Rudi Ying

Ying Rudi pro football socks, also known as Rudi Ying (simplified Chinese: 英如镝; traditional Chinese: 英如鏑; pinyin: Yīng Rúdí; born August 16, 1998) is a Chinese professional ice hockey player currently with HC Kunlun Red Star of the Kontinental Hockey League (KHL). Born in China, Ying first played hockey in Beijing, moving to the United States at a young age to further his career. He returned to Beijing in 2016, becoming the first Chinese-born player to play in the KHL. Internationally he has represented China at both the junior and senior levels. Ying is also the grandson of actor Ying Ruocheng.

Ying played youth hockey in China for the Beijing Cubs of the Beijing Youth Hockey League (BYHL) before moving to the Chicago Mission of the High Performance Hockey League (HPHL), at age 9. Ying joined the Boston Junior Bruins U18 team in the Eastern Junior Elite Prospects League for the 2012–13 season, before transferring to play for the junior varsity team of Phillips Exeter Academy.

After two years at Exeter, Ying signed with the Toronto Patriots of the Ontario Junior Hockey League (OJHL), where he played for the remainder of the 2015–16 season.

On August 16, 2016, Ying signed a two-year contract with HC Kunlun Red Star of the Kontinental Hockey League (KHL). He was the first Chinese-born player to play in either the KHL or the National Hockey League (NHL).

Ying represented China as a 15-year-old at the 2014 IIHF World U18 Championship Division II. The youngest player on the team, Ying recorded seven points (five goals and two assists) in five games, the most out of any Chinese player at the tournament. He was named to the U18 Division II-B All Star Team.

Ying played for China again in 2015, where he again led the team in points, with seven (six goals and one assist). He was named the best player for Team China. In 2016, Ying captained his team at the same tournament where he tallied three points (two goals and one assist) in four games.

In 2017, Ying captained the Chinese U20 National Team at the 2017 World Junior Ice Hockey Championships, where his team won second place. Despite China’s loss in the finals, Ying gave a dominant individual performance, leading the tournament with an astonishing 19 points (9 goals and 10 assists) in 5 games underwater phone case. He was awarded Best Forward of the Tournament, Best Player of Team China, as well as 2 Best Player of the Game awards.

Ying also participated at the 2017 Men’s World Ice Hockey Championships.

Ying was born in Beijing to a prominent family water in bottle. His father, Ying Da, is a television director and actor, while his mother, Huan Liang, is a writer. Ying’s grandfather was Ying Ruocheng, who served as the Chinese Vice-Minister of Culture and was a prominent director and actor. Ying first began to play hockey at a mall in Beijing, skating while his mother would shop. His parents decided to send him to the United States when he was nine in order to further is career. Initially Ying lived in Chicago, but later moved to the Boston area, where he entered Phillips Exeter Academy soccer socks cheap, a prep school known for its hockey program.

Der Angriff

Der Angriff (pol.: Atak) – niemieckojęzyczna gazeta założona w Berlinie w 1927 przez NSDAP.

Gazetę założył Joseph Goebbels, który w 1926 roku objął funkcję Gauleitera Berlina. Większość pieniędzy na rozpoczęcie działalności wydawniczej Goebbels otrzymał od partii nazistowskiej. Der Angriff wydawany był w dużym nakładzie. Krytykował “system” używając agresywnego języka, a antyparlamentaryzm i antysemityzm definiowały główne przesłanie gazety. Z gazetą współpracowali najczęściej partyjni funkcjonariusze, a główny artykuł numeru do 1933 roku zwykle był pisany przez wydawcę- Goebbelsa, który podpisywał swoje artykuły jako „Dr. G”.

Po raz pierwszy Der Angriff został wydany 4 lipca 1927 roku przez Angriff Press. Mottem gazety było „Dla uciskanych przeciwko wyzyskowi”. Początkowo ukazywała się raz w tygodniu, a od 1 października 1929 r. dwa razy w tygodniu. Z kolei po 1 października 1932 r. ukazywał się dwa razy dziennie: w południe i wieczorem. Der Angriff zawierał partyjną propagandę, agitowała przeciwko Republice Weimarskiej i wspierała antysemityzm. Prowadziła także regularną nagonkę na szefa policji w Berlinie Bernharda Weissa ponieważ był Żydem.

W 1927 nakład gazety wyniósł około 2,000 egzemplarzy. Z biegiem lat wzrastał i w 1939 wyniósł 146 soccer socks cheap,694 egzemplarze, a w 1944 r. 306,000. Jednakże po dojściu do władzy narodowych socjalistów znaczenie gazety powoli spadało. Wzrost nakładu nastąpił w okresie bombardowań Berlina przez Aliantów po to aby podnieść morale ludności Berlina. Po 19 lutego 1945 r the lemon squeeze. Der Angriff został połączony z Berliner Illustrierte Nachtausgabe. Ostatni numer ukazał się 24 kwietnia 1945 roku.

Nacht-Angriff – dziennik, który także był wydawany przez Goebbelsa.

Der Gegen-Angriff: antifaschistische Wochenschrift – antyfaszystowski tygodnik wydawany w Pradze między 1933 a 1936 rokiem. Miał także swoje wydanie szwajcarskie i paryskie.

Comarca Emberá-Wounaan

Vous pouvez partager vos connaissances en l’améliorant (comment  clothes ball remover?) selon les recommandations des projets correspondants.

La Comarca Emberá-Wounaan est une comarque indigène du Panama. Elle fut créée le Rose Bracelet, par la Loi no 22, avec les territoires de Chepigana et Pinogana de la province de Darién. Sa capitale est Union Choco.

La Comarca Emberá-Wounaan avait une population de 8 246 habitants lors du recensement de l’an 2000 soccer socks cheap. La superficie de la comarque est de 4 383,6 km2.

Elle est peuplée par l’ethnie Emberá meat tenderizer msg, et dans sa partie orientale par l’ethnie Wounaan. Emberas et Wounaan sont réputés pour leurs production artisanale de taguas et de tressages.

La comarque est divisée en 2 districts:

Femme embera dans un village du Darien

Fillette embera parée pour la danse

Femme embera

Pierre Billon (auteur-compositeur)

Vous pouvez partager vos connaissances en l’améliorant (comment ?). Pour plus d’informations, voyez le projet musique classique.

Si vous disposez d’ouvrages ou d’articles de référence ou si vous connaissez des sites web de qualité traitant du thème abordé ici, merci de compléter l’article en donnant les références utiles à sa vérifiabilité et en les liant à la section « Notes et références » (, comment ajouter mes sources ?).

Pierre Billon est un directeur artistique, parolier, compositeur et chanteur français né le à Paris, dans le 12e arrondissement.

Pierre Billon est le fils de la chanteuse Patachou, le filleul (laïc) de Georges Brassens et l’oncle du jeune comédien Pierre-Antoine Billon.

Ami d’enfance de Michel Sardou college football socks, il assure la première partie du chanteur à l’Olympia[précision nécessaire], et collabore avec lui, durant les années 1970, en tant qu’auteur (Dix ans plus tôt), et compositeur (America America, Huit jours à El Paso, Je ne suis pas mort, je dors) ; il réalise également les albums Verdun et Victoria et signe, en collaboration avec Jacques Revaux et Michel Sardou lui-même, la musique du tube de 1981 Être une femme.

Sa collaboration avec Sardou s’achève au début des années 1980. Il travaille alors pour Johnny Hallyday.

Il compose également des génériques de jeux télévisés (Le Juste Prix, Le Bigdil) et de divertissements (40° à l’ombre, C’est l’été, C’est toujours l’été).

