Diffeomorphometry is the metric study of imagery, shape and form in the discipline of
computational anatomy (CA) in
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 re ...
. The study of images in
computational anatomy rely on high-dimensional
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 ...
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 ide ...
which generate orbits of the form
, in which images
can be dense scalar
magnetic resonance
Magnetic resonance is a process by which a physical excitation (resonance) is set up via magnetism.
This process was used to develop magnetic resonance imaging and Nuclear magnetic resonance spectroscopy technology.
It is also being used to d ...
or
computed axial tomography images. For
deformable shapes these are the collection of
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
, points,
curves
A curve is a geometrical object in mathematics.
Curve(s) may also refer to:
Arts, entertainment, and media Music
* Curve (band), an English alternative rock music group
* ''Curve'' (album), a 2012 album by Our Lady Peace
* "Curve" (song), a ...
and
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s. The diffeomorphisms move the images and shapes through the orbit according to
which are defined as the
group actions of computational anatomy.
The orbit of shapes and forms is made into a metric space by inducing a metric on the group of diffeomorphisms. 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
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...
is constructed by defining a flow of diffeomorphisms
which connect the group elements from one to another, so for
then
. The metric between two coordinate systems or diffeomorphisms is then the shortest length or
geodesic flow
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
connecting them. The metric on the space associated to the geodesics is given by
. The metrics on the orbits
are inherited from the metric induced on the diffeomorphism group.
The group
is thusly made into a smooth
Riemannian manifold
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 ...
with Riemannian metric
associated to the tangent spaces at all
. 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 '' ...
satisfies at every point of the manifold
there is an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
inducing the norm on the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
that varies smoothly across
.
Oftentimes, the familiar
Euclidean metric
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occ ...
is not directly applicable because the patterns of shapes and images don't form a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. 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 is another. The method used 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 '' ...
is 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 diffeomorphisms group generated as Lagrangian and Eulerian flows
The diffeomorphisms in
computational anatomy are generated to satisfy the
Lagrangian and Eulerian specification of the flow fields,
, generated via the ordinary differential equation
with the Eulerian vector fields
in
for