Computational anatomy is an interdisciplinary field of
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical methods for modelling and simulation of biological structures.
The field is broadly defined and includes foundations in
anatomy
Anatomy () is the branch of biology concerned with the study of the structure of organisms and their parts. Anatomy is a branch of natural science that deals with the structural organization of living things. It is an old science, having its ...
,
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
and
pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
,
machine learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machine ...
,
computational mechanics
Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science (also called scientific computing) as a "third w ...
,
computational science
Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disc ...
,
biological imaging Biological imaging may refer to any imaging technique used in biology.
Typical examples include:
* Bioluminescence imaging, a technique for studying laboratory animals using luminescent protein
* Calcium imaging, determining the calcium status of a ...
,
neuroscience
Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, development ...
,
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
, and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
; it also has strong connections with
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
and
geometric mechanics Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid bodies to fluid mechanics to control theory.
Geometric mechanics applies principally to systems f ...
. Additionally, it complements newer, interdisciplinary fields like
bioinformatics
Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combi ...
and
neuroinformatics
Neuroinformatics is the field that combines informatics and neuroscience. Neuroinformatics is related with neuroscience data and information processing by artificial neural networks. There are three main directions where neuroinformatics has to be ...
in the sense that its interpretation uses metadata derived from the original sensor imaging modalities (of which
magnetic resonance imaging
Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio wave ...
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
Computational linguistics is an Interdisciplinarity, interdisciplinary field concerned with the computational modelling of natural language, as well as the study of appropriate computational approaches to linguistic questions. In general, comput ...
, a discipline that focuses on the linguistic structures rather than the
sensor
A sensor is a device that produces an output signal for the purpose of sensing a physical phenomenon.
In the broadest definition, a sensor is a device, module, machine, or subsystem that detects events or changes in its environment and sends ...
acting as the
transmission
Transmission may refer to:
Medicine, science and technology
* Power transmission
** Electric power transmission
** Propulsion transmission, technology allowing controlled application of power
*** Automatic transmission
*** Manual transmission
*** ...
and communication media.
In computational anatomy, the
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
group is used to study different coordinate systems via
coordinate transformations
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
as generated via the
Lagrangian and Eulerian velocities of flow in
. The
flows between coordinates in computational anatomy are constrained to be
geodesic flows satisfying
the principle of least action for the Kinetic energy of the flow. The kinetic energy is defined through a
Sobolev smoothness norm with strictly more than two generalized,
square-integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real number, real- or complex number, complex-valued measurable function for which the integral of the s ...
derivatives for each component of the flow velocity, which guarantees that the flows in
are diffeomorphisms.
It also implies that the
diffeomorphic shape momentum taken pointwise satisfying the
Euler-Lagrange equation for geodesics is determined by its neighbors through spatial derivatives on the velocity field. This separates the discipline from the case of
incompressible fluids for which momentum is a pointwise function of velocity. Computational anatomy intersects the study of
Riemannian manifolds
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
and nonlinear
global analysis In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector bundles. Global analysis uses techniques in infinite-dimensional manifold th ...
, 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
Morphometrics (from Greek μορϕή ''morphe'', "shape, form", and -μετρία ''metria'', "measurement") or morphometry refers to the quantitative analysis of ''form'', a concept that encompasses size and shape. Morphometric analyses are co ...
, with the distinction that Computational anatomy focuses on an infinite-dimensional space of
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
s transformed by a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
, hence the central use of the terminology
diffeomorphometry, the metric space study of coordinate systems via diffeomorphisms.
Genesis
At computational anatomy's heart is the comparison of shape by recognizing in one shape the other. This connects it to
D'Arcy Wentworth Thompson
Sir D'Arcy Wentworth Thompson CB FRS FRSE (2 May 1860 – 21 June 1948) was a Scottish biologist, mathematician and classics scholar. He was a pioneer of mathematical and theoretical biology, travelled on expeditions to the Bering Strait an ...
's developments
On Growth and Form
''On Growth and Form'' is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942.
The book covers many topi ...
which has led to scientific explanations of
morphogenesis
Morphogenesis (from the Greek ''morphê'' shape and ''genesis'' creation, literally "the generation of form") is the biological process that causes a cell, tissue or organism to develop its shape. It is one of three fundamental aspects of devel ...