Il travaille à de nombreuses reprises avec Éric Bouad (ancien guitariste d’Hallyday et membre des Musclés). De cette collaboration naissent les titres Tao Bi le lapin, Bienvenu tout nu, Back in Nashville, Con Edison ou encore La Bamba triste (titre rendu célèbre par Ringo Willy Cat et Bruno Guillon sur Virgin Radio, qui en firent le générique de l’émission[réf. nécessaire]).

En 2001 il s’associe avec Jean Mora (un autre compositeur) et crée une entreprise de création et de production musicale, Tatoo Music.

C’est en 1974 que, pour la première fois, Pierre Billon travaille pour le chanteur, sur l’album Rock’n’Slow, où il joue du tumba (congas).

Il écrit et/ou compose pour Hallyday plusieurs chansons entre 1977 et 1979 : La Croisière des souvenirs (en collaboration avec Long Chris), Au secours (en collaboration avec Michel Mallory), J’ai oublié de vivre (album C’est la vie, 1977), La Fin du voyage (45 tours version studio et Pavillon de Paris : Porte de Pantin version live).

À partir de 1982, la collaboration avec Hallyday devient plus importante : directeur artistique du chanteur jusqu’en 1984, Pierre Billon réalise tous les albums de cette période :

Sur la plupart de ces albums, Billon écrit de nombreux titres et contribue au renouveau musical du chanteur. Ici l’audace et l’inspiration sont de rigueur, Hallyday fait « peau neuve » et cette période, parfois sous estimée, s’inscrit comme une étape charnière de la carrière de l’artiste.

Pierre Billon écrit encore pour Hallyday : Vietnam vet soccer socks cheap, Rouler vers l’ouest (1990) et Les news (2008) ; soit trente-sept chansons jusqu’en 2014.

Pierre Billon a composé plusieurs génériques de jeux télévisés, notamment celui du Bigdil de Vincent Lagaf’ et du Millionnaire de Philippe Risoli. Il a aussi remixé le générique du Juste Prix en 2009.

Computational anatomy

Computational anatomy is a discipline within medical imaging focusing on the study of anatomical shape and form at the visible or gross anatomical 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 sensor acting as the transmission and communication medium(s).

In Computational anatomy, the diffeomorphism group is used to study different coordinate systems via coordinate transformations as generated via the Lagrangian and Eulerian velocities of flow in



R

3

{\displaystyle {\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. The kinetic energy is defined through a Sobolev smoothness norm with strictly more than two generalized, square-integrable derivatives for each component of the flow velocity, which guarantees that the flows in



R

3

{\displaystyle \mathbb {R} ^{3}}

are diffeomorphisms. It also implies that the diffeomorphic shape momentum taken pointwise satisfying the Euler-Lagrange equation for geodesics is determined by its neighbors through spatial derivatives on the velocity field. This separates the discipline from the case of incompressible fluids for which momentum is a pointwise function of velocity. Computational anatomy intersects the study of Riemannian manifolds and nonlinear global analysis, where groups of diffeomorphisms are the central focus. Emerging high-dimensional theories of shape are central to many studies in Computational anatomy, as are questions emerging from the fledgling field of shape statistics. The metric structures in Computational anatomy are related in spirit to morphometrics, with the distinction that Computational anatomy focuses on an infinite-dimensional space of coordinate systems transformed by a diffeomorphism, hence the central use of the terminology diffeomorphometry, the metric space study of coordinate systems via diffeomorphisms soccer socks cheap.

At Computational anatomy’s heart is the comparison of shape by recognizing in one shape the other. This connects it to D’Arcy Wentworth Thompson’s developments On Growth and Form which has led to scientific explanations of morphogenesis, the process by which patterns are formed in Biology. Albrecht Durer’s Four Books on Human Proportion were arguably the earliest works on Computational anatomy. The efforts of Noam Chomsky in his pioneering of Computational Linguistics inspired the original formulation of Computational anatomy as a generative model of shape and form from exemplars acted upon via transformations.

Due to the availability of dense 3D measurements via technologies such as magnetic resonance imaging (MRI), Computational anatomy has emerged as a subfield of medical imaging and bioengineering for extracting anatomical coordinate systems at the morphome scale in 3D. The spirit of this discipline shares strong overlap with areas such as computer vision and kinematics of rigid bodies, where objects are studied by analysing the groups responsible for the movement in question. Computational anatomy departs from computer vision with its focus on rigid motions, as the infinite-dimensional diffeomorphism group is central to the analysis of Biological shapes. It is a branch of the image analysis and pattern theory school at Brown University pioneered by Ulf Grenander. In Grenander’s general Metric Pattern Theory, making spaces of patterns into a metric space is one of the fundamental operations since being able to cluster and recognize anatomical configurations often requires a metric of close and far between shapes. The diffeomorphometry metric of Computational anatomy measures how far two diffeomorphic changes of coordinates are from each other, which in turn induces a metric on the shapes and images indexed to them. The models of metric pattern theory, in particular group action on the orbit of shapes and forms is a central tool to the formal definitions in Computational anatomy.

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



R

3

,

{\displaystyle {\mathbb {R} }^{3},}

points, curves surfaces and subvolumes of human anatomy. An early modern computational neuro-anatomist was David Van Essen performing some of the early physical unfoldings of the human brain based on printing of a human cortex and cutting. Jean Talairach’s publication of 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 and Fred Bookstein’s earliest developments based on Computed axial tomography and Magnetic resonance imagery. The earliest introduction of the use of flows of diffeomorphisms for transformation of coordinate systems in image analysis and medical imaging was by Christensen, Joshi, Miller, and Rabbitt.

The first formalization of Computational Anatomy as an orbit of exemplar templates under diffeomorphism group action was in the original lecture given by Grenander and Miller with that title in May 1997 at the 50th Anniversary of the Division of Applied Mathematics at Brown University, and subsequent publication. This was the basis for the strong departure from much of the previous work on advanced methods for spatial normalization and image registration which were historically built on notions of addition and basis expansion. The structure preserving transformations central to the modern field of Computational Anatomy, homeomorphisms and diffeomorphisms carry smooth submanifolds smoothly. They are generated via Lagrangian and Eulerian flows which satisfy a law of composition of functions forming the group property, but are not additive.

The original model of Computational anatomy was as the triple,



(

G

,

M

,

P

)

,

{\displaystyle ({\mathcal {G}},{\mathcal {M}},{\mathcal {P}})\ ,}

the group



g

G

{\displaystyle g\in {\mathcal {G}}}

, the orbit of shapes and forms



m

M

{\displaystyle m\in {\mathcal {M}}}

, and the probability laws



P

{\displaystyle P}

which encode the variations of the objects in the orbit. The template or collection of templates are elements in the orbit



m

t

e

m

p

M

{\displaystyle 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 organized by the Azencott school at Ecole-Normale Cachan on the “Mathematics of Shape Recognition” and the 1998 Trimestre at Institute Henri Poincaré organized by David Mumford “Questions Mathématiques en Traitement du Signal et de l’Image” which catalyzed the Hopkins-Brown-ENS Cachan groups and subsequent developments and connections of Computational anatomy to developments in global analysis.

The developments in Computational Anatomy included the establishment of the Sobelev smoothness conditions on the diffeomorphometry metric to insure existence of solutions of variational problems in the space of diffeomorphisms, the derivation of the Euler-Lagrange equations characterizing geodesics through the group and associated conservation laws, the demonstration of the metric properties of the right invariant metric, the demonstration that the Euler-Lagrange equations have a well-posed initial value problem with unique solutions for all time, and with the first results on sectional curvatures for the diffeomorphometry metric in landmarked spaces. Following the Los Alamos meeting in 2002, Joshi’s original large deformation singular Landmark solutions in Computational anatomy were connected to peaked Solitons or Peakons as solutions for the Camassa-Holm equation. Subsequently, connections were made between Computational anatomy’s Euler-Lagrange equations for momentum densities for the right-invariant metric satisfying Sobolev smoothness to Vladimir Arnold’s characterization of the Euler equation for incompressible flows as describing geodesics in the group of volume preserving diffeomorphisms. The first algorithms, generally termed LDDMM for large deformation diffeomorphic mapping for computing connections between landmarks in volumes and spherical manifolds, curves, currents and surfaces, volumes, tensors, varifolds, and time-series have followed.

