}$ becomes

$\min_{v: v = \dot \phi \circ \phi^{-1}} \frac{1}{2}\int_0^1 (Av_t|v_t) dt+\alpha \int_{{\mathbb R}^3} \| \phi_1 \cdot M(x) - M^\prime(x) \|_F^2 dx$

(Dense-TensorDTI-Matching)

• LDDMM matching based on the principal eigenvector of the diffusion tensor matrix

Denote the image $I(x), x \in {\mathbb R}^3$ taken as a unit vector field defined by the first eigenvector. The group action becomes

$\varphi \cdot I = \begin{cases} \frac{D_{\varphi^{-1}} \varphi I \circ \varphi^{-1} \|I \circ \varphi^{-1}\|}{\|D_{\varphi^{-1}} \varphi I \circ \varphi^{-1}\|} & I\circ \varphi \neq 0; \\ 0 & \text{otherwise.} \end{cases}$

with endpoint

$E(\phi_1) \doteq \alpha \int_{{\mathbb R}^3} \|\phi_1 \cdot I -I^\prime \|^2 \, dx+\beta \int_{{\mathbb R}^3} (\|\phi_1 \cdot I\|-\|I^\prime\|)^2 \, dx).$

The variational problem matching onto vector image $I^\prime (x), x \in {\mathbb R}^3$becomes

$\min_{v: \dot \phi \circ \phi^{-1}} \frac{1}{2} \int_0^1 (Av_t|v_t) \, dt + \alpha \int_{{\mathbb R}^3} \|\phi_1 \cdot I -I^\prime \|^2 \, dx+\beta \int_{{\mathbb R}^3} (\|\phi_1 \cdot I\|-\|I^\prime\|)^2 \, dx \ .$

Metamorphosis

Demonstrating metamorphosis allowing both diffeomorphic change in coordinate transformation as well as change in image intensity as associated to early Morphing technologies such as the Michael Jackson video. Notice the insertion of tumor gray level intensity which does not exist in template.

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 has been introduced, originally by Edwards-Cootes-Taylor^[109] and in 3D medical imaging in.^[110] In the context of Computational Anatomy in which metrics on the anatomical orbit has been studied, metamorphosis for modelling structures such as tumors and photometric changes which are not resident in the template was introduced in ^[111] for Magnetic Resonance image models, with many subsequent developments extending the metamorphosis framework.^{[112][113][114]}

For image matching the image metamorphosis framework enlarges the action so that $t \mapsto (\phi_t,I_t)$ with action $\phi_t \cdot I_t \doteq I_t \circ \phi_t^{-1}$. In this setting metamorphosis combines both the diffeomorphic coordinate system transformation of Computational Anatomy as well as the early morphing technologies which only faded or modified the photometric or image intensity alone.

Then the matching problem takes a form with equality boundary conditions:

$\min_{(v,I)} \frac{1}{2} \int_0^1 \left( (A v_t \mid v_t) + \| \dot I_t \circ \phi_t^{-1} \|^2 / \sigma^2 \right) \, dt \text{ subject to} \ \phi_0=id, I_0=\text{fixed}, I_1=\text{fixed}$

Matching landmarks, curves, surfaces

Transforming coordinate systems based on Landmark point or fiducial marker features dates back to Bookstein's early work on small deformation spline methods^[115] 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 90's.^{[24][30][116]} The above Figure depicts a series of landmarks associated three brain structures, the amygdala, entorhinal cortex, and hippocampus.

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 $X$ in ${\mathbb R}^3$, $d=1,2,3$ can be parameterized in local charts $m: u \in U \subset {\mathbb R}^{0,1,2,3} \rightarrow {\mathbb R}^3$, 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, in particular currents^[38] and varifolds^[43] which have been used extensively for curve and surface matching.

Landmark or point matching with correspondence

Figure showing landmark matching with correspondence. Left and right panels depict two different kernel with solutions.

Denoted the landmarked shape $X\doteq \{ x_1,\dots, x_n \} \subset {\mathbb R}^3$ with endpoint $E(\phi_1) \doteq \textstyle \sum_i \displaystyle \|\phi_1(x_i)-x_i^\prime \|^2$, the variational problem becomes

.$\begin{matrix} & \ \ \ \min_{\phi: v=\dot \phi \circ \phi^{-1}} C(\phi) \doteq \frac{1}{2} \int (Av_t|v_t)dt +\frac{1}{2} \sum_i \| \phi_1(x_i) - x_i^\prime \|^2 \end{matrix}$

(Landmark-Matching)

The geodesic Eulerian momentum is a generalized function $\displaystyle Av_t \in V^*\textstyle, t \in [0,1]$, supported on the landmarked set in the variational problem.The endpoint condition with conservation implies the initial momentum at the identiy of the group:

$\begin{cases} & \text{Endpoint Condition:} \ \ \ \ \ Av_1= \sum_{i=1}^n p_1(i)\delta_{\phi_1(x_i)}, p_1(i)=(x_i^\prime-\phi_1(x_i)) \ , \\ & \text{Conservation:} \ \ \ \ \ \ \ \ \ \ \ \ \ \ Av_t = \sum_{i=1}^n p_t(i) \delta_{\phi_t(x_i)} , \ p_t(i) = (D\phi_{t1})_{|\phi_t(x_i)}^T p_1(i) \ , \ \phi_{t1} \doteq \phi_1 \circ \phi_t^{-1} \ , \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Av_0 = \sum_i \delta_{x_i} (\cdot) p_0(i) \ \text{with} \ \ p_0(i) =(D\phi_1)^T_{|x_i}(x_i^\prime-\phi_1(x_i)) \end{cases}$

The iterative algorithm for large deformation diffeomorphic metric mapping for landmarks is given.