, the process by which
patterns
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated l ...
are formed in
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
.
Albrecht Durer's Four Books on Human Proportion were arguably the earliest works on computational anatomy. The efforts of
Noam Chomsky
Avram Noam Chomsky (born December 7, 1928) is an American public intellectual: a linguist, philosopher, cognitive scientist, historian, social critic, and political activist. Sometimes called "the father of modern linguistics", Chomsky is ...
in his pioneering of
computational linguistics
Computational linguistics is an Interdisciplinarity, interdisciplinary field concerned with the computational modelling of natural language, as well as the study of appropriate computational approaches to linguistic questions. In general, comput ...
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
Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio wave ...
(MRI), computational anatomy has emerged as a subfield of
medical imaging
Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to rev ...
and
bioengineering
Biological engineering or
bioengineering is the application of principles of biology and the tools of engineering to create usable, tangible, economically-viable products. Biological engineering employs knowledge and expertise from a number o ...
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
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ...
and
kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
of
rigid bodies
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
, where objects are studied by analysing the
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
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
Ulf Grenander (23 July 1923 – 12 May 2016) was a Swedish statistician and professor of applied mathematics at Brown University.
His early research was in probability theory, stochastic processes, time series analysis, and statistical theory (p ...
. In Grenander's general metric
pattern theory, making spaces of patterns into a
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
is one of the fundamental operations since being able to cluster and recognize anatomical configurations often requires a metric of close and far between shapes. The
diffeomorphometry metric of computational anatomy measures how far two diffeomorphic changes of coordinates are from each other, which in turn induces a
metric on the shapes and images indexed to them. The models of metric pattern theory, in particular group action on the orbit of shapes and forms is a central tool to the formal definitions in computational anatomy.
History
Computational anatomy is the study of shape and form at the
morphome
Morphome is one of the omes in biology to map and classify all the morphological features of species. Morphome is different from phenome in that it is the totality of morphological variants while phenome includes non-morphological variants.
See ...
or
gross anatomy
Gross anatomy is the study of anatomy at the visible or macroscopic level. The counterpart to gross anatomy is the field of histology, which studies microscopic anatomy. Gross anatomy of the human body or other animals seeks to understand the rel ...
millimeter, or
morphology
Morphology, from the Greek and meaning "study of shape", may refer to:
Disciplines
* Morphology (archaeology), study of the shapes or forms of artifacts
* Morphology (astronomy), study of the shape of astronomical objects such as nebulae, galaxies ...
scale, focusing on the study of sub-
manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Ne ...
of
points, curves surfaces and subvolumes of human anatomy.
An early modern computational neuro-anatomist was David Van Essen performing some of the early physical unfoldings of the human brain based on printing of a human cortex and cutting.
Jean Talairach's publication of
Talairach coordinates
Talairach coordinates, also known as Talairach space, is a 3-dimensional coordinate system (known as an 'atlas') of the human brain, which is used to brain mapping, map the location of brain structures independent from individual differences in the ...
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
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
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
In neuroimaging, spatial normalization is an image processing step, more specifically an image registration method. Human brains differ in size and shape, and one goal of spatial normalization is to deform human brain scans so one location in ...
and
image registration
Image registration is the process of transforming different sets of data into one coordinate system. Data may be multiple photographs, data from different sensors, times, depths, or viewpoints. It is used in computer vision, medical imaging, milit ...
which were historically built on notions of addition and basis expansion. The structure preserving transformations central to the modern field of Computational Anatomy,
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
s and
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
s carry smooth submanifolds smoothly. They are generated via
Lagrangian and Eulerian flows which
satisfy a law of composition of functions forming the group property, but are not additive.
The original model of computational anatomy was as the triple,
the group
, the orbit of shapes and forms
, and the probability laws
which encode the variations of the objects in the orbit. The template or collection of templates are elements in the orbit
of shapes.
The Lagrangian and Hamiltonian formulations of the equations of motion of computational anatomy took off post 1997 with several pivotal meetings including the 1997 Luminy meeting organized by the Azencott school at
Ecole-Normale Cachan on the "Mathematics of Shape Recognition" and the 1998 Trimestre at
Institute Henri Poincaré organized by
David Mumford
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...