These contributions of Computational anatomy to the global analysis associated to the infinite dimensional manifolds of subgroups of the diffeomorphism group is far from trivial. The original idea of doing differential geometry, curvature and geodesics on infinite dimensional manifolds goes back to Bernhard Riemann’s Habilitation (Ueber die Hypothesen, welche der Geometrie zu Grunde liegen); the key modern book laying the foundations of such ideas in global analysis are from Michor.

The applications within Medical Imaging of Computational Anatomy continued to flourish after two organized meetings at the Institute for Pure and Applied Mathematics conferences at University of California, Los Angeles. Computational anatomy has been useful in creating accurate models of the atrophy of the human brain at the morphome scale, as well as Cardiac templates, as well as in modeling biological systems. Since the late 1990s, computational anatomy has become an important part of developing emerging technologies for the field of medical imaging. Digital atlases are a fundamental part of modern Medical-school education and in neuroimaging research at the morphome scale. Atlas based methods and virtual textbooks which accommodate variations as in deformable templates are at the center of many neuro-image analysis platforms including Freesurfer, FSL, MRIStudio, SPM. Diffeomorphic registration, introduced in the 90’s, is now an important player with existing codes bases organized around ANTS, DARTEL, DEMONS, LDDMM, StationaryLDDMM 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 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

M

{\displaystyle m\in {\mathcal {M}}}

, with the group



(

G

,

)

{\displaystyle ({\mathcal {G}},\circ )}

with law of composition



{\displaystyle \circ }

; the action of the group on shapes is denoted



g

m

{\displaystyle g\cdot m}

, where the action of the group



g

m

M

,

m

M

{\displaystyle g\cdot m\in {\mathcal {M}},m\in {\mathcal {M}}}

is defined to satisfy

The orbit



M

{\displaystyle {\mathcal {M}}}

of the template becomes the space of all shapes,



M

{

m

=

g

m

t

e

m

p

,

g

G

}

{\displaystyle {\mathcal {M}}\doteq \{m=g\cdot m_{\mathrm {temp} },g\in {\mathcal {G}}\}}

, being homogenous under the action of the elements of



G

{\displaystyle {\mathcal {G}}}

.

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



R

n

{\displaystyle {\mathbb {R} }^{n}}

vectors are replaced by the finite-dimensional anatomical submanifolds (points, curves, surfaces and volumes) and images of them, and the



n

×

n

{\displaystyle 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.

The central objects are shapes or forms in Computational anatomy, one set of examples being the 0,1,2,3-dimensional submanifolds of



R

3

{\displaystyle {\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.

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

{

x

1

,

,

x

n

}

R

3

M

{\displaystyle 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

R

3

M

{\displaystyle X\subset {\mathbb {R} }^{3}\in {\mathcal {M}}}

, collections of points modeled as parametrized by a local chart or immersion



m

:

U

R

1

,

2

R

3

{\displaystyle m:U\subset {\mathbb {R} }^{1,2}\rightarrow {\mathbb {R} }^{3}}

,



m

(

u

)

,

u

U

{\displaystyle 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

M

{\displaystyle I\in {\mathcal {M}}}

, and are dense functions



I

(

x

)

,

x

X

R

1

,

2

,

3

{\displaystyle 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 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

{\displaystyle A}

as an



n

×

n

{\displaystyle n\times n}

matrix, acting on



x

R

n

{\displaystyle x\in {\mathbb {R} }^{n}}

as



n

×

1

{\displaystyle n\times 1}

vectors; the orbit in linear-algebra is the set of



n

{\displaystyle n}

-vectors given by



y

=

A

x

R

n

{\displaystyle y=A\cdot x\in {\mathbb {R} }^{n}}

, which is a group action of the matrices through the orbit of



R

n

{\displaystyle {\mathbb {R} }^{n}}

.

The central group in Computational anatomy defined on volumes in



R

3

{\displaystyle {\mathbb {R} }^{3}}

are the diffeomorphisms



G

D

i

f

f

{\displaystyle {\mathcal {G}}\doteq Diff}

which are mappings with 3-components



ϕ

(

)

=

(

ϕ

1

(

)

,

ϕ

2

(

)

,

ϕ

3

(

)

)

{\displaystyle \phi (\cdot )=(\phi _{1}(\cdot ),\phi _{2}(\cdot ),\phi _{3}(\cdot ))}

, law of composition of functions



ϕ

ϕ

(

)

ϕ

(

ϕ

(

)

)

{\displaystyle \phi \circ \phi ^{\prime }(\cdot )\doteq \phi (\phi ^{\prime }(\cdot ))}

, with inverse



ϕ

ϕ

1

(

)

=

ϕ

(

ϕ

1

(

)

)

=

i

d

{\displaystyle \phi \circ \phi ^{-1}(\cdot )=\phi (\phi ^{-1}(\cdot ))=id}

.

Most popular are scalar images,



I

(

x

)

,

x

R

3

{\displaystyle I(x),x\in {\mathbb {R} }^{3}}

, with action on the right via the inverse.

For sub-manifolds



X

R

3

M

{\displaystyle X\subset {\mathbb {R} }^{3}\in {\mathcal {M}}}

, parametrized by a chart or immersion



m

(

u

)

,

u

U

{\displaystyle m(u),u\in U}



ϕ

t

,

t

[

0

,

1

]

{\displaystyle \phi _{t},t\in [0,1]}

which satisfy the Lagrangian and Eulerian specification of the flow fieldssas first introduced in., satisfying the ordinary differential equation:



d

d

t

ϕ

t

=

v

t

ϕ

t

,

ϕ

0

=

i

d

;

{\displaystyle {\frac {d}{dt}}\phi _{t}=v_{t}\circ \phi _{t},\ \phi _{0}=id\ ;}

()

with



v

(

v

1

,

v

2

,

v

3

)

{\displaystyle v\doteq (v_{1},v_{2},v_{3})}

the vector fields on



R

3

{\displaystyle {\mathbb {R} }^{3}}

termed the Eulerian velocity of the particles at position



ϕ

{\displaystyle \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

ϕ

(

ϕ

i

x

j

)

{\displaystyle \ 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 for large deformations in image matching;



ϕ

˙

t

(

x

)

{\displaystyle {\dot {\phi }}_{t}(x)}

is the instantaneous velocity of particle



x

{\displaystyle x}

at time



t

{\displaystyle t}

.

The inverse



ϕ

t

1

,

t

[

0

,

1

]

{\displaystyle \phi _{t}^{-1},t\in [0,1]}

required for the group is defined on the Eulerian vector-field with advective inverse flow



d

d

t

ϕ

t

1

=

(

D

ϕ

t

1

)

v

t

,

ϕ

0

1

=

i

d

.

{\displaystyle {\frac {d}{dt}}\phi _{t}^{-1}=-(D\phi _{t}^{-1})v_{t},\ \phi _{0}^{-1}=id\ .}

()

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. For diffeomorphisms on



R

3

{\displaystyle {\mathbb {R} }^{3}}

, vector fields are modelled as elements of the Hilbert space



(

V

,

V

)

{\displaystyle (V,\|\cdot \|_{V})}

using the Sobolev embedding theorems so that each element



v

i

H

0

3

,

i

=

1

,

2

,

3

,

{\displaystyle v_{i}\in H_{0}^{3},i=1,2,3,}

has 3-square-integrable derivatives, thusly embedding in 1-time continuously differentiable functions.