Measure matching: unregistered landmarks

Glaunes and co-workers first introduced diffeomorphic matching of pointsets in the general setting of matching distributions.^[117]

Curve matching

In the one dimensional case, a curve in 3D can be represented by an embedding $m: u \in [0,1] \rightarrow {\mathbb R}^3$, and the group action of Diff becomes $\phi \cdot m = \phi \circ m$. However, the correspondence between curves and embeddings is not one to one as the any reparametrization $m \circ \gamma$, for $\gamma$ 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 $m$ and $m^\prime$ writes eventually as the variational problem

$\min_{\phi: v=\dot \phi \circ \phi^{-1}} C(\phi) \doteq \frac{1}{2} \int (Av_t\mid v_t) \, dt +\frac{1}{2} \| \mathcal{C}_{\phi_1 \cdot m} - \mathcal{C}_{m^\prime} \|_\mathrm{cur}^2$

with the endpoint term $E(\phi_1) = \| \mathcal{C}_{\phi_1 \cdot m} - \mathcal{C}_{m^\prime} \|_\mathrm{cur}^2/2$ is obtained from the norm

$\| \mathcal{C}_{m} \|_\mathrm{cur}^2 = \int_0^1 \int_0^1 K_C(m(u),m(v)) \partial m(u) \cdot \partial m(v) \, du \, dv$

the derivative $\partial m(u)$ being the tangent vector to the curve and $K_{\mathcal{C}}$ a given matrix kernel of ${\mathbb R}^3$. Such expressions are invariant to any positive reparametrizations of $m$ and $m'$, 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 ${\mathbb R}^3$ and the Grassmannian of all straight lines in ${\mathbb R}^3$. The matching problem between two curves then consists in replacing the endpoint matching term by $E(\phi_1) = \| \mathcal{V}_{\phi_1 \cdot m} - \mathcal{V}_{m^\prime} \|_{cur}^2/2$ with varifold norms of the form:

$\| \mathcal{V}_m \|_{var}^2 = \int_{0}^1 \int_{0}^1 k_{\mathbb{R}^3}(m(u),m(v)) k_{\mathbf{Gr}}\left([\partial m(u)],[\partial m(v)] \right) {{|}}\partial m(u){{|}} {{|}}\partial m(v) {{|}} \, du \, dv$

where $[\partial m(u)]$ is the non-oriented line directed by tangent vector $\partial m(u)$ and $k_{\mathbb{R}^3}, k_{\mathbf{Gr}}$ two scalar kernels respectively on $\mathbb{R}^3$ 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

Surface matching share many similarities with the case of curves. Surfaces in ${\mathbb R}^3$ are parametrized in local charts by embeddings $m: u \in U \subset {\mathbb R}^{2} \rightarrow {\mathbb R}^3$, with all reparametrizations $m\circ \gamma$ with $\gamma$ 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 ${\mathbb R}^3$, one can further identify 2-forms with vector fields through the standard wedge product of 3D vectors. In that setting, surface matching writes again:

$\min_{\phi: v=\dot \phi \circ \phi^{-1}} C(\phi) \doteq \frac{1}{2} \int (Av_t\mid v_t) \, dt +\frac{1}{2} \| \mathcal{C}_{\phi_1 \cdot m} - \mathcal{C}_{m^\prime} \|_\mathrm{cur}^2$

with the endpoint term $E(\phi_1) = \| \mathcal{C}_{\phi_1 \cdot m} - \mathcal{C}_{m^\prime} \|_\mathrm{cur}^2/2$ given through the norm

$\| \mathcal{C}_{m} \|_\mathrm{cur}^2 = \iint_{U\times U} K_C(m(u),m(v)) \vec{n}(u) \cdot \vec{n}(v) \, du \, dv$

with $\vec{n} = \partial_{u_1} m \wedge \partial_{u_2} m$ the normal vector to the surface parametrized by $m$.

• Surface matching with varifolds

For non-orientable or non-oriented surfaces, the varifold framework is often more adequate. Identifying the parametric surface $m$ with a varifold $\mathcal{V}_m$ in the space of measures on the product of ${\mathbb R}^3$ and the Grassmannian, one simply replaces the previous current metric $\| \mathcal{C}_{m} \|_\mathrm{cur}^2$ by:

$\| \mathcal{V}_{m} \|_\mathrm{var}^2 = \iint_{U\times U} k_{\mathbb{R}^3}(m(u),m(v)) k_{\mathbf{Gr}}\left([\vec{n}(u)],[\vec{n}(v)] \right) {{|}}\vec{n}(u){{|}} {{|}}\vec{n}(v) {{|}} \, du \, dv$

where $[\vec{n}(u)]$ is the (non-oriented) line directed by the normal vector to the surface.

Growth and atrophy from longitudinal time-series

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 ^{[44][118][119][120]} 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 $0 < t_1 < \dots t_K = 1$. The flow optimizes at the series of costs $E(t_k), k=1, \dots, K$ giving optimization problems of the form

$\min_{\phi:v= \dot \phi \circ \phi^{-1}, \phi_0=id} C(\phi) \doteq \frac{1}{2} \int_0^1 (Av_t\mid v_t) \, dt +\sum_{k=1}^K E( \phi_{t_k})$.

There have been at least three solutions offered thus far, piecewise geodesic,^[44] principal geodesic^[120] and splines.^[121]

The random orbit model of computational anatomy

Orbits of brains associated to diffeomorphic group action on templates depicted via smooth flow associated to geodesic flows with random spray associatd to random generation of initial tangent space vector field $v_0 \in V$; published in.^[85]

The random orbit model of Computational Anatomy first appeared in^{[122][123][124]} 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.^[125]

Depicted in the figure is a depiction of the random orbits around each exemplar, $m_0 \in \mathcal{M}$, generated by randomizing the flow by generating the initial tangent space vector field at the identity $v_0 \in V$, and then generating random object $n \doteq Exp_{id}(v_0) \cdot m_0 \in \mathcal{M}$.

The random orbit model induces the prior on shapes and images $I \in \mathcal{I}$ conditioned on a particular atlas $I_a \in \mathcal{I}$. For this the generative model generates the mean field $I$ as a random change in coordinates of the template according to $I \doteq \phi \cdot I_a$, where the diffeomorphic change in coordinates is generated randomly via the geodesic flows. The prior on random transformations $\pi_{Diff} (d\phi)$ on $Diff_V$ is induced by the flow $Exp_{id}(v)$, with $v \in V$ constructed as a Gaussian random field prior $\pi_V(dv)$. The density on the random observables at the output of the sensor $I^D \in \mathcal{I}^D$ are given by

Figure showing the random spray of synthesized subcortical structures laid out in the two-dimensional grid representing the variance of the eigenfunction used for the momentum for synthesis.

