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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 mome... | Wikipedia - Computational anatomy - History | 336 | 1,719 | null |
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 o... | Wikipedia - Computational anatomy - History | 217 | 1,081 | null |
Section: 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... | Wikipedia - Computational anatomy - The deformable template orbit model of computational anatomy | 255 | 929 | null |
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 {\... | Wikipedia - Computational anatomy - The deformable template orbit model of computational anatomy | 330 | 839 | null |
{\displaystyle (g\circ g^{\prime })\cdot m=g\cdot (g^{\prime }\cdot m)\in {\mathcal {M}}.} 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 homog... | Wikipedia - Computational anatomy - The deformable template orbit model of computational anatomy | 307 | 1,026 | null |
Section: 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 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 fun... | Wikipedia - Computational anatomy - Shapes and forms | 284 | 1,042 | null |
The landmarks 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 sur... | Wikipedia - Computational anatomy - Shapes and forms | 342 | 940 | null |
Section: Groups and group actions. 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 th... | Wikipedia - Computational anatomy - Groups and group actions | 224 | 857 | null |
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 i... | Wikipedia - Computational anatomy - Groups and group actions | 473 | 1,256 | null |
The central group in computational anatomy defined on volumes in R 3 {\displaystyle {\mathbb {R} }^{3}} are the diffeomorphisms G ≐ Diff {\displaystyle {\mathcal {G}}\doteq \operatorname {Diff} } which are mappings with 3-components φ ( ⋅ ) = ( φ 1 ( ⋅ ) , φ 2 ( ⋅ ) , φ 3 ( ⋅ ) ) {\displaystyle \varphi (\cdot )=(\varph... | Wikipedia - Computational anatomy - Groups and group actions | 346 | 837 | null |
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. φ ⋅ I ( x ) = I ∘ φ − 1 ( x ) , x ∈ R 3 . {\displaystyle \varphi \cdot I(x)=I\circ \varphi ^{-1}(x),x\in {\mathbb {R} }^{3}.} For sub-manifolds X ⊂ R 3 ∈ M {\displaystyle X\subset {... | Wikipedia - Computational anatomy - Groups and group actions | 266 | 661 | null |
Section: Lagrangian and Eulerian flows for generating diffeomorphisms. For the study of rigid body kinematics, the low-dimensional matrix Lie groups have been the central focus. The matrix groups are low-dimensional mappings, which are diffeomorphisms that provide one-to-one correspondences between coordinate systems, ... | Wikipedia - Computational anatomy - Lagrangian and Eulerian flows for generating diffeomorphisms | 315 | 1,266 | null |
The high-dimensional diffeomorphism groups used in Computational Anatomy are generated via smooth flows φ t , t ∈ [ 0 , 1 ] {\displaystyle \varphi _{t},t\in [0,1]} which satisfy the Lagrangian and Eulerian specification of the flow fields as first introduced in, satisfying the ordinary differential equation: with v ≐ (... | Wikipedia - Computational anatomy - Lagrangian and Eulerian flows for generating diffeomorphisms | 338 | 1,125 | null |
Section: 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. For diffeomorphisms on R 3 {\displaystyle {\mathb... | Wikipedia - Computational anatomy - The diffeomorphism group of computational anatomy | 215 | 732 | null |
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 has strictly greater than 2 generalized square-integrable spatial derivatives (thus v i... | Wikipedia - Computational anatomy - The diffeomorphism group of computational anatomy | 315 | 928 | null |
Section: 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... | Wikipedia - Computational anatomy - Diffeomorphometry: The metric space of shapes and forms | 305 | 1,262 | null |
Section: Diffeomorphometry: The metric space of shapes and forms > 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 φ ∈ Diff V {\displaystyle \varphi \in \ope... | Wikipedia - Computational anatomy - Diffeomorphometry: The metric space of shapes and forms > The right-invariant metric on diffeomorphisms | 186 | 530 | null |
Section: The action integral for Hamilton's principle on diffeomorphic flows. In classical mechanics the evolution of physical systems is described by solutions to the Euler–Lagrange equations associated to the Least-action principle of Hamilton. This is a standard way, for example of obtaining Newton's laws of motion ... | Wikipedia - Computational anatomy - The action integral for Hamilton's principle on diffeomorphic flows | 250 | 1,048 | null |
