Group actions in computational anatomy




Main article: 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.




Contents






  • 1 The orbit model of computational anatomy


  • 2 Several group actions in computational anatomy


    • 2.1 Submanifolds: organs, subcortical structures, charts, and immersions


    • 2.2 Scalar images such as MRI, CT, PET


    • 2.3 Oriented tangents on curves, eigenvectors of tensor matrices


    • 2.4 Tensor matrices


    • 2.5 Orientation Distribution Function and High Angular Resolution HARDI




  • 3 References





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.[1] The space of shapes are denoted m∈M{displaystyle min {mathcal {M}}}min {mathcal {M}}, with the group (G,∘){displaystyle ({mathcal {G}},circ )}({mathcal {G}},circ ) with law of composition ∘{displaystyle circ }circ ; the action of the group on shapes is denoted g⋅m{displaystyle gcdot m}{displaystyle gcdot m}, where the action of the group g⋅m∈M,m∈M{displaystyle gcdot min {mathcal {M}},min {mathcal {M}}}{displaystyle gcdot min {mathcal {M}},min {mathcal {M}}} is defined to satisfy


(g∘g′)⋅m=g⋅(g′⋅m)∈M.{displaystyle (gcirc g^{prime })cdot m=gcdot (g^{prime }cdot m)in {mathcal {M}}.}(gcirc g^{prime })cdot m=gcdot (g^{prime }cdot m)in {mathcal {M}}.

The orbit M{displaystyle {mathcal {M}}}{mathcal {M}} of the template becomes the space of all shapes, M≐{m=g⋅mtemp,g∈G}{displaystyle {mathcal {M}}doteq {m=gcdot m_{mathrm {temp} },gin {mathcal {G}}}}{displaystyle {mathcal {M}}doteq {m=gcdot m_{mathrm {temp} },gin {mathcal {G}}}}.



Several group actions in computational anatomy


Main article: Computational anatomy

The central group in CA defined on volumes in R3{displaystyle {mathbb {R} }^{3}}{mathbb {R} }^{3} are the diffeomorphism group G≐Diff{displaystyle {mathcal {G}}doteq mathrm {Diff} }{displaystyle {mathcal {G}}doteq mathrm {Diff} } which are mappings with 3-components ϕ(⋅)=(ϕ1(⋅),ϕ2(⋅),ϕ3(⋅)){displaystyle phi (cdot )=(phi _{1}(cdot ),phi _{2}(cdot ),phi _{3}(cdot ))}phi (cdot )=(phi _{1}(cdot ),phi _{2}(cdot ),phi _{3}(cdot )), law of composition of functions ϕ∘ϕ′(⋅)≐ϕ(ϕ′(⋅)){displaystyle phi circ phi ^{prime }(cdot )doteq phi (phi ^{prime }(cdot ))}{displaystyle phi circ phi ^{prime }(cdot )doteq phi (phi ^{prime }(cdot ))}, with inverse ϕ∘ϕ−1(⋅)=ϕ(ϕ−1(⋅))=id{displaystyle phi circ phi ^{-1}(cdot )=phi (phi ^{-1}(cdot ))=operatorname {id} }{displaystyle phi circ phi ^{-1}(cdot )=phi (phi ^{-1}(cdot ))=operatorname {id} }.



Submanifolds: organs, subcortical structures, charts, and immersions


For sub-manifolds X⊂R3∈M{displaystyle Xsubset {mathbb {R} }^{3}in {mathcal {M}}}{displaystyle Xsubset {mathbb {R} }^{3}in {mathcal {M}}}, parametrized by a chart or immersion m(u),u∈U{displaystyle m(u),uin U}{displaystyle m(u),uin U}, the diffeomorphic action the flow of the position



ϕ⋅m(u)≐ϕ∘m(u),u∈U{displaystyle phi cdot m(u)doteq phi circ m(u),uin U}{displaystyle phi cdot m(u)doteq phi circ m(u),uin U}.