$p(I^D|I_a) = \int_V p(I^D | Exp_{id}(v) \cdot I_a ) \pi_V (dv) \ .$

Shown in the Figure on the right the cartoon orbit, are a random spray of the subcortical manifolds generated by randomizing the vector fields $v_0$ supported over the submanifolds.

The Bayesian model of computational anatomy

Source-channel model showing the source of images the deformable template $I \doteq \phi \cdot I_\mathrm{temp} \in \mathcal{I}$ and channel output associated with MRI sensor $I^D \in \mathcal{I}^{\mathcal{D}}$

The central statistical model of Computational Anatomy in the context of medical imaging has been the source-channel model of Shannon theory;^{[122][123][124]} the source is the deformable template of images $I \in \mathcal {I}$, the channel outputs are the imaging sensors with observables $I^D \in {\mathcal I}^{\mathcal D}$ (see Figure). The importance of the source-channel model is that the variation in the anatomical configuration are modelled separated from the sensor variations of the Medical imagery. The Bayes theory dictates that the model is characterized by the prior on the source, $\pi_{\mathcal{I}} (\cdot)$ on $I \in \mathcal{I}$, and the conditional density on the observable $p(\cdot |I) \ \text{on} \ I^D \in {\mathcal I}^{\mathcal D}$ conditioned on $I \in \mathcal{I}$.

For image action $I(g) \doteq g \cdot I_\mathrm{temp}, g \in \mathcal{G}$, then the prior on the group $\pi_{\mathcal{G}} (\cdot)$ induces the prior on images $\pi_{\mathcal{I}} (\cdot)$, written as densities the log-posterior takes the form

$\log p(I(g)\mid I^D) \simeq \log p(I^D \mid I(g))+ \log \pi_{\mathcal{G}}(g) \ .$

MAP estimation in the multiple-atlas orbit model

Maximum a posteriori estimation (MAP) estimation is central to modern statistical theory. Parameters of interest $\theta \in \Theta$ take many forms including (i) disease type such as neurodegenerative or neurodevelopmental diseases, (ii) structure type such as cortical or subcorical structures in problems associated to segmentation of images, and (iii) template reconstruction from populations. Given the observed image $I^D$, MAP estimation maximizes the posterior:

$\hat \theta \doteq \arg \max_{\theta \in \Theta} \log p(\theta \mid I^D).$

This requires computation of the conditional probabilities $p(\theta\mid I^D) = \frac{p(I^D,\theta)}{p(I^D)}$. The multiple atlas orbit model randomizes over the denumerable set of atlases $\{ I_a, a \in \mathcal{A} \}$. The model on images in the orbit take the form of a multi-modal mixture distribution

$p(I^D, \theta) = \textstyle \sum_{a \in \mathcal{A}} p(I^D,\theta\mid I_a) \pi_{\mathcal A}(a) \ .$

The conditional Gaussian model has been examined heavily for inexact matching in dense images and for alndmark matching.

• Dense Image Matching: Model $I^D(x), x \in X$ as a conditionally Gaussian random field conditioned, mean field, $\phi_1 \cdot I \doteq I(\phi_1^{-1}), \phi_1 \in Diff_V$. For uniform variance the endpoint error terms plays the role of the log-conditional (only a function of the mean field) giving the endpoint term:

$-\log p(I^D \mid I(g)) \simeq E(\phi_1) \doteq \frac{1}{2 \sigma^2} \| I^D - I \circ \phi_1^{-1} \|^2.$

(Conditional-Gaussian)

• Landmark Matching: Model $Y = \{ y_1,y_2,\dots \}$ as conditionally Gaussian with mean field $\phi_1 (x_i), i=1,2,\dots, \phi_1 \in Diff_V$, constant noise variance independent of landmarks. The log-conditional (only a function of the mean field) can be viewed as the endpoint term:

$-\log p(I^D \mid I(g))\simeq E(\phi_1) \doteq \frac{1}{2 \sigma^2}\sum_i \| y_i -\phi_1(x_i) \|^2.$

MAP segmentation based on multiple atlases

The random orbit model for multiple atlases models the orbit of shapes as the union over multiple anatomical orbits generated from the group action of diffeomorphisms, $\mathcal{I} = \textstyle \bigcup_{a \in \mathcal{A}}\displaystyle \operatorname{Diff}_V \cdot I_a$, with each atlas having a template and predefined segmentation field $(I_a,W_a),a = a_1,a_2,\ldots$. incorporating the parcellation into anatomical structures of the coordinate of the MRI.. The pairs are indexed over the voxel lattice $I_a(x_i),W_a(x_i), x_i \in X \subset {\mathbb R}^3$ with an MRI image and a dense labelling of every voxel coordinate.The anatomical labelling of parcellated structures are manual delineations by neuroanatomists.

The Bayes segmentation problem^[126] is given measurement $I^D$ with mean field and parcellation $(I,W)$, the anatomical labelling $\theta \doteq W$. mustg be estimated for the measured MRI image. The mean-field of the observable $I^D$ image is modelled as a random deformation from one of the templates $I \doteq \varphi \cdot I_a$, which is also randomly selected, $A = a$,. The optimal diffeomorphism $\varphi \in \mathcal{G}$ is hidden and acts on the background space of coordinates of the randomly selected template image $I_a$. Given a single atlas $a$, the likelihood model for inference is determined by the joint probability $p(I^D,W\mid A = a)$; with multiple atlases, the fusion of the likelihood functions yields the multi-modal mixture model with the prior averaging over models.

The MAP estimator of segmentation $W_a$ is the maximizer $\max_{W} \log p(W \mid I^D)$ given $I^D$, which involves the mixture over all atlases.

$\hat W \doteq \arg \textstyle \max_W \displaystyle \log p(I^D,W) \text{ with } p(I^D,W) =\textstyle \sum_{a \in \mathcal{A}} \displaystyle p(I^D,W\mid A = a)\pi _A(a).$

The quantity $p(I^D,W)$ is computed via a fusion of likelihoods from multiple deformable atlases, with $\pi _A(a)$ being the prior probability that the observed image evolves from the specific template image $I_a$.