Section: The Euler–Lagrange equation on shape momentum for geodesics on the group of diffeomorphisms. Classical calculation of the Euler–Lagrange equation from Hamilton's principle requires the perturbation of the Lagrangian on the vector field in the kinetic energy with respect to first order perturbation of the flow.... | Wikipedia - Computational anatomy - The Euler–Lagrange equation on shape momentum for geodesics on the group of diffeomorphisms | 191 | 608 | null |
This requires adjustment by the Lie bracket of vector field, given by operator a d v : w ∈ V ↦ V {\displaystyle ad_{v}:w\in V\mapsto V} which involves the Jacobian given by a d v [ w ] ≐ [ v , w ] ≐ ( D v ) w − ( D w ) v ∈ V {\displaystyle ad_{v}[w]\doteq [v,w]\doteq (Dv)w-(Dw)v\in V} . Defining the adjoint a d v ∗ : V... | Wikipedia - Computational anatomy - The Euler–Lagrange equation on shape momentum for geodesics on the group of diffeomorphisms | 280 | 705 | null |
Defining the adjoint a d v ∗ : V ∗ → V ∗ , {\displaystyle ad_{v}^{*}:V^{*}\rightarrow V^{*},} then the first order variation gives the Eulerian shape momentum A v ∈ V ∗ {\displaystyle Av\in V^{*}} satisfying the generalized equation: meaning for all smooth w ∈ V , {\displaystyle w\in V,} ∫ X ( d d t A v t + a d v t ∗ (... | Wikipedia - Computational anatomy - The Euler–Lagrange equation on shape momentum for geodesics on the group of diffeomorphisms | 320 | 706 | null |
{\displaystyle \int _{X}\left({\frac {d}{dt}}Av_{t}+ad_{v_{t}}^{*}(Av_{t})\right)\cdot w\,dx=\int _{X}{\frac {d}{dt}}Av_{t}\cdot w\,dx+\int _{X}Av_{t}\cdot ((Dv_{t})w-(Dw)v_{t})dx=0.} Computational anatomy is the study of the motions of submanifolds, points, curves, surfaces and volumes. Momentum associated to points, ... | Wikipedia - Computational anatomy - The Euler–Lagrange equation on shape momentum for geodesics on the group of diffeomorphisms | 325 | 944 | null |
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 Li... | Wikipedia - Computational anatomy - The Euler–Lagrange equation on shape momentum for geodesics on the group of diffeomorphisms | 196 | 753 | null |
Section: 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 ... | Wikipedia - Computational anatomy - Riemannian exponential (geodesic positioning) and Riemannian logarithm (geodesic coordinates) | 178 | 671 | null |
The Riemannian exponential satisfies Exp i d ( v 0 ) = φ 1 {\displaystyle \operatorname {Exp} _{\rm {id}}(v_{0})=\varphi _{1}} for initial condition φ ˙ 0 = v 0 {\displaystyle {\dot {\varphi }}_{0}=v_{0}} , vector field dynamics φ ˙ t = v t ∘ φ t , t ∈ [ 0 , 1 ] {\displaystyle {\dot {\varphi }}_{t}=v_{t}\circ \varphi... | Wikipedia - Computational anatomy - Riemannian exponential (geodesic positioning) and Riemannian logarithm (geodesic coordinates) | 348 | 750 | null |
{\displaystyle w\in V} , ∫ X d d t A v t ⋅ w d x + ∫ X A v t ⋅ ( ( D v t ) w − ( D w ) v t ) d x = 0. {\displaystyle \ \ \ \int _{X}{\frac {d}{dt}}Av_{t}\cdot w\,dx+\int _{X}Av_{t}\cdot ((Dv_{t})w-(Dw)v_{t})\,dx=0.} Computing the flow v 0 {\displaystyle v_{0}} onto coordinates Riemannian logarithm, mapping L o g i d ( ... | Wikipedia - Computational anatomy - Riemannian exponential (geodesic positioning) and Riemannian logarithm (geodesic coordinates) | 271 | 594 | null |
{\displaystyle \ \ \ \int _{X}{\frac {d}{dt}}Av_{t}\cdot w\,dx+\int _{X}Av_{t}\cdot ((Dv_{t})w-(Dw)v_{t})\,dx=0.} Computing the flow v 0 {\displaystyle v_{0}} onto coordinates Riemannian logarithm, mapping L o g i d ( ⋅ ) : Diff V → V {\displaystyle Log_{\rm {id}}(\cdot ):\operatorname {Diff} _{V}\to V} at identity fro... | Wikipedia - Computational anatomy - Riemannian exponential (geodesic positioning) and Riemannian logarithm (geodesic coordinates) | 475 | 1,035 | null |
{\displaystyle \log _{\rm {id}}(\varphi )=v_{0}\ {\text{initial condition of EL geodesic}}{\dot {\varphi }}_{0}=v_{0},\varphi _{0}=id,\varphi _{1}=\varphi \ .} Extended to the entire group they become φ = Exp φ ( v 0 ∘ φ ) ≐ Exp i d ( v 0 ) ∘ φ {\displaystyle \varphi =\operatorname {Exp} _{\varphi }(v_{0}\circ \var... | Wikipedia - Computational anatomy - Riemannian exponential (geodesic positioning) and Riemannian logarithm (geodesic coordinates) | 323 | 855 | null |
{\displaystyle {\dot {\varphi }}_{t}=v_{t}\cdot \varphi _{t},\varphi _{0}={\rm {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... | Wikipedia - Computational anatomy - Hamiltonian formulation of computational anatomy | 347 | 766 | null |