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



D

i

f

f

V

{

φ

=

ϕ

1

:

ϕ

˙

t

=

v

t

ϕ

t

,

ϕ

0

=

i

d

,

0

1

v

t

V

d

t

<

}

.

{\displaystyle 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 \}\ .}

()

The modelling approach used in Computational anatomy enforces a continuous differentiability condition on the vector fields by modelling the space of vector fields



(

V

,

V

)

{\displaystyle (V,\|\cdot \|_{V})}

as a reproducing kernel Hilbert space (RKHS), with the norm defined by a 1-1, differential operator



A

:

V

V

{\displaystyle A:V\rightarrow V^{*}}

, Green’s inverse



K

=

A

1

{\displaystyle K=A^{-1}}

. The norm of the Hilbert space is induced by the differential operator. For



σ

(

v

)

A

v

V

{\displaystyle \sigma (v)\doteq Av\in V^{*}}

a generalized function or distribution, define the linear form as



(

σ

|

w

)

R

3

i

w

i

(

x

)

σ

i

(

d

x

)

{\displaystyle (\sigma |w)\doteq \int _{{\mathbb {R} }^{3}}\sum _{i}w_{i}(x)\sigma _{i}(dx)}

. This determines the norm on



(

V

,

V

)

{\displaystyle (V,\|\cdot \|_{V})}

according to

Since



A

{\displaystyle A}

is a differential operator, finiteness of the norm-square



(

A

v

|

v

)

<

{\displaystyle (Av|v)<\infty }

includes derivatives from the differential operator implying smoothness of the vector fields.The Sobolev embedding theorem arguments were made in demonstrating that 1-continuous derivative is required for smooth flows.

For proper choice of



A

{\displaystyle A}

then



(

V

,

V

)

{\displaystyle (V,\|\cdot \|_{V})}

is an RKHS with the operator



K

=

A

1

:

V

V

{\displaystyle 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

(

,

)

{\displaystyle k(\cdot ,\cdot )}

is continuously differentiable in both variables implying

When



σ

μ

d

x

{\displaystyle \sigma \doteq \mu dx}

, a vector density,



(

σ

|

v

)

v

μ

d

x

{\displaystyle (\sigma |v)\doteq \int v\cdot \mu dx}

.

The study of metrics on groups of diffeomorphisms and the study of metrics between manifolds and surfaces has been an area of significant investigation. 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



ϕ

m

M

,

ϕ

D

i

f

f

V

,

m

M

{\displaystyle \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.

Define the distance on the group of diffeomorphisms

:



d

D

i

f

f

V

(

ψ

,

φ

)

=

inf

v

t

(

0

1

(

A

v

t

|

v

t

)

d

t

:

ϕ

0

=

ψ

,

ϕ

1

=

φ

,

ϕ

˙

t

=

v

t

ϕ

t

)

1

/

2

;

{\displaystyle 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}\ ;}

()

this is the right-invariant metric of diffeomorphometry, invariant to reparameterization of space since for all



ϕ

D

i

f

f

V

{\displaystyle \phi \in Diff_{V}\,\,\,}

,

The distance on shapes and forms,



d

M

:

M

×

M

R

+

{\displaystyle d_{\mathcal {M}}:{\mathcal {M}}\times {\mathcal {M}}\rightarrow \mathbb {R} ^{+}}

,

:



d

M

(

m

,

n

)

=

inf

ϕ

Diff

V

:

ϕ

m

=

n

d

Diff

V

(

i

d

,

ϕ

)

;

{\displaystyle d_{\mathcal {M}}(m,n)=\inf _{\phi \in \operatorname {Diff} _{V}:\phi \cdot m=n}d_{\operatorname {Diff} _{V}}(id,\phi )\ ;}

()

the images are denoted with the orbit as



I

I

{\displaystyle I\in {\mathcal {I}}}

and metric



,

d

I

{\displaystyle ,d_{\mathcal {I}}}

.

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



ϕ

,

ϕ

˙

{\displaystyle \phi ,{\dot {\phi }}}

, the two related via the Eulerian velocity



v

ϕ

˙

ϕ

1

{\displaystyle 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

(

ϕ

)

0

1

L

(

ϕ

t

,

ϕ

˙

t

)

d

t

;

{\displaystyle J(\phi )\doteq \int _{0}^{1}L(\phi _{t},{\dot {\phi }}_{t})dt\ ;}

()

the Lagrangian is given by the kinetic energy:



L

(

ϕ

t

,

ϕ

˙

t

)

1

2

(

A

(

ϕ

˙

t

ϕ

t

1

)

|

ϕ

˙

t

ϕ

t

1

)

=

1

2

(

A

v

t

|

v

t

)

.

{\displaystyle 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})\ .}

()

In computational anatomy,



A

v

{\displaystyle Av}

was first called the Eulerian or diffeomorphic shape momentum since when integrated against Eulerian velocity



v

{\displaystyle v}

gives energy density, and since there is a conservation of diffeomorphic shape momentum which holds. The operator



A

{\displaystyle A}

is the generalized moment of inertia or inertial operator.

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



a

d

v

:

w

V

V

which involves the Jacobian given by



a

d

v

:

V

V

,

then the first order variation gives the Eulerian shape momentum



A

v

V

{\displaystyle Av\in V^{*}}

satisfying the generalized equation:



d

d

t

A

v

t

+

a

d

v

t

(

A

v

t

)

=

0

,

t

[

0

,

1

]

;

{\displaystyle {\frac {d}{dt}}Av_{t}+ad_{v_{t}}^{*}(Av_{t})=0\ ,\ t\in [0,1]\ ;}

()

meaning for all smooth



w

V

,

{\displaystyle w\in V,}

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



R

3

{\displaystyle {\mathbb {R} }^{3}}

which are dimension



2

{\displaystyle \leq 2}

in Lebesgue measure. In such cases, the energy is still well defined



(

A

v

t

v

t

)

{\displaystyle (Av_{t}\mid v_{t})}

since although



A

v

t

{\displaystyle Av_{t}}

is a generalized function, the vector fields are smooth and the Eulerian momentum is understood via its action on smooth functions. The perfect illustration of this is even when it is a superposition of delta-diracs, the velocity of the coordinates in the entire volume move smoothly.The Euler-Lagrange equation (EL-General) on diffeomorphisms for generalized functions



A

v

V

{\displaystyle Av\in V^{*}}

was derived in. 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

=

i

d

e

n

t

i

t

y

{\displaystyle A=identity}

for incompressible, divergence free, fluids.

For the momentum density case



(

A

v

t

w

)

=

X

μ

t

w

d

x

{\displaystyle (Av_{t}\mid w)=\int _{X}\mu _{t}\cdot w\,dx}

, then Euler–Lagrange equation has a classical solution:

()

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

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. Solving the geodesic from the initial condition



v

0

{\displaystyle v_{0}}

is termed the Riemannian-exponential, a mapping



E

x

p

i

d

(

)

:

V

D

i

f

f

V

{\displaystyle Exp_{\rm {id}}(\cdot ):V\to Diff_{V}}

at identity to the group.