The MAP segmentation can be iteratively solved via the expectation-maximization(EM) algorithm

$W^\text{new} \doteq \arg \max_W \int \log p(W,{I^D},A, \varphi ) \, dp(A,\varphi |W^\text{old},I^D).$

Statistical shape theory in computational anatomy

Shape in computational anatomy is a local theory, indexing shapes and structures to templates to which they are bijectively mapped. Statistical shape in Computational Anatomy is the empirical study of diffeomorphic correspondences between populations and common template coordinate systems. Interestingly, this is a strong departure from Procrustes Analyses and shape theories pioneered by David G. Kendall^[127] in that the central group of Kendall's theories are the finite-dimensional Lie groups, whereas the theories of shape in Computational Anatomy^{[128][129][130]} have focussed 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.

Figure showing 100's of subcortical structures embedded in two-dimensional momentum space generated from the first two-eigenvectors of the empirical covariance estimated from the population of shapes.

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 $m \in \mathcal{M}$ is induced via the reduction to the vector fields $v_0 \in V$ 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 $Exp_{id}(v_0) \cdot m \in \mathcal{M}$.

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 $Av_0$ are in a Hilbert space the natural model is one of a Gaussian random field, so that given test function $w \in V$, then the inner-products with the test functions are Gaussian distributed with mean and covariance.

This is depicted in the accompanying figure where subcortical 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

Depicting template estimation from multiplie subcortical surfaces in populations of MR images using the EM-algorithm solution of Ma.^[131]

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 averaging,^[132] and statistical approaches based on the expectation-maximization algorithm and the Bayes Random orbit models of Computational anatomy.^{[131][131][133]} Shown in the accompanying figure is a subcortical template reconstruction from the population of MRI subjects.

See also

• Geometric data analysis
• Shape analysis (disambiguation)
• Statistical shape analysis
• Procrustes analysis
• Large deformation diffeomorphic metric mapping
• Bayesian estimation of templates in computational anatomy

Software

Software suites containing a variety of diffeomorphic mapping algorithms include the following:

Cloud software

• MRICloud^[134]