{\displaystyle H(\varphi _{t},p_{t},v_{t})=\int _{X}p_{t}\cdot (v_{t}\circ \varphi _{t})\,dx-{\frac {1}{2}}\int _{X}Av_{t}\cdot v_{t}\,dx.} 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 ... | Wikipedia - Computational anatomy - Hamiltonian formulation of computational anatomy | 329 | 814 | null |
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 c... | Wikipedia - Computational anatomy - Hamiltonian formulation of computational anatomy | 179 | 643 | null |
Section: Hamiltonian formulation of computational anatomy > Stationarity of the Hamiltonian and kinetic energy along Euler–Lagrange. 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 ... | Wikipedia - Computational anatomy - Hamiltonian formulation of computational anatomy > Stationarity of the Hamiltonian and kinetic energy along Euler–Lagrange | 306 | 1,169 | null |
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(\varphi ,p)=\max _{v}H(\varphi ,p,v)} , equalling the Lagrangian-kinetic-energy and is stationary along EL-general. Defining the geodesic velocity at the identity v 0 =... | Wikipedia - Computational anatomy - Hamiltonian formulation of computational anatomy > Stationarity of the Hamiltonian and kinetic energy along Euler–Lagrange | 343 | 1,095 | null |
Section: The metric on geodesic flows of landmarks, surfaces, and volumes within the orbit. In computational anatomy the submanifolds are pointsets, curves, surfaces and subvolumes which are the basic primitives. The geodesic flows between the submanifolds determine the distance, and form the basic measuring and transp... | Wikipedia - Computational anatomy - The metric on geodesic flows of landmarks, surfaces, and volumes within the orbit | 320 | 976 | null |
Section: Conservation laws on diffeomorphic shape momentum for computational anatomy. Given the least-action there is a natural definition of momentum associated to generalized coordinates; the quantity acting against velocity gives energy. The field has studied two forms, the momentum associated to the Eulerian vector... | Wikipedia - Computational anatomy - Conservation laws on diffeomorphic shape momentum for computational anatomy | 205 | 1,066 | null |
Section: Geodesic interpolation of information between coordinate systems via variational problems. 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}} . The... | Wikipedia - Computational anatomy - Geodesic interpolation of information between coordinate systems via variational problems | 322 | 1,225 | null |
Section: Geodesic interpolation of information between coordinate systems via variational problems > Matching based on minimizing kinetic energy action with endpoint condition. min { C ( φ ) : v = φ ˙ ∘ φ − 1 , φ 0 = i d } ≐ 1 2 ∫ 0 1 ∫ X A v t ⋅ v t d x d t + E ( φ 1 ) {\displaystyle {\begin{aligned}&\min \left\{C(\va... | Wikipedia - Computational anatomy - Geodesic interpolation of information between coordinate systems via variational problems > Matching based on minimizing kinetic energy action with endpoint condition | 280 | 650 | null |
min { C ( φ ) : v = φ ˙ ∘ φ − 1 , φ 0 = i d } ≐ 1 2 ∫ 0 1 ∫ X A v t ⋅ v t d x d t + E ( φ 1 ) {\displaystyle {\begin{aligned}&\min \left\{C(\varphi ):v={\dot {\varphi }}\circ \varphi ^{-1},\varphi _{0}={\rm {id}}\right\}\\[5pt]\doteq {}&{\frac {1}{2}}\int _{0}^{1}\int _{X}Av_{t}\cdot v_{t}\,dx\,dt+E(\varphi _{1})\end{a... | Wikipedia - Computational anatomy - Geodesic interpolation of information between coordinate systems via variational problems > Matching based on minimizing kinetic energy action with endpoint condition | 421 | 833 | null |
{\displaystyle {\begin{cases}{\text{Euler conservation }}&\displaystyle {\frac {d}{dt}}Av_{t}+ad_{v_{t}}^{*}(Av_{t})=0,\ t\in [0,1)\ ,\\{\text{Boundary condition }}&\displaystyle \varphi _{0}={\rm {id}},Av_{1}=\left.-{\frac {\partial E(\varphi )}{\partial \varphi }}\right|_{\varphi =\varphi _{1}}\ .\end{cases}}} Conser... | Wikipedia - Computational anatomy - Geodesic interpolation of information between coordinate systems via variational problems > Matching based on minimizing kinetic energy action with endpoint condition | 310 | 840 | null |
min v 0 C ( v 0 ) ≐ 1 2 ∫ X A v 0 ⋅ v 0 d x + E ( E x p i d ( v 0 ) ⋅ I 0 ) min p 0 C ( p 0 ) = 1 2 ∫ X p 0 ⋅ K p 0 d x + E ( E x p id ( K p 0 ) ⋅ I 0 ) {\displaystyle {\begin{aligned}&\min _{v_{0}}C(v_{0})\doteq {\frac {1}{2}}\int _{X}Av_{0}\cdot v_{0}\,dx+E(\mathrm {Exp} _{\mathrm {id} }(v_{0})\cdot I_{0})\\[6pt]&\mi... | Wikipedia - Computational anatomy - Geodesic interpolation of information between coordinate systems via variational problems > Matching based on geodesic shooting | 330 | 641 | null |
