Diffusion MRI - Australian National...
Transcript of Diffusion MRI - Australian National...
![Page 1: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/1.jpg)
RedeemingtheClinicalPromiseof
DiffusionMRI
inSupportoftheNeurosurgicalWorkflow
LucFlorackTMCV,July262017,Hawaii,US
![Page 2: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/2.jpg)
EconomicCostofBrainDisordersinEurope2010:€798billion…
EuropeanJournalofNeurology2012,19:155–162
![Page 3: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/3.jpg)
(N)MRIscanner
![Page 4: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/4.jpg)
“everythingmustbemadeassimpleaspossible,butnotonebitsimpler”
aMributedtoAlbertEinstein
![Page 5: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/5.jpg)
operaPonalmodel:brain=constrainedwater
![Page 6: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/6.jpg)
operaPonalmodel:brain=constrainedwater
hypothesis
Pssuemicrostructureimpartsnon-randombarrierstowaterdiffusion
©C.Beaulieu,NMRBiomed.2002,vol.15,nr.7-8,DOI:10.1002/nbm.782
![Page 7: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/7.jpg)
operaPonalmodel:brain=constrainedwater
Riemannianparadigm
extrinsicdiffusiononEuclideanspace≈intrinsicgeometryofaRiemannianspace
![Page 8: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/8.jpg)
operaPonalmodel:brain=constrainedwater
Riemannianparadigm
extrinsicdiffusiononEuclideanspace≈intrinsicgeometryofaRiemannianspace
![Page 9: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/9.jpg)
operaPonalmodel:brain=constrainedwater
Riemannianparadigm
extrinsicdiffusiononEuclideanspace≈intrinsicgeometryofaRiemannianspace
![Page 10: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/10.jpg)
localgaugefigureRiemanngeometry
gaugefigure=unitsphere=indicatrix=Riemannianmetric=innerproduct
![Page 11: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/11.jpg)
localgaugefigureRiemanngeometry
![Page 12: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/12.jpg)
localgaugefigureRiemanngeometry
![Page 13: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/13.jpg)
localgaugefigureRiemanngeometry
![Page 14: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/14.jpg)
localgaugefigureRiemanngeometry
![Page 15: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/15.jpg)
length2=6
localgaugefigureRiemanngeometry
![Page 16: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/16.jpg)
length2=9
localgaugefigureRiemanngeometry
![Page 17: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/17.jpg)
geodesictractography
![Page 18: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/18.jpg)
‘short’geodesic
‘long’geodesic
Riemannianlength:Euclideanlength
5.0:6.0<1
7.5:6.0>1
geodesictractography
![Page 19: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/19.jpg)
DiffusionTensorImagingversus
localgaugefigure
![Page 20: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/20.jpg)
DiffusionTensorImagingphysicspipeline
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
![Page 21: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/21.jpg)
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
![Page 22: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/22.jpg)
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
![Page 23: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/23.jpg)
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
![Page 24: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/24.jpg)
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
![Page 25: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/25.jpg)
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
![Page 26: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/26.jpg)
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
DiffusionTensorImagingphysicspipeline
![Page 27: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/27.jpg)
E" = �1
2�~Bz
E# = +1
2�~Bz
�E = �~Bz = ~!Larmor
nuclearspinquantization→ Zeemansplitting
E = �µ ·B
![Page 28: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/28.jpg)
N"N#
= exp
�E
KT
�⇡ 1 +
�~Bz
KT
Boltzmannstatistics→magnetization(typicalclinical3TMRIscanner)
N" �N#N" +N#
⇡ �~Bz
2KT⇡ 10�5
Mz =N" �N#N" +N#
Mmax
⇡ (N" +N#)�2~2Bz
4KT
`lowsensitivitymodality’
`bigxsmall=measurable’
![Page 29: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/29.jpg)
Bloch-Torrey/Stejskal-Tanner/Basser-Majello-LeBihan:GaussiansignalaMenuaPoninq-space
Bloch-Torreyequation→DTI
@M?@t
= �i�(M?Bk �MkB?)�M?T2
+r ·DrM?
