“Draft:Variations in CA for Necessary Conditions”的意思、由来-开放百科全书

In CA the geodesics and their coordinates are generated by solving inexact matching problems calculating the geodesic

flows of diffeomorphisms from their start point onto targets. Inexact matching has been examined in many cases and have come to be called Large Deformation Diffeomorphic Metric Mapping (LDDMM) originally solved for landmarks with correspondence ^[1]^[2]^[3]^[4] and for dense image matching.^[5]^[6] These methods have been extended to landmarks without registration via measure matching,^[7] curves,^[8] surfaces,^[9]^[10] dense vector^[11] and tensor ^[12] imagery, and varifolds removing orientation.^[13] The objects being matched are generally modelled as noisey versions (see below Bayesian statistical modelling section) of the shapes and forms

. The geodesic position and geodesic coordinates associated to Log and Exp are solved as variational problems solving inexact matching of the coordinate systems using classical methods in optimal control.

Matching of coordinates results from introducing an endpoint condition

which measures the correspondence or registration of elements in the orbit under coordinate system transformation.

The necessary conditions for a minimizer of 1 satisfies

, for perturbation

, with fixed initial condition,

, with free endpoint. This gives the Euler equation interior to

, with a boundary condition at

where the target enters. Integrating by parts

gives

Dense image matching

a vector function. For dense images the action

implies we will requires the variation of the inverse

with respect to

for the chain rule calculation

. This requires the identity

following from the identity

. This is for such function spaces the generalization of the classic matrix perturbation of the inverse.

Landmark matching with correspondence

Landmark matching demonstrates the singularity of

as a generalized function or delta-distribution, since for landmarks,

all of boundary mass is conscentrated on singular subsets of the volume space. The optimizer of Control Problem 1 satisfies the weak Euler equation for delta-distributions:

LDDMM for image and landmark matching: perturbation in the vector fields

The original large deformation diffeomorphic metric mapping (LDDMM) algorithms took variations with respect to the vector field parameterization of the group, since

are in a vector spaces. Variations satisfy the necessary optimality conditions..