Section: Dense image matching in computational anatomy. Dense image matching has a long history now with the earliest efforts exploiting a small deformation framework. Large deformations began in the early 1990s, with the first existence to solutions to the variational problem for flows of diffeomorphisms for dense ima... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy | 195 | 858 | null |
Section: Dense image matching in computational anatomy > LDDMM dense image matching. 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 \varphi \cdot I\doteq I\circ \varphi ^{-1}} . Viewing this as an optimal control problem, the state of the... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > LDDMM dense image matching | 212 | 608 | null |
Viewing this as an optimal control problem, the state of the system is the diffeomorphic flow of coordinates φ t , t ∈ [ 0 , 1 ] {\displaystyle \varphi _{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 {\... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > LDDMM dense image matching | 295 | 741 | null |
μ 0 = ( I − I ′ ∘ φ 1 ) ∇ I | D φ 1 | . {\displaystyle {\begin{cases}{\text{Endpoint condition:}}&Av_{1}=\mu _{1}\,dx,\mu _{1}=(I\circ \varphi _{1}^{-1}-I^{\prime })\nabla (I\circ \varphi _{1}^{-1})\ ,\\[5pt]{\text{Conservation:}}&Av_{t}=\mu _{t}\,dx,\ \mu _{t}=(D\varphi _{t}^{-1})^{T}\mu _{0}\circ \varphi _{t}^{-1}|D\... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > LDDMM dense image matching | 309 | 614 | null |
Section: Dense image matching in computational anatomy > Hamiltonian LDDMM in the reduced advected state. Denote the Image I ( x ) , x ∈ X {\displaystyle I(x),x\in X} , with state q t ≐ I ∘ φ t − 1 {\displaystyle q_{t}\doteq I\circ \varphi _{t}^{-1}} and the dynamics related state and control given by the advective ter... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > Hamiltonian LDDMM in the reduced advected state | 231 | 635 | null |
Section: Dense image matching in computational anatomy > Diffusion tensor image matching in computational anatomy. Dense LDDMM tensor matching takes the images as 3x1 vectors and 3x3 tensors solving the variational problem matching between coordinate system based on the principle eigenvectors of the diffusion tensor MR... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > Diffusion tensor image matching in computational anatomy | 225 | 876 | null |
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. Denote the 3 × 3 {\displaystyle 3\times 3} tensor image M ( x ) , x ∈ R 3 {\displaystyle M(x),x\in {\mathbb {R}... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > Diffusion tensor image matching in computational anatomy | 332 | 998 | null |
{\displaystyle \varphi \cdot I={\begin{cases}{\frac {D_{\varphi ^{-1}}\varphi I\circ \varphi ^{-1}\|I\circ \varphi ^{-1}\|}{\|D_{\varphi ^{-1}}\varphi I\circ \varphi ^{-1}\|}}&I\circ \varphi \neq 0;\\0&{\text{otherwise.}}\end{cases}}} LDDMM matching based on the entire tensor matrix has group action becomes φ ⋅ M = ( λ... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > Diffusion tensor image matching in computational anatomy | 350 | 668 | null |
⟩ e ^ 1 ‖ D φ e 2 ‖ 2 − ⟨ e ^ 1 , D φ e 2 ⟩ 2 , e ^ 3 = e ^ 1 × e ^ 2 {\displaystyle {\begin{aligned}{\hat {e}}_{1}&={\frac {D\varphi e_{1}}{\|D\varphi e_{1}\|}}\ ,\ \ \ {\hat {e}}_{2}={\frac {D\varphi e_{2}-\langle {\hat {e}}_{1},D\varphi e_{2}\rangle {\hat {e}}_{1}}{\sqrt {\|D\varphi e_{2}\|^{2}-\langle {\hat {e}}_{1... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > Diffusion tensor image matching in computational anatomy | 290 | 547 | null |
Section: Dense image matching in computational anatomy > High angular resolution diffusion image (HARDI) matching in computational anatomy. High angular resolution diffusion imaging (HARDI) addresses the well-known limitation of DTI, that is, DTI can only reveal one dominant fiber orientation at each location. HARDI me... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > High angular resolution diffusion image (HARDI) matching in computational anatomy | 273 | 1,245 | null |
For the purpose of LDDMM ODF mapping, the square-root representation is chosen because it is one of the most efficient representations found to date as the various Riemannian operations, such as geodesics, exponential maps, and logarithm maps, are available in closed form. In the following, denote square-root ODF ( ODF... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > High angular resolution diffusion image (HARDI) matching in computational anatomy | 217 | 613 | null |