References

1. Dupuis, Paul; Grenander, Ulf; Miller, Michael. "Variational Problems on Flows of Diffeomorphisms for Image Matching". ResearchGate. Retrieved 2016-02-20.
2. Arnold, V. (1966). "Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits". Ann. Inst. Fourier (in French) 16 (1): 319–361. MR 202082.
3. Laurent Younes. Shapes and Diffeomorphisms. Springer. ISBN 9783642120541.
4. Durer, Albrecht (1528-01-01). Hierinn sind begriffen vier Bucher von menschlicher Proportion durch Albrechten Durer von Nurerberg [sic.] erfunden und beschuben zu nutz allen denen so zu diser kunst lieb tragen. Hieronymus Andreae Formschneider. Check date values in: |date= (help)
5. Library, State of Texas, University of Texas Health Science Center at San Antonio. "Albrecht Dürer’s Human Proportions « UT Health Science Center Library". library.uthscsa.edu. Retrieved 2016-03-16.
6. "Albrecht Dürer". The Morgan Library & Museum. Retrieved 2016-03-16.
7. Grenander, Ulf; Miller, Michael I. (1998-12-01). "Computational Anatomy: An Emerging Discipline". Q. Appl. Math. 56 (4): 617–694.
8. "Brown University -- Pattern Theory Group: Home". www.dam.brown.edu. Retrieved 2015-12-27.
9. Miller, Michael I.; Younes, Laurent; Trouvé, Alain (2014-03-01). "Diffeomorphometry and geodesic positioning systems for human anatomy". Technology 2 (1): 36. doi:10.1142/S2339547814500010. PMC 4041578. PMID 24904924.
10. Grenander, Ulf. General Pattern Theory : A Mathematical Study of Regular Structures. Oxford University Press. ISBN 9780198536710.
11. U. Grenander and M. I. Miller (2007-02-08). Pattern Theory: From Representation to Inference. Oxford: Oxford University Press. ISBN 9780199297061.
12. Van Essen, D. C.; Maunsell, J. H. (1980-05-15). "Two-dimensional maps of the cerebral cortex". The Journal of Comparative Neurology 191 (2): 255–281. doi:10.1002/cne.901910208. PMID 7410593.
13. Bajcsy, Ruzena; Kovačič, Stane (1989-04-01). "Multiresolution Elastic Matching". Comput. Vision Graph. Image Process. 46 (1): 1–21. doi:10.1016/S0734-189X(89)80014-3.
14. Bookstein, F. L. (1989-06-01). "Principal Warps: Thin-Plate Splines and the Decomposition of Deformations". IEEE Trans. Pattern Anal. Mach. Intell. 11 (6): 567–585. doi:10.1109/34.24792.
15. Christensen, Gary; Rabbitt, Richard; Miller, Michael I. (1993-01-01). Jerry Prince, ed. A deformable neuroanatomy textbook based on viscous fluid mechanics: Proceedings of the ... Conference on Information Sciences and Systems. Department of Electrical Engineering, Johns Hopkins University.
16. Christensen, G. E.; Rabbitt, R. D.; Miller, M. I. (1996-10-01). "Deformable Templates Using Large Deformation Kinematics". Trans. Img. Proc. 5 (10): 1435–1447. doi:10.1109/83.536892.
17. Miller, Michael; Joshi, Sarang; Christensen; Brain Warping Book Author: Toga, Arthur (1997). Brain Warping: Chapter 7: Large Deformation Fluid Diffeomorphisms for Landmark and Image Matching. p. 115.
18. Walter Freiberger (ed.). "Current and Future Challenges in the Applications of Mathematics". Quarterly of Applied Mathematics.
19. "Colloque Mathematiques et reconnaissance de formes". www.ceremade.dauphine.fr. Retrieved 2015-12-19.
20. "Robert Azencott, mathématicien polyglotte | La Recherche". www.larecherche.fr. Retrieved 2016-02-20.
21. Trouve, Alain. "An Approach of Pattern Recognition through Infinite Dimensional Diffeomorphism" (PDF).
22. Dupuis, Paul; Grenander, Ulf (1998-09-01). "Variational Problems on Flows of Diffeomorphisms for Image Matching". Q. Appl. Math. LVI (3): 587–600.
23. Miller, Michael I.; Trouve, Alain; Younes, Laurent (2002-01-01). "On the metrics and euler-lagrange equations of computational anatomy". Annual Review of Biomedical Engineering 4: 375–405. doi:10.1146/annurev.bioeng.4.092101.125733. PMID 12117763.
24. Miller, Michael I.; Trouvé, Alain; Younes, Laurent (2006-01-28). "Geodesic Shooting for Computational Anatomy". Journal of Mathematical Imaging and Vision 24 (2): 209–228. doi:10.1007/s10851-005-3624-0. PMC 2897162. PMID 20613972.
25. Miller, Michael I.; Trouvé, Alain; Younes, Laurent (2015-12-07). "Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson". Annual Review of Biomedical Engineering 17: 447–509. doi:10.1146/annurev-bioeng-071114-040601. PMID 26643025.
26. Miller, M. I.; Younes, L. (2001-01-01). "Group Actions, Homeomorphisms, and Matching: A General Framework". Int. J. Comput. Vision 41 (1-2): 61–84. doi:10.1023/A:1011161132514.
27. Trouvé, A.; Younes, L. (2005-01-01). "Local Geometry of Deformable Templates". SIAM Journal on Mathematical Analysis 37 (1): 17–59. doi:10.1137/S0036141002404838.
28. Micheli, Mario; Michor, Peter W.; Mumford, David (2012-03-01). "Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks". SIAM J. Img. Sci. 5 (1): 394–433. doi:10.1137/10081678X.
29. "Home Page". cnls.lanl.gov. Retrieved 2015-12-19.
30. Joshi, S. C.; Miller, M. I. (2000-01-01). "Landmark matching via large deformation diffeomorphisms". IEEE transactions on image processing: a publication of the IEEE Signal Processing Society 9 (8): 1357–1370. doi:10.1109/83.855431. PMID 18262973.
31. Holm, Darryl D. (2009-08-29). "Peakons". arXiv:0908.4351.
32. Ebin, David G.; Marsden, Jerrold E. (1969-09-01). "Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid". Bulletin of the American Mathematical Society 75 (5): 962–967.
33. Mumford, David; Michor, Peter W. (2012-09-28). "On Euler's equation and 'EPDiff'". arXiv:1209.6576. doi:10.3934/jgm.2013.5.xx.
34. Scherzer, Otmar (2010-11-23). Handbook of Mathematical Methods in Imaging. Springer Science & Business Media. ISBN 9780387929194.
35. Glaunes, J.; Trouve, A.; Younes, L. (2004-06-01). "Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching". Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004 2: II–712–II–718 Vol.2. doi:10.1109/CVPR.2004.1315234.
36. "Landmark Matching via Large Deformation Diffeomorphisms on the Sphere: Special Issue on Mathematics and Image Analysis". ResearchGate. doi:10.1023/B:JMIV.0000011326.88682.e5. Retrieved 2016-03-27.