"Questions Mathématiques en Traitement du Signal et de l'Image" which catalyzed the Hopkins-Brown-ENS Cachan groups and subsequent developments and connections of computational anatomy to developments in global analysis.
The developments in computational anatomy included the establishment of the Sobolev smoothness conditions on the diffeomorphometry metric to insure existence of solutions of
variational problems in the space of diffeomorphisms,
the derivation of the Euler-Lagrange equations characterizing geodesics through the group and associated conservation laws,
the demonstration of the metric properties of the right invariant metric,
the demonstration that the Euler-Lagrange equations have a well-posed initial value problem with unique solutions for all time, and with the first results on sectional curvatures for the diffeomorphometry metric in landmarked spaces. Following the Los Alamos meeting in 2002, Joshi's
original large deformation singular ''Landmark'' solutions in computational anatomy were connected to peaked
''solitons'' or
''peakons'' as solutions for the
Camassa-Holm equation. Subsequently, connections were made between computational anatomy's Euler-Lagrange equations for momentum densities for the right-invariant metric satisfying Sobolev smoothness to
Vladimir Arnold's[ characterization of the ]Euler equation
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
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
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
's Habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...
(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
The Institute for Pure and Applied Mathematics (IPAM) is an American mathematics institute funded by the National Science Foundation. The initial funding for the institute was approved in May 1999 and it was inaugurated in August, 2000.
IPAM ...
conferences at University of California, Los Angeles
The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the California St ...
. Computational anatomy has been useful in creating accurate models of the atrophy of the human brain at the morphome scale, as well as Cardiac templates, as well as in modeling biological systems. Since the late 1990s, computational anatomy has become an important part of developing emerging technologies for the field of medical imaging. Digital atlases are a fundamental part of modern Medical-school education and in neuroimaging research at the morphome scale. Atlas based methods and virtual textbooks which accommodate variations as in deformable templates are at the center of many neuro-image analysis platforms including Freesurfer, FSL, MRIStudio, SPM. Diffeomorphic registration, introduced in the 1990s, is now an important player with existing codes bases organized around ANTS, DARTEL, DEMONS, LDDMM, StationaryLDDMM, FastLDDMM, are examples of actively used computational codes for constructing correspondences between coordinate systems based on sparse features and dense images. Voxel-based morphometry
Voxel-based morphometry is a computational approach to neuroanatomy that measures differences in local concentrations of brain tissue, through a voxel-wise comparison of multiple brain images.
In traditional morphometry, volume of the whole brai ...
is an important technology built on many of these principles.
The deformable template orbit model of computational anatomy
The model of human anatomy is a deformable template, an orbit of exemplars under group action. Deformable template models have been central to Grenander's metric pattern theory, accounting for typicality via templates, and accounting for variability via transformation of the template. An orbit under group action as the representation of the deformable template is a classic formulation from differential geometry. The space of shapes are denoted , with the group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
with law of composition ; the action of the group on shapes is denoted , where the action of the group is defined to satisfy
:
The orbit of the template becomes the space of all shapes, , being homogenous under the action of the elements of .
The orbit model of computational anatomy is an abstract algebra - to be compared to linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
- since the groups act nonlinearly on the shapes. This is a generalization of the classical models of linear algebra, in which the set of finite dimensional vectors are replaced by the finite-dimensional anatomical submanifolds (points, curves, surfaces and volumes) and images of them, and the matrices of linear algebra are replaced by coordinate transformations based on linear and affine groups and the more general high-dimensional diffeomorphism groups.
Shapes and forms
The central objects are shapes or forms in computational anatomy, one set of examples being the 0,1,2,3-dimensional submanifolds of , a second set of examples being images generated via medical imaging
Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to rev ...
such as via magnetic resonance imaging
Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio wave ...
(MRI) and functional magnetic resonance imaging
Functional magnetic resonance imaging or functional MRI (fMRI) measures brain activity by detecting changes associated with blood flow. This technique relies on the fact that cerebral blood flow and neuronal activation are coupled. When an area o ...
. 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
The midbrain or mesencephalon is the forward-most portion of the brainstem and is associated with vision, hearing, motor control, sleep and wakefulness, arousal (alertness), and temperature regulation. The name comes from the Greek ''mesos'', " ...
or the gyral surface of the neocortex
The neocortex, also called the neopallium, isocortex, or the six-layered cortex, is a set of layers of the mammalian cerebral cortex involved in higher-order brain functions such as sensory perception, cognition, generation of motor commands, sp ...