In the following, denote square-root ODF ( ODF {\displaystyle {\sqrt {\text{ODF}}}} ) as ψ ( s ) {\displaystyle \psi ({\bf {s}})} , where ψ ( s ) {\displaystyle \psi ({\bf {s}})} is non-negative to ensure uniqueness and ∫ s ∈ S 2 ψ 2 ( s ) d s = 1 {\displaystyle \int _{{\bf {s}}\in {\mathbb {S} }^{2}}\psi ^{2}({\bf {s}... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > High angular resolution diffusion image (HARDI) matching in computational anatomy | 300 | 752 | null |
The variational problem for matching assumes that two ODF volumes can be generated from one to another via flows of diffeomorphisms φ t {\displaystyle \varphi _{t}} , which are solutions of ordinary differential equations φ ˙ t = v t ( φ t ) , t ∈ [ 0 , 1 ] , {\displaystyle {\dot {\varphi }}_{t}=v_{t}(\varphi _{t}),t\i... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > High angular resolution diffusion image (HARDI) matching in computational anatomy | 375 | 968 | null |
Denote the action of the diffeomorphism on template as φ 1 ⋅ ψ t e m p ( s , x ) {\displaystyle \varphi _{1}\cdot \psi _{\mathrm {temp} }({\bf {s}},x)} , s ∈ S 2 {\displaystyle {\bf {s}}\in {\mathbb {S} }^{2}} , x ∈ X {\displaystyle x\in X} are respectively the coordinates of the unit sphere, S 2 {\displaystyle {{\math... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > High angular resolution diffusion image (HARDI) matching in computational anatomy | 452 | 1,063 | null |
The group action of the diffeomorphism on the template is given according to φ 1 ⋅ ψ ( x ) ≐ ( D φ 1 ) ψ ∘ φ 1 − 1 ( x ) , x ∈ X {\displaystyle \varphi _{1}\cdot \psi (x)\doteq (D\varphi _{1})\psi \circ \varphi _{1}^{-1}(x),x\in X} , where ( D φ 1 ) {\displaystyle (D\varphi _{1})} is the Jacobian of the affine-transfor... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > High angular resolution diffusion image (HARDI) matching in computational anatomy | 558 | 1,137 | null |
{\displaystyle (D\varphi _{1})\psi \circ \varphi _{1}^{-1}(x)={\sqrt {\frac {\det {{\bigl (}D_{\varphi _{1}^{-1}}\varphi _{1}{\bigr )}^{-1}}}{\left\|{{\bigl (}D_{\varphi _{1}^{-1}}\varphi _{1}{\bigr )}^{-1}}{\bf {s}}\right\|^{3}}}}\quad \psi \left({\frac {(D_{\varphi _{1}^{-1}}\varphi _{1}{\bigr )}^{-1}{\bf {s}}}{\|(D_... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > High angular resolution diffusion image (HARDI) matching in computational anatomy | 361 | 761 | null |
{\displaystyle C(v)=\inf _{v:{\dot {\varphi }}_{t}=v_{t}\circ \varphi _{t},\varphi _{0}={\rm {id}}}\int _{0}^{1}\int _{X}Av_{t}\cdot v_{t}\,dx\,dt+\lambda \int _{x\in \Omega }\|\log _{(D\varphi _{1})\psi _{\mathrm {temp} }\circ \varphi _{1}^{-1}(x)}(\psi _{\mathrm {targ} }(x))\|_{(D\varphi _{1})\psi _{\mathrm {temp} }\... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > High angular resolution diffusion image (HARDI) matching in computational anatomy | 348 | 610 | null |
_{1}}=\cos ^{-1}\langle \psi _{1},\psi _{2}\rangle =\cos ^{-1}\left(\int _{{\bf {s}}\in {\mathbb {S} }^{2}}\psi _{1}({\bf {s}})\psi _{2}({\bf {s}})d{\bf {s}}\right),} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the normal dot product between points in the sphere under the L 2 {\displaystyle \mathrm... | Wikipedia - Computational anatomy - Dense image matching in computational anatomy > High angular resolution diffusion image (HARDI) matching in computational anatomy | 234 | 649 | null |
Section: Metamorphosis. 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, ori... | Wikipedia - Computational anatomy - Metamorphosis | 287 | 1,220 | null |
In this setting metamorphosis combines both the diffeomorphic coordinate system transformation of computational anatomy as well as the early morphing technologies which only faded or modified the photometric or image intensity alone. Then the matching problem takes a form with equality boundary conditions: min ( v , I ... | Wikipedia - Computational anatomy - Metamorphosis | 272 | 687 | null |
Section: Matching landmarks, curves, surfaces. Transforming coordinate systems based on Landmark point or fiducial marker features dates back to Bookstein's early work on small deformation spline methods for interpolating correspondences defined by fiducial points to the two-dimensional or three-dimensional background ... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces | 338 | 1,426 | null |
Section: Matching landmarks, curves, surfaces > Landmark or point matching with correspondence. 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(\varphi _{1})\doteq \... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Landmark or point matching with correspondence | 242 | 646 | null |