Scalar images such as MRI, CT, PET


Most popular are scalar images, I(x),x∈R3{displaystyle I(x),xin {mathbb {R} }^{3}}{displaystyle I(x),xin {mathbb {R} }^{3}}, with action on the right via the inverse.



ϕ⋅I(x)=I∘ϕ−1(x),x∈R3{displaystyle phi cdot I(x)=Icirc phi ^{-1}(x),xin {mathbb {R} }^{3}}{displaystyle phi cdot I(x)=Icirc phi ^{-1}(x),xin {mathbb {R} }^{3}}.


Oriented tangents on curves, eigenvectors of tensor matrices


Many different imaging modalities are being used with various actions. For images such that I(x){displaystyle I(x)}{displaystyle I(x)} is a three-dimensional vector then



φ⋅I=((Dφ)I)∘φ−1,{displaystyle varphi cdot I=((Dvarphi ),I)circ varphi ^{-1},}{displaystyle varphi cdot I=((Dvarphi ),I)circ varphi ^{-1},}

φ⋆I=((DφT)−1I)∘φ−1{displaystyle varphi star I=((Dvarphi ^{T})^{-1}I)circ varphi ^{-1}}{displaystyle varphi star I=((Dvarphi ^{T})^{-1}I)circ varphi ^{-1}}



Tensor matrices


Cao et al.
[2]
examined actions for mapping MRI images measured via diffusion tensor imaging and represented via there principle eigenvector.
For tensor fields a positively oriented orthonormal basis
I(x)=(I1(x),I2(x),I3(x)){displaystyle I(x)=(I_{1}(x),I_{2}(x),I_{3}(x))}{displaystyle I(x)=(I_{1}(x),I_{2}(x),I_{3}(x))}
of R3{displaystyle {mathbb {R} }^{3}}{mathbb {R} }^{3}, termed frames, vector cross product denoted I1×I2{displaystyle I_{1}times I_{2}}{displaystyle I_{1}times I_{2}} then


φ⋅I=(DφI1‖DφI1‖,(DφT)−1I3×DφI1‖(DφT)−1I3×DφI1‖,(DφT)−1I3‖(DφT)−1I3‖)∘φ−1 ,{displaystyle varphi cdot I=left({frac {Dvarphi I_{1}}{|Dvarphi ,I_{1}|}},{frac {(Dvarphi ^{T})^{-1}I_{3}times Dvarphi ,I_{1}}{|(Dvarphi ^{T})^{-1}I_{3}times Dvarphi ,I_{1}|}},{frac {(Dvarphi ^{T})^{-1}I_{3}}{|(Dvarphi ^{T})^{-1}I_{3}|}}right)circ varphi ^{-1} ,}{displaystyle varphi cdot I=left({frac {Dvarphi I_{1}}{|Dvarphi ,I_{1}|}},{frac {(Dvarphi ^{T})^{-1}I_{3}times Dvarphi ,I_{1}}{|(Dvarphi ^{T})^{-1}I_{3}times Dvarphi ,I_{1}|}},{frac {(Dvarphi ^{T})^{-1}I_{3}}{|(Dvarphi ^{T})^{-1}I_{3}|}}right)circ varphi ^{-1} ,}

The Fr'enet frame of three orthonormal vectors, I1{displaystyle I_{1}}{displaystyle I_{1}} deforms as a tangent, I3{displaystyle I_{3}}{displaystyle I_{3}} deforms like
a normal to the plane generated by I1×I2{displaystyle I_{1}times I_{2}}{displaystyle I_{1}times I_{2}}, and I3{displaystyle I_{3}}{displaystyle I_{3}}. H is uniquely constrained by the
basis being positive and orthonormal.


For 3×3{displaystyle 3times 3}{displaystyle 3times 3} non-negative symmetric matrices, an action would become φ⋅I=(DφIDφT)∘φ−1{displaystyle varphi cdot I=(Dvarphi ,IDvarphi ^{T})circ varphi ^{-1}}{displaystyle varphi cdot I=(Dvarphi ,IDvarphi ^{T})circ varphi ^{-1}}.