S(x, q, ⌧) = S0(x) exp(�⌧q ·D(x)q)
D(x) = �1
⌧
r2q ln
S(x, q, ⌧)
S0(x)
![Page 30: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/30.jpg)
Bloch-Torreyequation→DTIfurtherreading
![Page 31: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/31.jpg)
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
geodesictractographymathematicspipeline
![Page 32: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/32.jpg)
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
geodesictractographymathematicspipeline
![Page 33: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/33.jpg)
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
geodesictractographymathematicspipeline
![Page 34: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/34.jpg)
physicsinanutshell:
nuclearspinquantization→Zeemansplitting→Boltzmannstatistics→magnetization→Bloch-Torreyequation→DTI
mathematicsinanutshell:
DTI→ localgaugefigure→geodesictractography
geodesictractographymathematicspipeline
![Page 35: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/35.jpg)
DTI→ localgaugefigure
DTIsignalmodel:S(x, q, ⌧) = S0(x)e
�⌧q
TD(x)q
Riemannmetric: G(⇠, ⇠) |x
= ⇠
TG(x)⇠
Fusteretal.: G(x).
= D
adj(x)
Lengletetal./O’Donnelletal.: G(x).
= D
inv(x)
![Page 36: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/36.jpg)
Riemannmetric:lengths&anglesG(v, v) = kvk2
Levi-Civitaconnection:paralleltransportrx
x = 0
Christoffelsymbols:“pseudo-forces”(relativetolocalcoordinateframes)x
i +X
jk
�ijk(x) x
jx
k = 0
localgaugefigure→ tractography
![Page 37: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/37.jpg)
Riemannmetric:lengths&anglesG(v, v) = kvk2
Levi-Civitaconnection:paralleltransportrx
x = 0
Christoffelsymbols:“pseudo-forces”(relativetolocalcoordinateframes)x
i +X
jk
�ijk(x) x
jx
k = 0
localgaugefigure→ tractography
![Page 38: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/38.jpg)
Riemannmetric:lengths&anglesG(v, v) = kvk2
Levi-Civitaconnection:paralleltransportrx
x = 0
Christoffelsymbols:“pseudo-forces”(relativetolocalcoordinateframes)x
i +X
jk
�ijk(x) x
jx
k = 0
localgaugefigure→ tractography
![Page 39: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/39.jpg)
Euclideangeodesic
Riemannmetric:lengths&anglesG(v, v) = kvk2
Levi-Civitaconnection:paralleltransportrx
x = 0
Christoffelsymbols:“pseudo-forces”(relativetolocalcoordinateframes)x
i +X
jk
�ijk(x) x
jx
k = 0
localgaugefigure→ tractography
![Page 40: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/40.jpg)
Riemanniangeodesic
Riemannmetric:lengths&anglesG(v, v) = kvk2
Levi-Civitaconnection:paralleltransportrx
x = 0
Christoffelsymbols:“pseudo-forces”(relativetolocalcoordinateframes)x
i +X
jk
�ijk(x) x
jx
k = 0
localgaugefigure→ tractography
![Page 41: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/41.jpg)
Riemann-DTIparadigm:
• neuralfiberbundlescorrespondtorelativelyshortgeodesicsinaRiemannian‘brainspace’
• theRiemannianstructurecanbeinferredfromDTI
![Page 42: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/42.jpg)
geodesiccompleteness=
redundantconnecPons
ellipsoidalgaugefigure=
poorangularresoluPon
Riemann-DTIparadigmpros&cons
pro:pixels→geodesiccongruences
con:destructiveinterferenceoforientationpreferences