The endpoint condition with conservation implies the initial momentum at the identity of the group: { Endpoint condition: A v 1 = ∑ i = 1 n p 1 ( i ) δ φ 1 ( x i ) , p 1 ( i ) = ( x i ′ − φ 1 ( x i ) ) , Conservation: A v t = ∑ i = 1 n p t ( i ) δ φ t ( x i ) , p t ( i ) = ( D φ t 1 ) | φ t ( x i ) T p 1 ( i ) , φ t 1 ... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Landmark or point matching with correspondence | 337 | 691 | null |
p_{t}(i)=(D\varphi _{t1})_{|\varphi _{t}(x_{i})}^{T}p_{1}(i)\ ,\ \varphi _{t1}\doteq \varphi _{1}\circ \varphi _{t}^{-1}\ ,\\[5pt]&Av_{0}=\sum _{i}\delta _{x_{i}}(\cdot )p_{0}(i)\ {\text{ with }}p_{0}(i)=(D\varphi _{1})_{|x_{i}}^{T}(x_{i}^{\prime }-\varphi _{1}(x_{i}))\end{cases}}} The iterative algorithm for large def... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Landmark or point matching with correspondence | 222 | 381 | null |
Section: Matching landmarks, curves, surfaces > Measure matching: unregistered landmarks. 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 pred... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Measure matching: unregistered landmarks | 247 | 788 | null |
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: ‖ μ m ‖ m e a 2 = ∑ i , j = 1 n ρ i ρ j k ( x i , x j ) {\displaystyle \|\mu _{m}\|_{\mathrm {mea} }^{2}=\sum _{i,j=1}^{n}\rho _{i}\rho... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Measure matching: unregistered landmarks | 342 | 768 | null |
Section: Matching landmarks, curves, surfaces > Curve matching. 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 \varphi \cdot m=\varphi \ci... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Curve matching | 261 | 1,050 | null |
Matching of two curves m {\displaystyle m} and m ′ {\displaystyle m^{\prime }} writes eventually as the variational problem min φ : v = φ ˙ ∘ φ − 1 C ( φ ) ≐ 1 2 ∫ ( A v t ∣ v t ) d t + 1 2 ‖ C φ 1 ⋅ m − C m ′ ‖ c u r 2 {\displaystyle \min _{\varphi :v={\dot {\varphi }}\circ \varphi ^{-1}}C(\varphi )\doteq {\frac {1}{2... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Curve matching | 350 | 707 | null |
C ( m ( u ) , m ( v ) ) ∂ m ( u ) ⋅ ∂ m ( v ) d u d v {\displaystyle \|{\mathcal {C}}_{m}\|_{\mathrm {cur} }^{2}=\int _{0}^{1}\int _{0}^{1}K_{C}(m(u),m(v))\partial m(u)\cdot \partial m(v)\,du\,dv} the derivative ∂ m ( u ) {\displaystyle \partial m(u)} being the tangent vector to the curve and K C {\displaystyle K_{\mat... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Curve matching | 273 | 798 | null |
The matching problem between two curves then consists in replacing the endpoint matching term by E ( φ 1 ) = ‖ V φ 1 ⋅ m − V m ′ ‖ cur 2 / 2 {\displaystyle E(\varphi _{1})=\|{\mathcal {V}}_{\varphi _{1}\cdot m}-{\mathcal {V}}_{m^{\prime }}\|_{\text{cur}}^{2}/2} with varifold norms of the form: ‖ V m ‖ v a r 2 = ∫ 0 1 ∫... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Curve matching | 350 | 729 | null |
Section: Matching landmarks, curves, surfaces > Surface matching. 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 {\mathb... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Surface matching | 230 | 823 | null |
In that setting, surface matching writes again: min φ : v = φ ˙ ∘ φ − 1 C ( φ ) ≐ 1 2 ∫ ( A v t ∣ v t ) d t + 1 2 ‖ C φ 1 ⋅ m − C m ′ ‖ c u r 2 {\displaystyle \min _{\varphi :v={\dot {\varphi }}\circ \varphi ^{-1}}C(\varphi )\doteq {\frac {1}{2}}\int (Av_{t}\mid v_{t})\,dt+{\frac {1}{2}}\|{\mathcal {C}}_{\varphi _{1}\c... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Surface matching | 350 | 676 | null |
v {\displaystyle \|{\mathcal {C}}_{m}\|_{\mathrm {cur} }^{2}=\iint _{U\times U}K_{C}(m(u),m(v)){\vec {n}}(u)\cdot {\vec {n}}(v)\,du\,dv} with 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} . This surface... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Surface matching | 227 | 594 | null |
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}\|_{\ma... | Wikipedia - Computational anatomy - Matching landmarks, curves, surfaces > Surface matching | 349 | 708 | null |
Section: Growth and atrophy from longitudinal time-series. There are many settings in which there are a series of measurements, a time-series to which the underlying coordinate systems will be matched and flowed onto. This occurs for example in the dynamic growth and atrophy models and motion tracking such as have been... | Wikipedia - Computational anatomy - Growth and atrophy from longitudinal time-series | 150 | 650 | null |