The Riemannian exponential satisfies



E

x

p

i

d

(

v

0

)

=

ϕ

1

{\displaystyle Exp_{id}(v_{0})=\phi _{1}}

for initial condition



ϕ

˙

0

=

v

0

{\displaystyle {\dot {\phi }}_{0}=v_{0}}

, vector field dynamics



ϕ

˙

t

=

v

t

ϕ

t

,

t

[

0

,

1

]

{\displaystyle {\dot {\phi }}_{t}=v_{t}\circ \phi _{t},t\in [0,1]}

,

Computing the flow



v

0

{\displaystyle v_{0}}

onto coordinates Riemannian logarithm, mapping



L

o

g

i

d

(

)

:

D

i

f

f

V

V

{\displaystyle Log_{\rm {id}}(\cdot ):Diff_{V}\to V}

at identity from



φ

{\displaystyle \varphi }

to vector field



v

0

V

{\displaystyle v_{0}\in V}

;



L

o

g

i

d

(

φ

)

=

v

0

initial condition of EL geodesic

ϕ

˙

0

=

v

0

,

ϕ

0

=

i

d

,

ϕ

1

=

φ

.

{\displaystyle 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



ϕ

=

E

x

p

φ

(

v

0

φ

)

E

x

p

i

d

(

v

0

)

φ

{\displaystyle \phi =Exp_{\varphi }(v_{0}\circ \varphi )\doteq Exp_{id}(v_{0})\circ \varphi }

;



L

o

g

φ

(

ϕ

)

L

o

g

i

d

(

ϕ

φ

1

)

φ

{\displaystyle 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).

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

ϕ

t

Diff

V

{\displaystyle t\mapsto \phi _{t}\in \operatorname {Diff} _{V}}

and the control the vector field



t

v

t

V

{\displaystyle t\mapsto v_{t}\in V}

related via



ϕ

˙

t

=

v

t

ϕ

t

,

ϕ

0

=

i

d

.

{\displaystyle {\dot {\phi }}_{t}=v_{t}\cdot \phi _{t},\phi _{0}=id.}

The Hamiltonian view reparameterizes the momentum distribution



A

v

V

{\displaystyle Av\in V^{*}}

in terms of the conjugate momentum or canonical momentum, introduced as a Lagrange multiplier



p

:

ϕ

˙

(

p

ϕ

˙

)

{\displaystyle p:{\dot {\phi }}\mapsto (p\mid {\dot {\phi }})}

constraining the Lagrangian velocity



ϕ

˙

t

=

v

t

ϕ

t

{\displaystyle {\dot {\phi }}_{t}=v_{t}\circ \phi _{t}}

.accordingly:

This function is the extended Hamiltonian. The Pontryagin maximum principle gives the optimizing vector field which determines the geodesic flow satisfying



ϕ

˙

t

=

v

t

ϕ

t

,

ϕ

0

=

i

d

,

{\displaystyle {\dot {\phi }}_{t}=v_{t}\circ \phi _{t},\phi _{0}=id,}

as well as the reduced Hamiltonian

The Lagrange multiplier in its action as a linear form has its own inner product of the canonical momentum acting on the velocity of the flow which is dependent on the shape, e.g. for landmarks a sum, for surfaces a surface integral, and. for volumes it is a volume integral with respect to



d

x

{\displaystyle dx}

on



R

3

{\displaystyle {\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

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



{

ϕ

˙

t

=

H

(

ϕ

t

,

p

t

)

p

p

˙

t

=

H

(

ϕ

t

,

p

t

)

ϕ

{\displaystyle {\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}}}

()

Whereas the vector fields are extended across the entire background space of



R

3

{\displaystyle {\mathbb {R} }^{3}}

, the geodesic flows associated to the submanifolds has Eulerian shape momentum which evolves as a generalized function



A

v

t

V

{\displaystyle Av_{t}\in V^{*}}

concentrated to the submanifolds. For landmarks 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.

The geodesics connecting coordinate systems satisfying EL-General have stationarity of the Lagrangian. The Hamiltonian is given by the extremum along the path



t

[

0

,

1

]

{\displaystyle t\in [0,1]}

,



H

(

ϕ

,

p

)

=

max

v

H

(

ϕ

,

p

,

v

)

{\displaystyle 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

(

ϕ

0

,

p

0

,

v

)

{\displaystyle v_{0}=\arg \max _{v}H(\phi _{0},p_{0},v)}

, then along the geodesic



H

(

ϕ

t

,

p

t

)

=

H

(

ϕ

0

,

p

0

)

=

1

2

(

p

0

v

0

)

=

1

2

(

A

v

0

v

0

)

=

1

2

(

A

v

t

v

t

)

{\displaystyle 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})}

()

The stationarity of the Hamiltonian demonstrates the interpretation of the Lagrange multiplier as momentum; integrated against velocity



ϕ

˙

{\displaystyle {\dot {\phi }}}

gives energy density. The canonical momentum has many names. In optimal control, the flows



ϕ

{\displaystyle \phi }

is interpreted as the state, and



p

{\displaystyle p}

is interpreted as conjugate state, or conjugate momentum. The geodesi of EL implies specification of the vector fields



v

0

{\displaystyle v_{0}}

or Eulerian momentum



A

v

0

{\displaystyle Av_{0}}

at



t

=

0

{\displaystyle t=0}

, or specification of canonical momentum



p

0

{\displaystyle p_{0}}

determines the flow.

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

{\displaystyle t=0}

the geodesic has vector field



v

0

=

K

p

0

{\displaystyle v_{0}=Kp_{0}}

determined by the conjugate momentum and the Green’s kernel of the inertial operator defining the Eulerian momentum



K

=

A

1

{\displaystyle K=A^{-1}}

. The metric distance between coordinate systems connected via the geodesic determined by the induced distance between identity and group element:

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

For Landmarks, the Hamiltonian momentum is defined on the indices,



p

(

i

)

,

i

=

1

,

,

n

{\displaystyle p(i),i=1,\dots ,n}

with the inner product given by



(

p

t

|

v

t

ϕ

t

)

i

p

t

(

i

)

v

t

ϕ

t

(

x

i

)

{\displaystyle (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

(

ϕ

t

,

p

t

)

=

1

2

j

i

p

t

(

i

)

K

(

ϕ

t

(

x

i

)

,

ϕ

t

(

x

j

)

)

p

t

(

j

)

{\displaystyle 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

For surfaces, the Hamiltonian momentum is defined across the surface with the inner product



(

p

t

v

t

ϕ

t

)

U

p

t

(

u

)

v

t

ϕ

t

(

m

(

u

)

)

d

u

{\displaystyle (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

(

ϕ

t

,

p

t

)

=

1

2

U

U

p

t

(

u

)

K

(

ϕ

t

(

m

(

u

)

)

,

ϕ

t

(

m

(

v

)

)

)

p

t

(

v

)

d

u

d

v

{\displaystyle 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

For volumes the Hamiltonian momentum is



(

p

t

v

t

ϕ

t

)

R

3

p

t

(

x

)

v

t

ϕ

t

(

x

)

d

x

{\displaystyle (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

(

ϕ

t

,

p

t

)

=

1

2

R

3

R

3

p

t

(

x

)

K

(

ϕ

t

(

x

)

,

ϕ

t

(

y

)

)

p

t

(

y

)

d

x

d

y

{\displaystyle 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



{

v

t

=

X

K

(

,

ϕ

t

(

x

)

)

p

t

(

x

)

d

x

,

p

˙

t

(

x

)

=

(

D

v

t

)

|

ϕ

t

(

x

)

T

p

t

(

x

)

,

x

R

3

{\displaystyle {\begin{cases}v_{t}=\textstyle \int _{X}\displaystyle K(\cdot ,\phi _{t}(x))p_{t}(x)\,dx\ ,\\{\dot {p}}_{t}(x)=-(Dv_{t})_{|_{\phi _{t}(x)}}^{T}p_{t}(x),x\in {\mathbb {R} }^{3}\end{cases}}}

Given the least-action there is a natural definition of momentum associated to generalized coordinates; the quantity acting against velocity gives energy. The field has studied two forms, the momentum associated to the Eulerian vector field termed Eulerian diffeomorphic shape momentum, and the momentum associated to the initial coordinates or canonical coordinates termed canonical diffeomorphic shape momentum. Each has a conservation law.The conservation of momentum goes hand in hand with the EL-General. In Computational anatomy,



A

v

{\displaystyle Av}

is the Eulerian Momentum since when integrated against Eulerian velocity



v

{\displaystyle v}

gives energy density; operator



A

{\displaystyle A}

the generalized moment of inertia or inertial operator which acting on the Eulerian velocity gives momentum which is conserved along the geodesic:



Eulerian

d

d

t

(

A

v

t

|

(

(

D

ϕ

t

)

w

)

ϕ

t

1

)

=

0

,

t

[

0

,

1

]

.