For mapping MRI DTI images[3][4] (tensors), then eigenvalues are preserved with the diffeomorphism rotating eigenvectors and preserves the eigenvalues.
Given eigenelements
{λi,ei,i=1,2,3}{displaystyle {lambda _{i},e_{i},i=1,2,3}}{displaystyle {lambda _{i},e_{i},i=1,2,3}}, then the action becomes


φ⋅I≐(λ1e^1e^1T+λ2e^2e^2T+λ3e^3e^3T)∘φ−1{displaystyle varphi cdot Idoteq (lambda _{1}{hat {e}}_{1}{hat {e}}_{1}^{T}+lambda _{2}{hat {e}}_{2}{hat {e}}_{2}^{T}+lambda _{3}{hat {e}}_{3}{hat {e}}_{3}^{T})circ varphi ^{-1}}{displaystyle varphi cdot Idoteq (lambda _{1}{hat {e}}_{1}{hat {e}}_{1}^{T}+lambda _{2}{hat {e}}_{2}{hat {e}}_{2}^{T}+lambda _{3}{hat {e}}_{3}{hat {e}}_{3}^{T})circ varphi ^{-1}}

e^1=Dφe1‖Dφe1‖ ,e^2=Dφe2−⟨e^1,(Dφe2⟩e^1‖Dφe2−⟨e^1,(Dφe2⟩e^1‖ , e^3≐e^1×e^2 .{displaystyle {hat {e}}_{1}={frac {Dvarphi e_{1}}{|Dvarphi e_{1}|}} ,{hat {e}}_{2}={frac {Dvarphi e_{2}-langle {hat {e}}_{1},(Dvarphi e_{2}rangle {hat {e}}_{1}}{|Dvarphi e_{2}-langle {hat {e}}_{1},(Dvarphi e_{2}rangle {hat {e}}_{1}|}} , {hat {e}}_{3}doteq {hat {e}}_{1}times {hat {e}}_{2} .}{displaystyle {hat {e}}_{1}={frac {Dvarphi e_{1}}{|Dvarphi e_{1}|}} ,{hat {e}}_{2}={frac {Dvarphi e_{2}-langle {hat {e}}_{1},(Dvarphi e_{2}rangle {hat {e}}_{1}}{|Dvarphi e_{2}-langle {hat {e}}_{1},(Dvarphi e_{2}rangle {hat {e}}_{1}|}} , {hat {e}}_{3}doteq {hat {e}}_{1}times {hat {e}}_{2} .}


Orientation Distribution Function and High Angular Resolution HARDI


Further information: Computational_anatomy § Diffusion_tensor_image_matching_in_computational_anatomy, and LDDMM § LDDMM ODF

Orientation distribution function (ODF) characterizes the angular profile of the diffusion probability density function of water molecules and can be reconstructed from High Angular Resolution Diffusion Imaging (HARDI). The ODF is a probability density function defined on a unit sphere, S2{displaystyle {mathbb {S} }^{2}}{displaystyle {mathbb {S} }^{2}}. In the field of information geometry,[5] the space of ODF forms a Riemannian manifold with the Fisher-Rao metric. 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{displaystyle {sqrt {text{ODF}}}}{displaystyle {sqrt {text{ODF}}}}) as ψ(s){displaystyle psi ({bf {s}})}{displaystyle psi ({bf {s}})}, where ψ(s){displaystyle psi ({bf {s}})}{displaystyle psi ({bf {s}})} is non-negative to ensure uniqueness and ∫s∈S2ψ2(s)ds=1{displaystyle int _{{bf {s}}in {mathbb {S} }^{2}}psi ^{2}({bf {s}})d{bf {s}}=1}{displaystyle int _{{bf {s}}in {mathbb {S} }^{2}}psi ^{2}({bf {s}})d{bf {s}}=1}.