The generic time-series matching problem considers the series of times is 0 < t 1 < ⋯ < t K = 1 {\displaystyle 0<t_{1}<\cdots <t_{K}=1} . The flow optimizes at the series of costs E ( t k ) , k = 1 , … , K {\displaystyle E(t_{k}),k=1,\ldots ,K} giving optimization problems of the form min φ : v = φ ˙ ∘ φ − 1 , φ 0 = i ... | Wikipedia - Computational anatomy - Growth and atrophy from longitudinal time-series | 297 | 688 | null |
Section: The random orbit model of computational anatomy. 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 fo... | Wikipedia - Computational anatomy - The random orbit model of computational anatomy | 308 | 1,170 | null |
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 accordi... | Wikipedia - Computational anatomy - The random orbit model of computational anatomy | 350 | 1,041 | null |
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 ∣ Exp i d ( v ) ⋅ I a ) π V ( d v ) . {\displaystyle p(I^{D}\mid I_{a})=\int _{V}p(I^{D}\mid \operatorname {Exp} _{\rm {id}}(v)\cdot I_{a})\pi _{V}(dv)\ .}... | Wikipedia - Computational anatomy - The random orbit model of computational anatomy | 204 | 521 | null |
Section: The Bayesian model of computational anatomy. 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 imag... | Wikipedia - Computational anatomy - The Bayesian model of computational anatomy | 154 | 624 | null |
Section: Statistical shape theory in computational anatomy. Shape in computational anatomy is a local theory, indexing shapes and structures to templates to which they are bijectively mapped. Statistical shape in computational anatomy is the empirical study of diffeomorphic correspondences between populations and commo... | Wikipedia - Computational anatomy - Statistical shape theory in computational anatomy | 337 | 1,535 | null |
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... | Wikipedia - Computational anatomy - Statistical shape theory in computational anatomy | 157 | 743 | null |
Article: Connectomics. Connectomics is the production and study of connectomes, which are comprehensive maps of connections within an organism's nervous system. Study of neuronal wiring diagrams looks at how they contribute to the health and behavior of an organism. There are two very different types of connectomes; mi... | Wikipedia - Connectomics - Summary | 336 | 1,699 | null |
Section: Methods > Macroscale Connectomics. Macroscale connectomes are commonly collected using diffusion-weighted magnetic resonance imaging (DW-MRI) and functional magnetic resonance imaging (fMRI). DW-MRI datasets can span the entire brain, imaging white matter between the cortex and subcortex, providing information... | Wikipedia - Connectomics - Methods > Macroscale Connectomics | 334 | 1,598 | null |
Section: Methods > Macroscale Connectomics > Stimulation. Techniques that actively manipulate the brain, often called neuromodulation, can provide insights into the connectome. For example, transcranial magnetic stimulation (TMS) is a non-invasive neuromodulation technique that applies strong magnetic pulses between sc... | Wikipedia - Connectomics - Methods > Macroscale Connectomics > Stimulation | 306 | 1,634 | null |
Section: Methods > Macroscale Connectomics > Electrophysiological Methods. Electrophysiological methods measure the difference in signals from different parts of the brain to estimate the connectivity between them, a process that requires a low signal-to-noise ratio to maintain the accuracy of the measurements and suff... | Wikipedia - Connectomics - Methods > Macroscale Connectomics > Electrophysiological Methods | 162 | 875 | null |
Section: Methods > Microscale Connectomics. Microscale connectomes focuses on resolving individual cell-to-cell connectivity within much smaller volumes of nervous system tissue. The most common method for neural circuit reconstruction is chemical brain preservation followed by 3D electron microscopy, which offers sing... | Wikipedia - Connectomics - Methods > Microscale Connectomics | 326 | 1,638 | null |
X-ray nanotomography using a synchrotron source can now reach <100 nm resolution, and can theoretically continue to improve. Unlike EM, this technique does not require the tissue being imaged to be stained with heavy metals or to be physically sectioned. Conventional light microscopy is constrained by light diffraction... | Wikipedia - Connectomics - Methods > Microscale Connectomics | 256 | 1,283 | null |
Section: Methods > Software. In addition to advanced microscopy techniques, connectomics heavily relies on software analysis tools and machine learning pipelines for reconstructing and analyzing neural networks. These tools are designed to process and interpret the vast amounts of data generated by volume electron micr... | Wikipedia - Connectomics - Methods > Software | 245 | 1,335 | null |