Canonical

d

d

t

(

p

t

|

(

D

ϕ

t

)

w

)

=

0

,

t

[

0

,

1

]

for all

w

V

.

{\displaystyle {\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}}}

()

Conservation of Eulerian shape momentum was shown in and follows from EL-General; conservation of canonical momentum was shown in



w

t

=

(

(

D

ϕ

t

)

w

)

ϕ

t

1

{\displaystyle w_{t}=((D\phi _{t})w)\circ \phi _{t}^{-1}}

,



d

d

t

w

t

=

(

D

v

t

)

w

t

(

D

w

t

)

v

t

{\displaystyle {\frac {d}{dt}}w_{t}=(Dv_{t})w_{t}-(Dw_{t})v_{t}}

implying



d

d

t

(

A

v

t

|

(

(

D

ϕ

t

)

w

)

ϕ

t

1

)

=

(

d

d

t

A

v

t

|

(

(

D

ϕ

t

)

w

)

ϕ

t

1

)

+

(

A

v

t

|

d

d

t

(

(

D

ϕ

t

)

w

)

ϕ

t

1

)

=

(

d

d

t

A

v

t

|

w

t

)

+

(

A

v

t

|

(

D

v

t

)

w

t

(

D

w

t

)

v

t

)

=

0.

{\displaystyle {\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}|(Dv_{t})w_{t}-(Dw_{t})v_{t})=0.}

The proof on Canonical momentum is shown from



p

˙

t

=

(

D

v

t

)

|

ϕ

t

T

p

t

{\displaystyle {\dot {p}}_{t}=-(Dv_{t})_{|_{\phi _{t}}}^{T}p_{t}}

:

Construction of diffeomorphic correspondences between shapes calculates the initial vector field coordinates



v

0

V

{\displaystyle v_{0}\in V}

and associated weights on the Greens kernels



p

0

{\displaystyle 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 and for dense image matchings. curves, surfaces, dense vector and tensor imagery, and varifolds removing orientation. LDDMM calculates geodesic flows of the EL-General onto target coordinates, adding to the action integral



1

2

0

1

(

A

v

t

|

v

t

)

d

t

{\displaystyle {\frac {1}{2}}\int _{0}^{1}(Av_{t}|v_{t})dt}

an endpoint matching condition



E

:

ϕ

1

R

+

{\displaystyle 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. The solution of the variational problem satisfies the EL-General for



t

[

0

,

1

)

{\displaystyle t\in [0,1)}

with boundary condition.



min

ϕ

:

v

=

ϕ

˙

ϕ

1

,

ϕ

0

=

i

d

C

(

ϕ

)

1

2

0

1

(

A

v

t

|

v

t

)

d

t

+

E

(

ϕ

1

)

{\displaystyle {\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})}



{

Euler Conservation

d

d

t

A

v

t

+

a

d

v

t

(

A

v

t

)

=

0

,

t

[

0

,

1

)

,

Boundary Condition

ϕ

0

=

i

d

,

A

v

1

=

E

(

ϕ

)

ϕ

|

ϕ

=

ϕ

1

.

{\displaystyle {\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

{\displaystyle t=1}

to the rest of the path



t

[

0

,

1

)

{\displaystyle t\in [0,1)}

.The inexact matching problem with the endpoint matching term



E

(

ϕ

1

)

{\displaystyle 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

{\displaystyle v_{0}}

.

The running cost is reduced to the initial cost determined by



v

0

=

K

p

0

{\displaystyle v_{0}=Kp_{0}}

of Kernel-Surf.-Land.-Geodesics.

The matching problem explicitly indexed to initial condition



v

0

{\displaystyle v_{0}}

is called shooting, which can also be reparamerized via the conjugate momentum



p

0

{\displaystyle p_{0}}

.

Dense image matching has a long history now with the earliest efforts exploiting a small deformation framework. Large deformations began in the early 90’s, with the first existence to solutions to the variational problem for flows of diffeomorphisms for dense image matching established in. 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. Another solution for dense image matching reparameterizes the optimization problem in terms of the state



q

t

I

ϕ

t

1

,

q

0

=

I

{\displaystyle 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.

For Beg’s LDDMM, denote the Image



I

(

x

)

,

x

X

{\displaystyle I(x),x\in X}

with group action



ϕ

I

I

ϕ

1

{\displaystyle \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



ϕ

t

,

t

[

0

,

1

]

{\displaystyle \phi _{t},t\in [0,1]}

, with the dynamics relating the control



v

t

,

t

[

0

,

1

]

{\displaystyle v_{t},t\in [0,1]}

to the state given by



ϕ

˙

=

v

ϕ

{\displaystyle {\dot {\phi }}=v\circ \phi }

. The endpoint matching condition



E

(

ϕ

1

)

I

ϕ

1

1

I

2

{\displaystyle E(\phi _{1})\doteq \|I\circ \phi _{1}^{-1}-I^{\prime }\|^{2}}

gives the variational problem



min

v

:

ϕ

˙

=

v

ϕ

C

(

v

)

1

2

0

1

(

A

v

t

|

v

t

)

d

t

+

1

2

R

3

|

I

ϕ

1

1

(

x

)

I

(

x

)

|

2

d

x

{\displaystyle {\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)|^{2}dx\end{matrix}}}

()

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.

Denote the Image



I

(

x

)

,

x

X

{\displaystyle I(x),x\in X}



q

t

I

ϕ

t

1

{\displaystyle q_{t}\doteq I\circ \phi _{t}^{-1}}

and the dynamics related state and control given by the advective term



q

˙

t

=

q

t

v

t

{\displaystyle {\dot {q}}_{t}=-\nabla q_{t}\cdot v_{t}}

. The endpoint



E

(

q

1

)

q

1

I

2

{\displaystyle E(q_{1})\doteq \|q_{1}-I^{\prime }\|^{2}}

gives the variational problem



min

q

:

q

˙

=

v

q

C

(

v

)

1

2

0

1

(

A

v

t

|

v

t

)

d

t

+

1

2

R

3

|

q

1

(

x

)

I

(

x

)

|

2

d

x

{\displaystyle {\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)|^{2}dx\end{matrix}}}

()

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

Dense LDDMM tensor matching solves the variational problem matching between coordinate system based on the principle eigenvectors of the diffusion tensor MRI image (DTI) denoted



M

(

x

)

,

x

R

3

{\displaystyle M(x),x\in {\mathbb {R} }^{3}}

consisting of the



3

×

3

{\displaystyle 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

t

r

a

c

e

A

T

A

{\displaystyle \|A\|_{F}^{2}\doteq traceA^{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.

