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Diffeomorphometry
Diffeomorphometry is the metric study of imagery, shape and form in the discipline of computational anatomy (CA) in medical imaging. The study of images in computational anatomy rely on high-dimensional diffeomorphism groups \varphi \in \operatorname_V which generate orbits of the form \mathcal \doteq \ , in which images I \in \mathcal can be dense scalar magnetic resonance or computed axial tomography images. For deformable shapes these are the collection of manifolds \mathcal \doteq \ , points, curves and surfaces. The diffeomorphisms move the images and shapes through the orbit according to (\varphi,I)\mapsto \varphi \cdot I 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 Computation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Anatomy
Computational anatomy is an interdisciplinary field of biology 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, applied mathematics and pure mathematics, machine learning, computational mechanics, computational science, biological 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Anatomy
Computational anatomy is an interdisciplinary field of biology 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, applied mathematics and pure mathematics, machine learning, computational mechanics, computational science, biological 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Metric And Lie-Bracket Interpretation Of The Euler Equation On Geodesics
Computational anatomy (CA) is the study of shape and form in medical imaging. The study of deformable shapes in computational anatomy rely on high-dimensional diffeomorphism groups \varphi \in \operatorname_V which generate orbits of the form \mathcal \doteq \ . In CA, this orbit is in general considered a smooth Riemannian manifold since at every point of the manifold m \in \mathcal there is an inner product inducing the norm \, \cdot \, _m on the tangent space that varies smoothly from point to point in the manifold of shapes m \in \mathcal . This is generated by viewing the group of diffeomorphisms \varphi \in \operatorname_V as a Riemannian manifold with \, \cdot \, _\varphi , associated to the tangent space at \varphi \in\operatorname_V . This induces the norm and metric on the orbit m \in \mathcal under the action from the group of diffeomorphisms. The diffeomorphisms group generated as Lagrangian and Eulerian flows The diffeomorphisms in computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Deformation Diffeomorphic Metric Mapping
Large deformation diffeomorphic metric mapping (LDDMM) is a specific suite of algorithms used for diffeomorphic mapping and manipulating dense imagery based on diffeomorphic metric mapping within the academic discipline of computational anatomy, to be distinguished from its precursor based on diffeomorphic mapping. The distinction between the two is that diffeomorphic metric maps satisfy the property that the length associated to their flow away from the identity induces a metric on the group of diffeomorphisms, which in turn induces a metric on the orbit of shapes and forms within the field of Computational Anatomy. The study of shapes and forms with the metric of diffeomorphic metric mapping is called diffeomorphometry. A diffeomorphic mapping system is a system designed to map, manipulate, and transfer information which is stored in many types of spatially distributed medical imagery. Diffeomorphic mapping is the underlying technology for mapping and analyzing information m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 manifolds M and N, a differentiable map f \colon M \rightarrow N is called a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. They are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a subset X of a manifold M and a subset Y of a manifold N, a function f:X\to Y is said to be smooth if for all p in X there is a neighbor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 consisting of points, curves, surfaces and subvolumes,. This generalized the ideas of the more familiar orbits of linear algebra which are linear vector spaces. Medical images are scalar and tensor images from medical imaging. The group actions are used to define models of human shape which accommodate variation. These orbits are deformable templates as originally formulated more abstractly in pattern theory. The orbit model of computational anatomy The central model of human anatomy in computational anatomy is a Groups and group action, a classic formulation from differential geometry. The orbit is called the space of shapes and forms. The space of shapes are denoted m \in \mathcal , with the group (\mathcal, \circ ) with law of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 waves to generate images of the organs in the body. MRI does not involve X-rays or the use of ionizing radiation, which distinguishes it from CT and PET scans. MRI is a medical application of nuclear magnetic resonance (NMR) which can also be used for imaging in other NMR applications, such as NMR spectroscopy. MRI is widely used in hospitals and clinics for medical diagnosis, staging and follow-up of disease. Compared to CT, MRI provides better contrast in images of soft-tissues, e.g. in the brain or abdomen. However, it may be perceived as less comfortable by patients, due to the usually longer and louder measurements with the subject in a long, confining tube, though "Open" MRI designs mostly relieve this. Additionally, implants and oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 brain or its subparts is measured by drawing regions of interest (ROIs) on images from brain scanning and calculating the volume enclosed. However, this is time consuming and can only provide measures of rather large areas. Smaller differences in volume may be overlooked. The value of VBM is that it allows for comprehensive measurement of differences, not just in specific structures, but throughout the entire brain. VBM registers every brain to a template, which gets rid of most of the large differences in brain anatomy among people. Then the brain images are smoothed so that each voxel represents the average of itself and its neighbors. Finally, the image volume is compared across brains at every voxel. However, VBM can be sensitive to vari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Software Suite
A software suite (also known as an application suite) is a collection of computer programs (application software, or programming software) of related functionality, sharing a similar user interface and the ability to easily exchange data with each other. Features Advantages * Less costly than buying individual packages * Identical or very similar GUI * Designed to interface with each other * Helps the learning curve of the user Disadvantages * Not all purchased features are always used by the user * Takes a significant amount of disk space (bloatware), as compared to buying only the needed packages * Requires effort to use the packages together Types * Office suites, such as Microsoft Office * Internet suites * Graphics suite, such as Adobe Creative Cloud * IDEs, such as Eclipse, and Visual Studio See also * Application software * Package (package management system) * Runtime environment In computer programming, a runtime system or runtime environment is a sub-system that ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin Maximum Principle
Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It states that it is necessary for any optimal control along with the optimal state trajectory to solve the so-called Hamiltonian system, which is a two-point boundary value problem, plus a maximum condition of the control Hamiltonian. These necessary conditions become sufficient under certain convexity conditions on the objective and constraint functions. The maximum principle was formulated in 1956 by the Russian mathematician Lev Pontryagin and his students, and its initial application was to the maximization of the terminal speed of a rocket. The result was derived using ideas from the classical calculus of variations. After a slight perturbation of the optimal control, one considers the first-order term of a Taylor expansion with respect to the pert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Function For The Three-variable Laplace Equation
In physics, the Green's function (or fundamental solution) for Laplace's equation in three variables is used to describe the response of a particular type of physical system to a point source. In particular, this Green's function arises in systems that can be described by Poisson's equation, a partial differential equation (PDE) of the form : \nabla^2u(\mathbf) = f(\mathbf) where \nabla^2 is the Laplace operator in \mathbb^3, f(\mathbf) is the source term of the system, and u(\mathbf) is the solution to the equation. Because \nabla^2 is a linear differential operator, the solution u(\mathbf) to a general system of this type can be written as an integral over a distribution of source given by f(\mathbf): : u(\mathbf) = \int_ G(\mathbf,\mathbf)f(\mathbf)d\mathbf' where the Green's function for Laplace's equation in three variables G(\mathbf,\mathbf) describes the response of the system at the point \mathbf to a point source located at \mathbf: :\nabla^2 G(\mathbf,\mathbf) = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