Section: Comparative connectomics. Comparative connectomics is a subfield in neuroscience that focuses on comparing the connectomes, or neural network maps, across different species, developmental stages, or pathological states. This comparative approach aims to uncover fundamental principles of brain organization and ... | Wikipedia - Connectomics - Comparative connectomics | 161 | 903 | null |
Section: Plasticity of the connectome. At the beginning of the connectome project, it was thought that the connections between neurons were unchangeable once established and that only individual synapses could be altered. However, recent evidence suggests that connectivity is also subject to change, termed neuroplastic... | Wikipedia - Connectomics - Plasticity of the connectome | 246 | 1,237 | null |
Section: Plasticity of the connectome > Macroscale rewiring. Evidence for macroscale rewiring mostly comes from research on grey and white matter density, which could indicate new connections or changes in axon density. Direct evidence for this level of rewiring comes from primate studies, using viral tracing to map th... | Wikipedia - Connectomics - Plasticity of the connectome > Macroscale rewiring | 164 | 826 | null |
Section: Model systems > Caenorhabditis Elegans. The C. elegans roundworm has a simple nervous system of 302 neurons and 5000 synaptic connections, (as compared to the human brain which has 100 billion neurons and more than 100 trillion chemical synapses). It was the first of the very few animals in which a full connec... | Wikipedia - Connectomics - Model systems > Caenorhabditis Elegans | 198 | 1,028 | null |
Section: Model systems > Fruit Fly. Within the last decade, largely owing to technological advancements in EM data collection and image processing, multiple synapse-scale connectome datasets have been generated for the fruit fly Drosophila melanogaster in its adult and larval forms. The full fly connectome contains on ... | Wikipedia - Connectomics - Model systems > Fruit Fly | 311 | 1,374 | null |
Section: Model systems > Mouse. An online database known as MouseLight displays over 1000 neurons mapped in the mouse brain based on a collective database of sub-micron resolution images of these brains. This platform illustrates the thalamus, hippocampus, cerebral cortex, and hypothalamus based on single-cell projecti... | Wikipedia - Connectomics - Model systems > Mouse | 226 | 1,093 | null |
Section: Applications. Macroscale and microscale connectomics have very different applications. Macroscale connectomics has furthered our understanding of various brain networks including visual, brainstem, and language networks, among others. Microscale connectomics, on the other hand, concentrates on mechanistic expl... | Wikipedia - Connectomics - Applications | 346 | 1,904 | null |
Specifically, studies on different brain disorders such as schizophrenia and bipolar disorder with a focus on the connectomics involved reveal information. Both of these disorders have a similar genetic origin, and research found that those with higher polygenic scores for schizophrenia and bipolar disorder have lower ... | Wikipedia - Connectomics - Applications | 345 | 2,008 | null |
This is an expanding field and there is room for greater application to mental health disorders and brain malfunction, in which current research is building on neural networks and the psychopathology involved. Human connectomes have an individual variability, which can be measured with the cumulative distribution funct... | Wikipedia - Connectomics - Applications | 176 | 854 | null |
Section: Comparison to genomics. The recent advancements in the field of connectomics have sparked conversation around its relation to the field of genomics. Recently, scientists in the field have highlighted the parallels between this project and large-scale genomics initiatives. Additionally, they have referenced the... | Wikipedia - Connectomics - Comparison to genomics | 300 | 1,780 | null |
Section: As a network or graph. A connectome can be viewed as a graph, and the rich tools, definitions and algorithms of graph theory and network science can be applied to these graphs. In case of a micro-scale connectome, the nodes of this network (or graph) are the neurons, and the edges correspond to the synapses be... | Wikipedia - Connectomics - As a network or graph | 325 | 1,498 | null |
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