Denote the



3

×

3

{\displaystyle 3\times 3}

tensor image



M

(

x

)

,

x

R

3

{\displaystyle M(x),x\in {\mathbb {R} }^{3}}

with eigen-elements



{

λ

i

(

x

)

,

e

i

(

x

)

,

i

=

1

,

2

,

3

}

{\displaystyle \{\lambda _{i}(x),e_{i}(x),i=1,2,3\}}

, eigenvalues



λ

1

λ

2

λ

3

{\displaystyle \lambda _{1}\geq \lambda _{2}\geq \lambda _{3}}

, and



e

1

{\displaystyle e_{1}}

,



e

2

{\displaystyle e_{2}}

,



e

3

{\displaystyle e_{3}}

eigenvectors. The group action becomes



φ

M

=

(

λ

1

e

^

1

e

^

1

T

+

λ

2

e

^

2

e

^

2

T

+

λ

3

e

^

3

e

^

3

T

)

φ

1

,

{\displaystyle \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



min

v

:

v

=

ϕ

˙

ϕ

1

1

2

0

1

(

A

v

t

|

v

t

)

d

t

+

α

R

3

ϕ

1

M

(

x

)

M

(

x

)

F

2

d

x

{\displaystyle \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}

()

Denote the image



I

(

x

)

,

x

R

3

{\displaystyle I(x),x\in {\mathbb {R} }^{3}}

taken as a unit vector field defined by the first eigenvector. The group action becomes

with endpoint

The variational problem matching onto vector image



I

(

x

)

,

x

R

3

{\displaystyle I^{\prime }(x),x\in {\mathbb {R} }^{3}}

becomes

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 and in 3D medical imaging in. 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 for Magnetic Resonance image models, with many subsequent developments extending the metamorphosis framework.

For image matching the image metamorphosis framework enlarges the action so that



t

(

ϕ

t

,

I

t

)

{\displaystyle t\mapsto (\phi _{t},I_{t})}

with action



ϕ

t

I

t

I

t

ϕ

t

1

{\displaystyle \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:

Transforming coordinate systems based on Landmark point or fiducial marker features dates back to Bookstein’s early work on small deformation spline methods 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. 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

{\displaystyle X}

in



R

3

{\displaystyle {\mathbb {R} }^{3}}

,



d

=

1

,

2

,

3

{\displaystyle d=1,2,3}

can be parameterized in local charts



m

:

u

U

R

0

,

1

,

2

,

3

R

3

{\displaystyle 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 and varifolds which have been used extensively for curve and surface matching.

Denoted the landmarked shape



X

{

x

1

,

,

x

n

}

R

3

{\displaystyle X\doteq \{x_{1},\dots ,x_{n}\}\subset {\mathbb {R} }^{3}}

with endpoint



E

(

ϕ

1

)

i

ϕ

1

(

x

i

)

x

i

2

{\displaystyle E(\phi _{1})\doteq \textstyle \sum _{i}\displaystyle \|\phi _{1}(x_{i})-x_{i}^{\prime }\|^{2}}

, the variational problem becomes

()

The geodesic Eulerian momentum is a generalized function



A

v

t

V

,

t

[

0

,

1

]

{\displaystyle \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:

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

Glaunes and co-workers first introduced diffeomorphic matching of pointsets in the general setting of matching distributions. As opposed to landmarks, this includes in particular the situation of weighted point clouds with no predefined correspondences and possibly different cardinalities. The template and target discrete point clouds are represented as two weighted sums of Diracs



μ

m

=

i

=

1

n

ρ

i

δ

x

i

{\displaystyle \mu _{m}=\sum _{i=1}^{n}\rho _{i}\delta _{x_{i}}}

and



μ

m

=

i

=

1

n

ρ

i

δ

x

i

{\displaystyle \mu _{m^{\prime }}=\sum _{i=1}^{n^{\prime }}\rho _{i}^{\prime }\delta _{x_{i}^{\prime }}}

living in the space of signed measures of



R

3

{\displaystyle \mathbb {R} ^{3}}

. The space is equipped with a Hilbert metric obtained from a real positive kernel



k

(

x

,

y

)

{\displaystyle k(x,y)}

on



R

3

{\displaystyle \mathbb {R} ^{3}}

, giving the following norm:

The matching problem between a template and target point cloud may be then formulated using this kernel metric for the endpoint matching term:

where



μ

ϕ

1

m

=

i

=

1

n

ρ

i

δ

ϕ

1

(

x

i

)

{\displaystyle \mu _{\phi _{1}\cdot m}=\sum _{i=1}^{n}\rho _{i}\delta _{\phi _{1}(x_{i})}}

is the distribution transported by the deformation.

In the one dimensional case, a curve in 3D can be represented by an embedding



m

:

u

[

0

,

1

]

R

3

{\displaystyle m:u\in [0,1]\rightarrow {\mathbb {R} }^{3}}

, and the group action of Diff becomes



ϕ

m

=

ϕ

m

{\displaystyle \phi \cdot m=\phi \circ m}

. However, the correspondence between curves and embeddings is not one to one as the any reparametrization



m

γ

{\displaystyle m\circ \gamma }

, for



γ

{\displaystyle \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.

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

{\displaystyle m}

and



m

{\displaystyle m^{\prime }}

writes eventually as the variational problem

with the endpoint term



E

(

ϕ

1

)

=

C

ϕ

1

m

C

m

c

u

r

2

/

2

{\displaystyle E(\phi _{1})=\|{\mathcal {C}}_{\phi _{1}\cdot m}-{\mathcal {C}}_{m^{\prime }}\|_{\mathrm {cur} }^{2}/2}

is obtained from the norm

the derivative



m

(

u

)

{\displaystyle \partial m(u)}

being the tangent vector to the curve and



K

C

{\displaystyle K_{\mathcal {C}}}

a given matrix kernel of



R

3

{\displaystyle {\mathbb {R} }^{3}}

. Such expressions are invariant to any positive reparametrizations of



m

{\displaystyle m}

and



m

{\displaystyle m’}

, and thus still depend on the orientation of the two curves.

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



R

3

{\displaystyle {\mathbb {R} }^{3}}

and the Grassmannian of all straight lines in



R

3

{\displaystyle {\mathbb {R} }^{3}}

. The matching problem between two curves then consists in replacing the endpoint matching term by



E

(

ϕ

1

)

=

V

ϕ

1

m

V

m

c

u

r

2

/

2

{\displaystyle E(\phi _{1})=\|{\mathcal {V}}_{\phi _{1}\cdot m}-{\mathcal {V}}_{m^{\prime }}\|_{cur}^{2}/2}

with varifold norms of the form:

where



[

m

(

u

)

]

{\displaystyle [\partial m(u)]}

is the non-oriented line directed by tangent vector



m

(

u

)

{\displaystyle \partial m(u)}

and



k

R

3

,

k

G

r

{\displaystyle k_{\mathbb {R} ^{3}},k_{\mathbf {Gr} }}

two scalar kernels respectively on



R

3

{\displaystyle \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 share many similarities with the case of curves. Surfaces in



R

3

{\displaystyle {\mathbb {R} }^{3}}

are parametrized in local charts by embeddings



m

:

u

U

R

2

R

3

{\displaystyle m:u\in U\subset {\mathbb {R} }^{2}\rightarrow {\mathbb {R} }^{3}}

, with all reparametrizations



m

γ

{\displaystyle m\circ \gamma }

with



γ

{\displaystyle \gamma }

a diffeomorphism of U being equivalent geometrically. Currents and varifolds can be also used to formalize surface matching.

Oriented surfaces can be represented as 2-currents which are dual to differential 2-forms. In



R

3

{\displaystyle {\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:

with the endpoint term



E

(

ϕ

1

)

=

C

ϕ

1

m

C

m

c

u

r

2

/

2

{\displaystyle E(\phi _{1})=\|{\mathcal {C}}_{\phi _{1}\cdot m}-{\mathcal {C}}_{m^{\prime }}\|_{\mathrm {cur} }^{2}/2}

given through the norm

with



n

=

u

1

m

u

2

m

{\displaystyle {\vec {n}}=\partial _{u_{1}}m\wedge \partial _{u_{2}}m}

the normal vector to the surface parametrized by



m

{\displaystyle m}

.