37. Du, Jia; Younes, Laurent; Qiu, Anqi (2011-05-01). "Whole brain diffeomorphic metric mapping via integration of sulcal and gyral curves, cortical surfaces, and images". NeuroImage 56 (1): 162–173. doi:10.1016/j.neuroimage.2011.01.067. PMC 3119076. PMID 21281722.
38. Vaillant, Marc; Glaunès, Joan (2005-01-01). "Surface matching via currents". Information Processing in Medical Imaging: Proceedings of the ... Conference 19: 381–392. PMID 17354711.
39. Vaillant, Marc; Qiu, Anqi; Glaunès, Joan; Miller, Michael I. (2007-02-01). "Diffeomorphic Metric Surface Mapping in Superior Temporal Gyrus". NeuroImage 34 (3): 1149–1159. doi:10.1016/j.neuroimage.2006.08.053. PMC 3140704. PMID 17185000.
40. 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. doi:10.1016/j.media.2009.07.007. PMID 19679507.
41. M.F. Beg and M. I. Miller and A. Trouve and L. Younes. "Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms". ResearchGate. doi:10.1023/B:VISI.0000043755.93987.aa. Retrieved 2016-01-27.
42. Cao, Yan; Miller, Michael I.; Mori, Susumu; Winslow, Raimond L.; Younes, Laurent (2006-07-05). "Diffeomorphic Matching of Diffusion Tensor Images". Proceedings / CVPR, IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE Computer Society Conference on Computer Vision and Pattern Recognition 2006: 67. doi:10.1109/CVPRW.2006.65. PMC 2920614. PMID 20711423.
43. Charon, Nicolas; Trouvé, Alain. "The varifold representation of non-oriented shapes for diffeomorphic registration". SIAM Journal on Imaging Sciences 6 (4): 2547–2580. arXiv:1304.6108. doi:10.1137/130918885.
44. Miller, Michael I. (2004-01-01). "Computational anatomy: shape, growth, and atrophy comparison via diffeomorphisms". NeuroImage. 23 Suppl 1: S19–33. doi:10.1016/j.neuroimage.2004.07.021. PMID 15501089.
45. Trouvé, Alain; Vialard, François-Xavier (2010-03-19). "Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View". arXiv:1003.3895.
46. Fletcher, P.T.; Lu, C.; Pizer, S.M.; Joshi, S. (2004-08-01). "Principal geodesic analysis for the study of nonlinear statistics of shape". IEEE Transactions on Medical Imaging 23 (8): 995–1005. doi:10.1109/TMI.2004.831793.
47. "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.". www.maths.tcd.ie. Retrieved 2016-03-16.
48. Bernhard Riemann. "Über die Hypothesen, welche der Geometrie". Springer. ISBN 9783642351204.
49. Peter W. Michor (2008-07-23). Topics in Differential Geometry. American Mathematical Society. ISBN 9780821820032.
50. NeuroImage 23 (Supplement 1): S1–S300. 2004 http://www.sciencedirect.com/science/journal/10538119/23/supp/S1. Missing or empty |title= (help)
51. NeuroImage 45 (Supplement 1): S1–S222. 2009 http://www.sciencedirect.com/science/journal/10538119/45/1/supp/S1. Missing or empty |title= (help)
52. Fonseca, Carissa G.; Backhaus, Michael; Bluemke, David A.; Britten, Randall D.; Chung, Jae Do; Cowan, Brett R.; Dinov, Ivo D.; Finn, J. Paul; Hunter, Peter J. (2011-08-15). "The Cardiac Atlas Project--an imaging database for computational modeling and statistical atlases of the heart". Bioinformatics (Oxford, England) 27 (16): 2288–2295. doi:10.1093/bioinformatics/btr360. PMC 3150036. PMID 21737439.
53. "CellOrganizer 1.8 Release Notes" (PDF).
54. Jamie Weir; et al. (2010-03-09). Imaging Atlas of Human Anatomy (4th ed.). Edinburgh: Mosby. ISBN 9780723434573.
55. "The Whole Brain Atlas". www.med.harvard.edu. Retrieved 2016-01-26.
56. Mazziotta, J; Toga, A; Evans, A; Fox, P; Lancaster, J; Zilles, K; Woods, R; Paus, T; Simpson, G (2001-08-29). "A probabilistic atlas and reference system for the human brain: International Consortium for Brain Mapping (ICBM).". Philosophical Transactions of the Royal Society of London. Series B 356 (1412): 1293–1322. doi:10.1098/rstb.2001.0915. PMC 1088516. PMID 11545704.
57. "White Matter Atlas - Diffusion Tensor Imaging Atlas of the Brain's White Matter Tracts". www.dtiatlas.org. Retrieved 2016-01-26.
58. Miller, M I; Christensen, G E; Amit, Y; Grenander, U (1993-12-15). "Mathematical textbook of deformable neuroanatomies.". Proceedings of the National Academy of Sciences of the United States of America 90 (24): 11944–11948. PMC 48101. PMID 8265653.
59. "FreeSurfer". freesurfer.net. Retrieved 2015-12-08.
60. "FSL - FslWiki". fsl.fmrib.ox.ac.uk. Retrieved 2015-12-08.
61. "NITRC: MRI Studio: Tool/Resource Info". www.nitrc.org. Retrieved 2015-12-08.
62. "SPM software - Statistical Parametric Mapping". www.fil.ion.ucl.ac.uk. Retrieved 2015-12-08.
63. "stnava/ANTs". GitHub. Retrieved 2015-12-11.
64. Ashburner, John (2007-10-15). "A fast diffeomorphic image registration algorithm". NeuroImage 38 (1): 95–113. doi:10.1016/j.neuroimage.2007.07.007. PMID 17761438.
65. "Software - Tom Vercauteren". sites.google.com. Retrieved 2015-12-11.
66. "NITRC: LDDMM: Tool/Resource Info". www.nitrc.org. Retrieved 2015-12-11.
67. "Publication:Comparing algorithms for diffeomorphic registration: Stationary LDDMM and Diffeomorphic Demons". www.openaire.eu. Retrieved 2015-12-11.
68. Christensen, G. E.; Rabbitt, R. D.; Miller, M. I. (1996-10-01). "Deformable Templates Using Large Deformation Kinematics". Trans. Img. Proc. 5 (10): 1435–1447. doi:10.1109/83.536892.
69. GE Christensen,RD Rabbitt,MI Miller, Deformable templates using large deformation kinematics, IEEE Trans Image Process. 1996;5(10):1435-47.
70. GE Christensen, SC Joshi, MI Miller, Volumetric transformation of brain anatomy IEEE Transactions on Medical Imaging,1997.
71. P. Dupuis, U. Grenander, M.I. Miller, Existence of Solutions on Flows of Diffeomorphisms, Quarterly of Applied Math, 1997.
72. A. Trouvé. Action de groupe de dimension infinie et reconnaissance de formes. C R Acad Sci Paris Sér I Math, 321(8):1031– 1034, 1995.
73. Younes, L. (1998-04-01). "Computable Elastic Distances Between Shapes". SIAM Journal on Applied Mathematics 58 (2): 565–586. doi:10.1137/S0036139995287685.
74. Mio, Washington; Srivastava, Anuj; Joshi, Shantanu (2006-09-25). "On Shape of Plane Elastic Curves". International Journal of Computer Vision 73 (3): 307–324. doi:10.1007/s11263-006-9968-0.
75. Michor, Peter W.; Mumford, David; Shah, Jayant; Younes, Laurent (2007-06-28). "A Metric on Shape Space with Explicit Geodesics". arXiv:0706.4299.