The perturbation in the vector field requires the identity

which implies

Take the variation in the vector fields

using the chain rule

which gives the first variation

References

1. ^Joshi 1357–1370
2. ^{{Cite book|title = Geodesic Interpolating Splines|url = http://dl.acm.org/citation.cfm?id=646596.756898|journal = Proceedings of the Third International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition|date = 2001-01-01|location = London, UK, UK|isbn = 978-3-540-42523-6|pages = 513–527|series = EMMCVPR '01|first = Vincent|last = Camion|first2 = Laurent|last2 = Younes}}
3. ^{{Cite journal|title = Statistics on diffeomorphisms via tangent space representations|pmid = 15501085|journal = NeuroImage|date = 2004-01-01|issn = 1053-8119|pages = S161–169|volume = 23 Suppl 1|doi = 10.1016/j.neuroimage.2004.07.023|first = M.|last = Vaillant|first2 = M. I.|last2 = Miller|first3 = L.|last3 = Younes|first4 = A.|last4 = Trouvé|citeseerx = 10.1.1.132.6802}}
4. ^{{Cite journal|title = A hamiltonian particle method for diffeomorphic image registration|pmid = 17633716|journal = Information Processing in Medical Imaging: Proceedings of the ... Conference|date = 2007-01-01|issn = 1011-2499|pages = 396–407|volume = 20|first = Stephen|last = Marsland|first2 = Robert|last2 = McLachlan}}
5. ^{{Cite journal|title = Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms|journal = International Journal of Computer Vision|volume = 61|issue = 2|pages = 139–157|url = https://www.researchgate.net/publication/220660081 |accessdate = 2015-11-22|doi = 10.1023/B:VISI.0000043755.93987.aa|year = 2005|last1 = Beg|first1 = M. Faisal|last2 = Miller|first2 = Michael I|last3 = Trouvé|first3 = Alain|last4 = Younes|first4 = Laurent}}
6. ^{{Cite journal|title = Diffeomorphic 3D Image Registration via Geodesic Shooting Using an Efficient Adjoint Calculation|journal = Int. J. Comput. Vision|date = 2012-04-01|issn = 0920-5691|pages = 229–241|volume = 97|issue = 2|doi = 10.1007/s11263-011-0481-8|first = François-Xavier|last = Vialard|first2 = Laurent|last2 = Risser|first3 = Daniel|last3 = Rueckert|first4 = Colin J.|last4 = Cotter}}
7. ^{{Cite book|title = L.: Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching|volume = 2|pages = 712–718|chapter-url = https://www.researchgate.net/publication/4082354|website = ResearchGate|accessdate = 2015-11-25|doi = 10.1109/CVPR.2004.1315234|chapter = Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching|year = 2004|last1 = Glaunes|first1 = J|last2 = Trouve|first2 = A|last3 = Younes|first3 = L|isbn = 978-0-7695-2158-9|citeseerx = 10.1.1.158.4209}}
8. ^{{Cite journal|title = Large Deformation Diffeomorphic Metric Curve Mapping|pmc = 2858418|journal = International Journal of Computer Vision|date = 2008-12-01|issn = 0920-5691|pmid = 20419045|pages = 317–336|volume = 80|issue = 3|doi = 10.1007/s11263-008-0141-9|first = Joan|last = Glaunès|first2 = Anqi|last2 = Qiu|first3 = Michael I.|last3 = Miller|first4 = Laurent|last4 = Younes}}
9. ^"Vaillant 1149–1159"
10. ^{{Cite journal|title = Surface matching via currents|journal = Proceedings of Information Processing in Medical Imaging (IPMI 2005), Number 3565 in Lecture Notes in Computer Science|date = 2005-01-01|pages = 381–392|first = Marc|last = Vaillant|first2 = Joan|last2 = Glaunès|citeseerx = 10.1.1.88.4666}}
11. ^{{Cite book|title = Large deformation diffeomorphic metric mapping of fiber orientations|url = http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1544880&url=http%253A%252F%252Fieeexplore.ieee.org%252Fiel5%252F10347%252F32976%252F01544880.pdf%253Farnumber%253D1544880|journal = Tenth IEEE International Conference on Computer Vision, 2005. ICCV 2005|date = 2005-10-01|pages = 1379–1386 Vol. 2|volume = 2|doi = 10.1109/ICCV.2005.132|first = Yan|last = Cao|first2 = M.I.|last2 = Miller|first3 = R.L.|last3 = Winslow|first4 = L.|last4 = Younes|isbn = 978-0-7695-2334-7|citeseerx = 10.1.1.158.1582}}
12. ^{{Cite journal|title = Large deformation diffeomorphic metric mapping of vector fields|url = http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1501927&url=http%253A%252F%252Fieeexplore.ieee.org%252Fiel5%252F42%252F32229%252F01501927|journal = IEEE Transactions on Medical Imaging|date = 2005-09-01|issn = 0278-0062|pmc = 2848689|pmid = 17427733|pages = 1216–1230|volume = 24|issue = 9|doi = 10.1109/TMI.2005.853923|first = Yan|last = Cao|first2 = M.I.|last2 = Miller|first3 = R.L.|last3 = Winslow|first4 = L.|last4 = Younes|citeseerx = 10.1.1.157.8377}}
13. ^{{Cite journal|title = The Varifold Representation of Nonoriented Shapes for Diffeomorphic Registration|journal = SIAM Journal on Imaging Sciences|date = 2013-01-01|pages = 2547–2580|volume = 6|issue = 4|doi = 10.1137/130918885|first = N.|last = Charon|first2 = A.|last2 = Trouvé|arxiv = 1304.6108 }}

Registering and Matching Information Across Coordiate Systems in CA via Variations

Dense image matching

Landmark matching with correspondence

LDDMM for image and landmark matching: perturbation in the vector fields

References