; subvolumes correspond to subregions of the human body, the heart
The heart is a muscular organ in most animals. This organ pumps blood through the blood vessels of the circulatory system. The pumped blood carries oxygen and nutrients to the body, while carrying metabolic waste such as carbon dioxide t ...
, the thalamus
The thalamus (from Greek θάλαμος, "chamber") is a large mass of gray matter located in the dorsal part of the diencephalon (a division of the forebrain). Nerve fibers project out of the thalamus to the cerebral cortex in all directions, ...
, the kidney.
The landmarks are a collections of points with no other structure, delineating important fiducials within human shape and form (see associated landmarked image).
The sub-manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
shapes such as surfaces are collections of points modeled as parametrized by a local chart or immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux
* Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
, (see Figure showing shapes as mesh surfaces).
The images such as MR images or DTI images , and are dense functions
are scalars, vectors, and matrices (see Figure showing scalar image).
Groups and group actions
Group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
s and group actions
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
are familiar to the Engineering community with the universal popularization and standardization of linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
as a basic model for analyzing signals and systems in mechanical engineering
Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and ...
, electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
and applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
. In linear algebra the matrix groups (matrices with inverses) are the central structure, with group action defined by the usual definition of as an matrix, acting on as vectors; the orbit in linear-algebra is the set of -vectors given by , which is a group action of the matrices through the orbit of .
The central group in computational anatomy defined on volumes in are the diffeomorphisms
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ma ...
which are mappings with 3-components , law of composition of functions , with inverse .
Most popular are scalar images, , with action on the right via the inverse.
:.
For sub-manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s , parametrized by a chart or immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux
* Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
, the diffeomorphic action the flow of the position
:.
Several group actions in computational anatomy
Group actions are central to Riemannian geometry and defining orbits (control theory).
The orbits of computational anatomy consist of anatomical shapes and medical images; the anatomical shapes are submanifolds of differential geometry consi ...
have been defined.
Lagrangian and Eulerian flows for generating diffeomorphisms
For the study of rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
, the low-dimensional matrix Lie groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
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 In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fait ...
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 which satisfy the Lagrangian and Eulerian specification of the flow field
__NOTOC__
In classical field theories, the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an indi ...
s as first introduced in., satisfying the ordinary differential equation:
with the vector fields on termed the Eulerian velocity of the particles at position of the flow. The vector fields are functions in a function space, modelled as a smooth Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
space of high-dimension, with the Jacobian of the flow a high-dimensional field in a function space as well, rather than a low-dimensional matrix as in the matrix groups. Flows were first introduced for large deformations in image matching; is the instantaneous velocity of particle at time .
The inverse required for the group is defined on the Eulerian vector-field with advective inverse 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.[P. Dupuis, U. Grenander, M.I. Miller, Existence of Solutions on Flows of Diffeomorphisms, Quarterly of Applied Math, 1997.
][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.] For diffeomorphisms on , vector fields are modelled as elements of the Hilbert space using the Sobolev embedding theorems so that each element has strictly greater than 2 generalized square-integrable spatial derivatives (thus is sufficient), yielding 1-time continuously differentiable functions.
The diffeomorphism group are flows with vector fields absolutely integrable in Sobolev norm:
where
with the linear operator mapping to the dual space , with the integral calculated by integration by parts when is a generalized function in the dual space.
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. The diffeomorphometry metric measures how close and far two shapes or images are from each other; the metric length is the shortest length of the flow which carries one coordinate system into the other.
Oftentimes, the familiar Euclidean metric is not directly applicable because the patterns of shapes and images don't form a vector space. In the Riemannian orbit model of computational anatomy, diffeomorphisms acting on the forms don't act linearly. There are many ways to define metrics, and for the sets associated to shapes the Hausdorff metric In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metri ...
is another. The method we use to induce the Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
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
this is the right-invariant metric of diffeomorphometry, invariant to reparameterization of space since for all ,
:.
The metric on shapes and forms
The distance on shapes and forms,,
the images are denoted with the orbit as and metric .