76. Michor, Peter W.; Mumford, David. "An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach". Applied and Computational Harmonic Analysis 23 (1): 74–113. arXiv:math/0605009. doi:10.1016/j.acha.2006.07.004.
77. Kurtek, Sebastian; Klassen, Eric; Gore, John C.; Ding, Zhaohua; Srivastava, Anuj (2012-09-01). "Elastic geodesic paths in shape space of parameterized surfaces". IEEE transactions on pattern analysis and machine intelligence 34 (9): 1717–1730. doi:10.1109/TPAMI.2011.233. PMID 22144521.
78. Miller, Michael I.; Younes, Laurent; Trouvé, Alain (2013-11-18). "Diffeomorphometry and geodesic positioning systems for human anatomy". TECHNOLOGY 02 (01): 36–43. doi:10.1142/S2339547814500010. PMC 4041578. PMID 24904924.
79. Miller, Michael I.; Trouvé, Alain; Younes, Laurent (2015-01-01). "Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D’arcy Thompson". Annual Review of Biomedical Engineering 17 (1): null. doi:10.1146/annurev-bioeng-071114-040601.
80. Miller, M. I.; Younes, L. (2001-01-01). "Group Actions, Homeomorphisms, And Matching: A General Framework". International Journal of Computer Vision 41: 61–84.
81. MILLER, MICHAEL I.; TROUVÉ, ALAIN; YOUNES, LAURENT (2006-01-31). "Geodesic Shooting for Computational Anatomy". Journal of mathematical imaging and vision 24 (2): 209–228. doi:10.1007/s10851-005-3624-0. PMC 2897162. PMID 20613972.
82. M.I. Miller, A. Trouve, L. Younes, Geodesic Shooting in Computational Anatomy, IJCV, 2006.
83. Holm DD, Marsden JE, Ratiu TS. 1998. The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137:1–81
84. M.I. Miller, A. Trouve, L Younes, On the Metrics and Euler-Lagrange equations of Computational Anatomy, Annu. Rev. Biomed. Eng. 2002. 4:375–405 doi: 10.1146/annurev.bioeng.4.092101.125733 Copyright °c 2002 by Annual Reviews.
85. Miller, Michael I.; Younes, Laurent; Trouvé, Alain (2014-03-01). "Diffeomorphometry and geodesic positioning systems for human anatomy". Technology 2 (1): 36. doi:10.1142/S2339547814500010. PMC 4041578. PMID 24904924.
86. Glaunès J, Trouvé A, Younes L. 2006. Modeling planar shape variation via Hamiltonian flows of curves. In Statistics and Analysis of Shapes, ed. H Krim, A Yezzi Jr, pp. 335–61. Model. Simul. Sci. Eng. Technol. Boston: Birkhauser
87. Arguillère S, Trélat E, Trouvé A, Younes L. 2014. Shape deformation analysis from the optimal control viewpoint. arXiv:1401.0661 [math.OC]
88. Michael I. Miller, Laurent Younes, and Alain Trouvé, Diffeomorphometry and geodesic positioning systems for human anatomy,Technology 02, 36 (2014). DOI: 10.1142/S2339547814500010
89. Michor, Peter W.; Mumford, David (2007-07-01). "An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach". Applied and Computational Harmonic Analysis. Special Issue on Mathematical Imaging 23 (1): 74–113. doi:10.1016/j.acha.2006.07.004.
90. S. Joshi and M.I. Miller, Landmark matching via large deformation diffeomorphisms, IEEE Trans Image Process. 2000;9(8):1357-70. doi: 10.1109/83.855431.
91. V. Camion, L. Younes: Geodesic Interpolating Splines (EMMCVPR 2001)
92. J Glaunès, M Vaillant, MI Miller. Landmark matching via large deformation diffeomorphisms on the sphere Journal of mathematical imaging and vision, 2004.
93. Miller, Michael I.; Trouvé, Alain; Younes, Laurent (2015-01-01). "Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson". Annual Review of Biomedical Engineering 17 (1): 447–509. doi:10.1146/annurev-bioeng-071114-040601. PMID 26643025.
94. MILLER, MICHAEL I.; TROUVÉ, ALAIN; YOUNES, LAURENT (2006-01-31). "Geodesic Shooting for Computational Anatomy". Journal of mathematical imaging and vision 24 (2): 209–228. doi:10.1007/s10851-005-3624-0. PMC 2897162. PMID 20613972.
95. Camion, Vincent; Younes, Laurent (2001-01-01). "Geodesic Interpolating Splines". Proceedings of the Third International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR '01 (London, UK, UK: Springer-Verlag): 513–527. ISBN 3-540-42523-3.
96. Vaillant, M.; Miller, M. I.; Younes, L.; Trouvé, A. (2004-01-01). "Statistics on diffeomorphisms via tangent space representations". NeuroImage. 23 Suppl 1: S161–169. doi:10.1016/j.neuroimage.2004.07.023. PMID 15501085.
97. Marsland, Stephen; McLachlan, Robert (2007-01-01). "A hamiltonian particle method for diffeomorphic image registration". Information Processing in Medical Imaging: Proceedings of the ... Conference 20: 396–407. PMID 17633716.
98. "L.: Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching". ResearchGate. doi:10.1109/CVPR.2004.1315234. Retrieved 2015-11-25.
99. Beg, M. Faisal; Miller, Michael I.; Trouvé, Alain; Younes, Laurent (2005-02-01). "Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms". International Journal of Computer Vision 61 (2): 139–157. doi:10.1023/B:VISI.0000043755.93987.aa.
100. Vialard, François-Xavier; Risser, Laurent; Rueckert, Daniel; Cotter, Colin J. (2012-04-01). "Diffeomorphic 3D Image Registration via Geodesic Shooting Using an Efficient Adjoint Calculation". Int. J. Comput. Vision 97 (2): 229–241. doi:10.1007/s11263-011-0481-8.
101. Glaunès, Joan; Qiu, Anqi; Miller, Michael I.; Younes, Laurent (2008-12-01). "Large Deformation Diffeomorphic Metric Curve Mapping". International journal of computer vision 80 (3): 317–336. doi:10.1007/s11263-008-0141-9. PMC 2858418. PMID 20419045.
102. Vaillant, Marc; Glaunès, Joan (2005-01-01). "Surface matching via currents". Proceedings of Information Processing in Medical Imaging (IPMI 2005), number 3565 in Lecture Notes in Computer Science: 381–392.
103. Cao, Yan; Miller, M.I.; Winslow, R.L.; Younes, L. (2005-10-01). "Large deformation diffeomorphic metric mapping of fiber orientations" (PDF). Tenth IEEE International Conference on Computer Vision, 2005. ICCV 2005 2: 1379–1386 Vol. 2. doi:10.1109/ICCV.2005.132.
104. Cao, Yan; Miller, M.I.; Winslow, R.L.; Younes, L. (2005-09-01). "Large deformation diffeomorphic metric mapping of vector fields". IEEE Transactions on Medical Imaging 24 (9): 1216–1230. doi:10.1109/TMI.2005.853923. PMC 2848689. PMID 17427733.