Denote diffeomorphic transformation as ϕ{displaystyle phi }phi . Group action of diffeomorphism on ψ(s){displaystyle psi ({bf {s}})}{displaystyle psi ({bf {s}})}, ϕ⋅ψ{displaystyle phi cdot psi }{displaystyle phi cdot psi }, needs to guarantee the non-negativity and ∫s∈S2ϕ⋅ψ2(s)ds=1{displaystyle int _{{bf {s}}in {mathbb {S} }^{2}}phi cdot psi ^{2}({bf {s}})d{bf {s}}=1}{displaystyle int _{{bf {s}}in {mathbb {S} }^{2}}phi cdot psi ^{2}({bf {s}})d{bf {s}}=1}. Based on the derivation in,[6] this group action is defined as


(Dϕ)ψ∘ϕ−1(x)=det(Dϕ−1ϕ)−1‖(Dϕ−1ϕ)−1s‖3ψ((Dϕ−1ϕ)−1s‖(Dϕ−1ϕ)−1s‖,ϕ−1(x)),{displaystyle {begin{aligned}(Dphi )psi circ phi ^{-1}(x)={sqrt {frac {det {{bigl (}D_{phi ^{-1}}phi {bigr )}^{-1}}}{left|{{bigl (}D_{phi ^{-1}}phi {bigr )}^{-1}}{bf {s}}right|^{3}}}}quad psi left({frac {(D_{phi ^{-1}}phi {bigr )}^{-1}{bf {s}}}{|(D_{phi ^{-1}}phi {bigr )}^{-1}{bf {s}}|}},phi ^{-1}(x)right),end{aligned}}}{displaystyle {begin{aligned}(Dphi )psi circ phi ^{-1}(x)={sqrt {frac {det {{bigl (}D_{phi ^{-1}}phi {bigr )}^{-1}}}{left|{{bigl (}D_{phi ^{-1}}phi {bigr )}^{-1}}{bf {s}}right|^{3}}}}quad psi left({frac {(D_{phi ^{-1}}phi {bigr )}^{-1}{bf {s}}}{|(D_{phi ^{-1}}phi {bigr )}^{-1}{bf {s}}|}},phi ^{-1}(x)right),end{aligned}}}

where (Dϕ){displaystyle (Dphi )}{displaystyle (Dphi )} is the Jacobian of ϕ{displaystyle phi }phi .



References





  1. ^ Miller, Michael I.; Younes, Laurent; Trouvé, Alain (2014-03-01). "Diffeomorphometry and geodesic positioning systems for human anatomy". Technology. 2 (1): 36. doi:10.1142/S2339547814500010. ISSN 2339-5478. PMC 4041578. PMID 24904924..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  2. ^ Cao Y1, Miller MI, Winslow RL, Younes, Large deformation diffeomorphic metric mapping of vector fields. IEEE Trans Med Imaging. 2005 Sep;24(9):1216-30.


  3. ^ Alexander, D. C.; Pierpaoli, C.; Basser, P. J.; Gee, J. C. (2001-11-01). "Spatial transformations of diffusion tensor magnetic resonance images". IEEE Transactions on Medical Imaging. 20 (11): 1131–1139. doi:10.1109/42.963816. ISSN 0278-0062. PMID 11700739.


  4. ^ Cao, Yan; Miller, Michael I.; Mori, Susumu; Winslow, Raimond L.; Younes, Laurent (2006-07-05). Diffeomorphic Matching of Diffusion Tensor Images. Proceedings / CVPR, IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE Computer Society Conference on Computer Vision and Pattern Recognition. 2006. p. 67. doi:10.1109/CVPRW.2006.65. ISBN 978-0-7695-2646-1. ISSN 1063-6919. PMC 2920614. PMID 20711423.


  5. ^ Amari, S (1985). Differential-Geometrical Methods in Statistics. Springer.


  6. ^ Du, J; Goh, A; Qiu, A (2012). "Diffeomorphic metric mapping of high angular resolution diffusion imaging based on Riemannian structure of orientation distribution functions". IEEE Trans Med Imaging. 31 (5): 1021–1033. doi:10.1109/TMI.2011.2178253. PMID 22156979.









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