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 Hamilton may refer to:
People
* Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname
** The Duke of Hamilton, the premier peer of Scotland
** Lord Hamilt ...
. This is a standard way, for example of obtaining Newton's laws of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
of free particles. More generally, the Euler-Lagrange equations can be derived for systems of generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 3 ...
. The Euler-Lagrange equation in computational anatomy describes the geodesic shortest path flows between coordinate systems of the diffeomorphism metric. In computational anatomy the generalized coordinates are the flow of the diffeomorphism and its Lagrangian velocity , the two related via the Eulerian velocity .
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function ...
for generating the Euler-Lagrange equation requires the action integral on the Lagrangian given by
the Lagrangian is given by the kinetic energy:
Diffeomorphic or Eulerian shape momentum
In computational anatomy, was first called the Eulerian or diffeomorphic shape momentum since when integrated against Eulerian velocity gives energy density, and since there is a conservation of diffeomorphic shape momentum which holds. The operator is the generalized moment of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
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
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function ...
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 which involves the Jacobian given by
:.
Defining the adjoint then the first order variation gives the Eulerian shape momentum satisfying the generalized equation:
meaning for all smooth
:
Computational anatomy is the study of the motions of submanifolds, points, curves, surfaces and volumes.
Momentum associated to points, curves and surfaces are all singular, implying the momentum is concentrated on subsets of which are dimension in Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
. In such cases, the energy is still well defined since although is a generalized function, the vector fields are smooth and the Eulerian momentum is understood via its action on smooth functions. The perfect illustration of this is even when it is a superposition of delta-diracs, the velocity of the coordinates in the entire volume move smoothly. The Euler-Lagrange equation () on diffeomorphisms for generalized functions was derived in.[M.I. Miller, A. Trouve, L. Younes, Geodesic Shooting in Computational Anatomy, IJCV, 2006.] 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 for incompressible, divergence free, fluids.
Diffeomorphic shape momentum: a classical vector function
For the momentum density case , 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.
Riemannian exponential (geodesic positioning) and Riemannian logarithm (geodesic coordinates)
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 is termed the Riemannian-exponential, a mapping at identity to the group.
The Riemannian exponential satisfies for initial condition , vector field dynamics ,
* for classical equation diffeomorphic shape momentum , , then
:
* for generalized equation, then ,,
:
Computing the flow onto coordinates Riemannian logarithm, mapping at identity from to vector field ;
Extended to the entire group they become
; .
These are inverses of each other for unique solutions of Logarithm; the first is called geodesic positioning, the latter geodesic coordinates (see exponential map, Riemannian geometry for the finite dimensional version).The geodesic metric is a local flattening of the Riemannian coordinate system (see figure).
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 and the control the vector field related via The Hamiltonian view
reparameterizes the momentum distribution in terms of the ''conjugate momentum or'' ''canonical momentum, i''ntroduced as a Lagrange multiplier constraining the Lagrangian velocity .accordingly:
:
This function is the extended Hamiltonian. The Pontryagin maximum principle gives the optimizing vector field which determines the geodesic flow satisfying as well as the reduced Hamiltonian
:
The Lagrange multiplier in its action as a linear form has its own inner product of the canonical momentum acting on the velocity of the flow which is dependent on the shape, e.g. for landmarks a sum, for surfaces a surface integral, and. for volumes it is a volume integral with respect to on . In all cases the Greens kernels carry weights which are the canonical momentum evolving according to an ordinary differential equation which corresponds to EL but is the geodesic reparameterization in canonical momentum. The optimizing vector field is given by
:
with dynamics of canonical momentum reparameterizing the vector field along the geodesic
Stationarity of the Hamiltonian and kinetic energy along Euler–Lagrange
Whereas the vector fields are extended across the entire background space of , the geodesic flows associated to the submanifolds has Eulerian shape momentum which evolves as a generalized function concentrated to the submanifolds. For landmarks[V. Camion, L. Younes: Geodesic Interpolating Splines (EMMCVPR 2001)
][J Glaunès, M Vaillant, MI Miller. Landmark matching via large deformation diffeomorphisms on the sphere
Journal of mathematical imaging and vision, 2004.
] 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 have stationarity of the Lagrangian. The Hamiltonian is given by the extremum along the path