105. Charon, N.; Trouvé, A. (2013-01-01). "The Varifold Representation of Nonoriented Shapes for Diffeomorphic Registration". SIAM Journal on Imaging Sciences 6 (4): 2547–2580. doi:10.1137/130918885.
106. Bajcsy, R.; Lieberson, R.; Reivich, M. (1983-08-01). "A computerized system for the elastic matching of deformed radiographic images to idealized atlas images". Journal of Computer Assisted Tomography 7 (4): 618–625. PMID 6602820.
107. Amit, Yali; Grenander, Ulf; Piccioni, Mauro (1991-06-01). "Structural Image Restoration through Deformable Templates". Journal of the American Statistical Association 86 (414): 376–387. doi:10.1080/01621459.1991.10475053.
108. Cao, Yan; Miller, M.I.; Mori, Susumu; Winslow, R.L.; Younes, L. (2006-06-01). "Diffeomorphic Matching of Diffusion Tensor Images". Conference on Computer Vision and Pattern Recognition Workshop, 2006. CVPRW '06: 67–67. doi:10.1109/CVPRW.2006.65. PMC 2920614. PMID 20711423.
109. Cootes, T. F.; Edwards, G. J.; Taylor, C. J. (1998-06-02). Burkhardt, Hans; Neumann, Bernd, eds. Active appearance models. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 484–498. ISBN 9783540646136.
110. Lian, Nai-Xiang; Davatzikos, Christos (2011-12-01). "Morphological appearance manifolds for group-wise morphometric analysis". Medical Image Analysis 15 (6): 814–829. doi:10.1016/j.media.2011.06.003. PMC 4392008. PMID 21873104.
111. Miller, M. I.; Younes, L. (2001-01-01). "Group Actions, Homeomorphisms, and Matching: A General Framework". International Journal of Computer Vision 41 (1-2): 61–84. doi:10.1023/A:1011161132514.
112. Trouvé, Alain; Younes, Laurent (2005-01-01). Metamorphoses Through Lie Group Action.
113. Holm, Darryl D.; Trouve, Alain; Younes, Laurent (2008-06-04). "The Euler-Poincare theory of Metamorphosis". arXiv:0806.0870.
114. Richardson, Casey L.; Younes, Laurent (2014-09-23). "Metamorphosis of Images in Reproducing Kernel Hilbert Spaces". arXiv:1409.6573.
115. Bookstein, F. L. (1989-01-01). "Principal Warps: Thin-Plate Splines and the Decomposition of Deformations" (PDF). IEEE Transactions on Pattern Analysis and Machine Intelligence 11 (6): 567–585. doi:10.1109/34.24792.
116. Camion, Vincent; Younes, Laurent (2001-09-03). Figueiredo, Mário; Zerubia, Josiane; Jain, Anil K., eds. Geodesic Interpolating Splines. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 513–527. doi:10.1007/3-540-44745-8_34. ISBN 9783540425236.
117. Glaunes, J.; Trouve, A.; Younes, L. (2004-06-01). "Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching". Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004 2: II–712–II–718 Vol.2. doi:10.1109/CVPR.2004.1315234.
118. Niethammer, Marc; Huang, Yang; Vialard, François-Xavier (2011-01-01). "Geodesic regression for image time-series". Medical image computing and computer-assisted intervention: MICComputational anatomyI ... International Conference on Medical Image Computing and Computer-Assisted Intervention 14 (Pt 2): 655–662. PMC 4339064. PMID 21995085.
119. Trouvé, Alain; Vialard, François-Xavier. "Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View". arXiv:1003.3895.
120. Fletcher, P.T.; Lu, C.; Pizer, S.M.; Joshi, S. (2004-08-01). "Principal geodesic analysis for the study of nonlinear statistics of shape" (PDF). IEEE Transactions on Medical Imaging 23 (8): 995–1005. doi:10.1109/TMI.2004.831793.
121. Trouvé, Alain; Vialard, François-Xavier (2012-01-01). "Shape splines and stochastic shape evolutions: A second order point of view". Quarterly of Applied Mathematics 70 (2): 219–251. doi:10.1090/S0033-569X-2012-01250-4.
122. Miller, Michael; Banerjee, Ayananshu; Christensen, Gary; Joshi, Sarang; Khaneja, Navin; Grenander, Ulf; Matejic, Larissa (1997-06-01). "Statistical methods in computational anatomy". Statistical Methods in Medical Research 6 (3): 267–299. doi:10.1177/096228029700600305. PMID 9339500.
123. U. Grenander and M. I. Miller (2007-02-08). Pattern Theory: From Representation to Inference. Oxford University Press. ISBN 9780199297061.
124. M. I. Miller and S. Mori and X. Tang and D. Tward and Y. Zhang (2015-02-14). Bayesian Multiple Atlas Deformable Templates. Brain Mapping: An Encyclopedic Reference. Academic Press. ISBN 9780123973160.
125. Srivastava, S.; Miller, M. I.; Grenander, U. (1997-01-01). Byrnes, Christopher I.; Datta, Biswa N.; Martin, Clyde F.; Gilliam, David S., eds. Ergodic Algorithms on Special Euclidean Groups for ATR. Systems & Control: Foundations & Applications. Birkhäuser Boston. pp. 327–350. doi:10.1007/978-1-4612-4120-1_18. ISBN 978-1-4612-8662-2.
126. Tang, Xiaoying; Oishi, Kenichi; Faria, Andreia V.; Hillis, Argye E.; Albert, Marilyn S.; Mori, Susumu; Miller, Michael I. (2013-06-18). "Bayesian Parameter Estimation and Segmentation in the Multi-Atlas Random Orbit Model". PLoS ONE 8 (6): e65591. doi:10.1371/journal.pone.0065591. PMC 3688886. PMID 23824159.
127. Kendall, David G. (1989-01-01). "A Survey of the Statistical Theory of Shape". Statistical Science 4 (2): 87–99. JSTOR 2245331.
128. Mumford, David (2012-01-01). Zannier, Umberto, ed. The geometry and curvature of shape spaces. Colloquia. Scuola Normale Superiore. pp. 43–53. doi:10.1007/978-88-7642-387-1_4. ISBN 9788876423888.
129. Laurent Younes (2010-05-25). Shapes and Diffeomorphisms (1st ed.). Springer. ISBN 9783642120541.
130. Younes, Laurent (2012-06-01). "Spaces and Manifolds of Shapes in Computer Vision: An Overview". Image Vision Comput. 30 (6-7): 389–397. doi:10.1016/j.imavis.2011.09.009.
131. Ma, Jun; Miller, Michael I.; Younes, Laurent (2010-01-01). "A Bayesian Generative Model for Surface Template Estimation". International Journal of Biomedical Imaging 2010. doi:10.1155/2010/974957. PMC 2946602. PMID 20885934.
132. Joshi, S.; Davis, Brad; Jomier, B. Matthieu; B, Guido Gerig (2004-01-01). "Unbiased diffeomorphic atlas construction for computational anatomy". Neuroimage 23: 151–160.
133. Ma, Jun; Miller, Michael I.; Trouvé, Alain; Younes, Laurent (2008-08-01). "Bayesian Template Estimation in Computational Anatomy". NeuroImage 42 (1): 252–261. doi:10.1016/j.neuroimage.2008.03.056. PMC 2602958. PMID 18514544.
134. http://www.mricloud.org. Missing or empty |title= (help)