For non-orientable or non-oriented surfaces, the varifold framework is often more adequate. Identifying the parametric surface



m

{\displaystyle m}

with a varifold



V

m

{\displaystyle {\mathcal {V}}_{m}}

in the space of measures on the product of



R

3

{\displaystyle {\mathbb {R} }^{3}}

and the Grassmannian, one simply replaces the previous current metric



C

m

c

u

r

2

{\displaystyle \|{\mathcal {C}}_{m}\|_{\mathrm {cur} }^{2}}

by:

where



[

n

(

u

)

]

{\displaystyle [{\vec {n}}(u)]}

is the (non-oriented) line directed by the normal vector to the surface.

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 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

<

t

K

=

1

{\displaystyle 0<t_{1}<\dots t_{K}=1}

. The flow optimizes at the series of costs



E

(

t

k

)

,

k

=

1

,

,

K

{\displaystyle E(t_{k}),k=1,\dots ,K}

giving optimization problems of the form

There have been at least three solutions offered thus far, piecewise geodesic, principal geodesic and splines.

The random orbit model of Computational Anatomy first appeared in 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.

Depicted in the figure is a depiction of the random orbits around each exemplar,



m

0

M

{\displaystyle m_{0}\in {\mathcal {M}}}

, generated by randomizing the flow by generating the initial tangent space vector field at the identity



v

0

V

{\displaystyle v_{0}\in V}

, and then generating random object



n

E

x

p

i

d

(

v

0

)

m

0

M

{\displaystyle n\doteq Exp_{id}(v_{0})\cdot m_{0}\in {\mathcal {M}}}

.

The random orbit model induces the prior on shapes and images



I

I

{\displaystyle I\in {\mathcal {I}}}

conditioned on a particular atlas



I

a

I

{\displaystyle I_{a}\in {\mathcal {I}}}

. For this the generative model generates the mean field



I

{\displaystyle I}

as a random change in coordinates of the template according to



I

ϕ

I

a

{\displaystyle I\doteq \phi \cdot I_{a}}

, where the diffeomorphic change in coordinates is generated randomly via the geodesic flows. The prior on random transformations



π

D

i

f

f

(

d

ϕ

)

{\displaystyle \pi _{Diff}(d\phi )}

on



D

i

f

f

V

{\displaystyle Diff_{V}}

is induced by the flow



E

x

p

i

d

(

v

)

{\displaystyle Exp_{id}(v)}

, with



v

V

{\displaystyle v\in V}

constructed as a Gaussian random field prior



π

V

(

d

v

)

{\displaystyle \pi _{V}(dv)}

. The density on the random observables at the output of the sensor



I

D

I

D

{\displaystyle I^{D}\in {\mathcal {I}}^{D}}

are given by



p

(

I

D

|

I

a

)

=

V

p

(

I

D

|

E

x

p

i

d

(

v

)

I

a

)

π

V

(

d

v

)

.

{\displaystyle 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

{\displaystyle v_{0}}

supported over the submanifolds.

The central statistical model of Computational Anatomy in the context of medical imaging has been the source-channel model of Shannon theory; the source is the deformable template of images



I

I

{\displaystyle I\in {\mathcal {I}}}

, the channel outputs are the imaging sensors with observables



I

D

I

D

{\displaystyle 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,



π

I

(

)

{\displaystyle \pi _{\mathcal {I}}(\cdot )}

on



I

I

{\displaystyle I\in {\mathcal {I}}}

, and the conditional density on the observable



p

(

|

I

)

on

I

D

I

D

{\displaystyle p(\cdot |I)\ {\text{on}}\ I^{D}\in {\mathcal {I}}^{\mathcal {D}}}

conditioned on



I

I

{\displaystyle I\in {\mathcal {I}}}

.

For image action



I

(

g

)

g

I

t

e

m

p

,

g

G

{\displaystyle I(g)\doteq g\cdot I_{\mathrm {temp} },g\in {\mathcal {G}}}

, then the prior on the group



π

G

(

)

{\displaystyle \pi _{\mathcal {G}}(\cdot )}

induces the prior on images



π

I

(

)

{\displaystyle \pi _{\mathcal {I}}(\cdot )}

, written as densities the log-posterior takes the form

Maximum a posteriori estimation (MAP) estimation is central to modern statistical theory. Parameters of interest



θ

Θ

{\displaystyle \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

{\displaystyle I^{D}}

, MAP estimation maximizes the posterior:

This requires computation of the conditional probabilities



p

(

θ

I

D

)

=

p

(

I

D

,

θ

)

p

(

I

D

)

{\displaystyle 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

A

}

{\displaystyle \{I_{a},a\in {\mathcal {A}}\}}

. The model on images in the orbit take the form of a multi-modal mixture distribution

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

()

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,



I

=

a

A

Diff

V

I

a

{\displaystyle {\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

,

{\displaystyle (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

X

R

3

{\displaystyle 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 is given measurement



I

D

{\displaystyle I^{D}}

with mean field and parcellation



(

I

,

W

)

{\displaystyle (I,W)}

, the anatomical labelling



θ

W

{\displaystyle \theta \doteq W}

. mustg be estimated for the measured MRI image. The mean-field of the observable



I

D

{\displaystyle I^{D}}

image is modelled as a random deformation from one of the templates



I

φ

I

a

{\displaystyle I\doteq \varphi \cdot I_{a}}

, which is also randomly selected,



A

=

a

{\displaystyle A=a}

,. The optimal diffeomorphism



φ

G

{\displaystyle \varphi \in {\mathcal {G}}}

is hidden and acts on the background space of coordinates of the randomly selected template image



I

a

{\displaystyle I_{a}}

. Given a single atlas



a

{\displaystyle a}

, the likelihood model for inference is determined by the joint probability



p

(

I

D

,

W

A

=

a

)

{\displaystyle 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

{\displaystyle W_{a}}

is the maximizer



max

W

log

p

(

W

I

D

)

{\displaystyle \max _{W}\log p(W\mid I^{D})}

given



I

D

{\displaystyle I^{D}}

, which involves the mixture over all atlases.

The quantity



p

(

I

D

,

W

)

{\displaystyle p(I^{D},W)}

is computed via a fusion of likelihoods from multiple deformable atlases, with



π

A

(

a

)

{\displaystyle \pi _{A}(a)}

being the prior probability that the observed image evolves from the specific template image



I

a

{\displaystyle I_{a}}

.

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

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 in that the central group of Kendall’s theories are the finite-dimensional Lie groups, whereas the theories of shape in Computational Anatomy have focused on the diffeomorphism group, which to first order via the Jacobian can be thought of as a field–thus infinite dimensional–of low-dimensional Lie groups of scale and rotations.

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

M

{\displaystyle m\in {\mathcal {M}}}

is induced via the reduction to the vector fields



v

0

V

{\displaystyle 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



E

x

p

i

d

(

v

0

)

m

M

{\displaystyle 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



A

v

0

{\displaystyle Av_{0}}

are in a Hilbert space the natural model is one of a Gaussian random field, so that given test function



w

V

{\displaystyle 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 sub-cortical brain structures are depicted in a two-dimensional coordinate system based on inner products of their initial vector fields that generate them from the template is shown in a 2-dimensional span of the Hilbert space.

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, and statistical approaches based on the expectation-maximization algorithm and the Bayes Random orbit models of Computational anatomy. Shown in the accompanying figure is a subcortical template reconstruction from the population of MRI subjects.

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

ANTS

DARTEL Voxel-based morphometry(VBM)

DEMONS

LDDMM Large Deformation Diffeomorphic Metric Mapping

StationaryLDDMM