![Page 43: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/43.jpg)
specificmodel(DTI): S(x, q, ⌧) = S0(x) exp(�D(x, q, ⌧))
D(x, q, ⌧) = ⌧q
TD(x) q
with6d.o.f.’sperpointsample
⬍6d.o.f.’soflocalgaugefigure
S(x, q, ⌧) =1X
k=0
S
i1...ik(x, ⌧)�i1...ik(q)genericmodel(HARDI):
D(x, q, ⌧) =1X
k=0
D
i1...ik(x, ⌧) i1...ik(q)
or ∞ d.o.f.’sperpointsample
beyondtheRiemann-DTIparadigmaparadigmshift
![Page 44: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/44.jpg)
beyondtheRiemann-DTIparadigmaparadigmshift
DTI ⊂ HARDI
⬍ ⬍
Riemanngeometry
⊂ Finslergeometry
![Page 45: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/45.jpg)
beyondtheRiemann-DTIparadigmaparadigmshift
DTI ⊂ HARDI
⬍ ⬍
Riemanngeometry
⊂ Finslergeometry
![Page 46: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/46.jpg)
beyondtheRiemann-DTIparadigmaparadigmshift
DTI ⊂ HARDI
⬍ ⬍
Riemanngeometry
⊂ Finslergeometry…
![Page 47: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/47.jpg)
beyondtheRiemann-DTIparadigmaparadigmshift
DTI ⊂ HARDI
⬍ ⬍
Riemanngeometry
⊂ Finslergeometry
![Page 48: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/48.jpg)
Finslergeometryheuristics
w
x
![Page 49: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/49.jpg)
wosculatingindicatrices
basemanifold: x 2 R3 (x,w) 2 R3 ⇥ S2→ (spherebundle)
x
basemanifold: x 2 R3 → (slittangentbundle)(x, ⇠) 2 R3 ⇥ TR3\{0}
Finslergeometryheuristics
![Page 50: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/50.jpg)
familyofosculatingindicatrices⬍
single(convex)indicatrix
gaugefigure=unitsphere=indicatrix=Finslermetric≠innerproduct
Finslergeometryheuristics
![Page 51: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/51.jpg)
in: Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data, Springer 2014
connections
Cartan tensor
HV-splittingFinslergeometry
axiomatics
![Page 52: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/52.jpg)
distance: d(x1, x2) = inf
⇢Z
�F (�(t), �(t)) dt
���� � 2 C
1([t1, t2],R3) , �(t1) = x1 , �(t2) = x2
�
Finslermetric:F
2(x, ⇠) = gij(x, ⇠)⇠i⇠
j , gij(x, ⇠) =1
2@⇠i@⇠jF
2(x, ⇠)
Cijk(x, ⇠) =1
2@⇠kgij(x, ⇠) =
1
4@⇠i⇠j⇠kF
2(x, ⇠)Cartantensor:
Finslergeometryaxiomatics
Riemannianlimit:F
2(x, ⇠) = gij(x)⇠i⇠
j , gij(x) =1
2@⇠i@⇠jF
2(x, ⇠)
Cijk(x, ⇠) = 0Riemannianlimit: (Deicke’sTheorem)
![Page 53: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/53.jpg)
Deicke’sTheorem:
SpaceisRiemannianifftheCartantensorvanishes.
Finslergeometryaxiomatics
![Page 54: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/54.jpg)
HARDI~Finsleriangeometry(generalizednorm):
F (x,�y) = |�|F (x, y)
F
⇤(x,�q) = |�|F ⇤(x, q)
DTI~Riemanniangeometry(innerproductnorm):
F (x, y) =qy
TDinv(x)y
F
⇤(x, q) =pq
TD(x)q
Finsler-DTIparadigmgeodesictractography
![Page 55: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/55.jpg)
Finsler-DTIparadigmgeodesictractography
Finslermetric:lengthsGv(v, v) = kvk2F
Chern-Rund(orother)connection:paralleltransportrx
x = 0
formalChristoffelsymbols:“pseudo-forces”x
i +X
jk
�ijk(x, x) x
jx
k = 0
![Page 56: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/56.jpg)
• neuralfiberbundlescorrespondtorelativelyshortgeodesicsinthe3-dimensional‘horizontalpart’ofa5-dimensional`brainspace’furnishedwithaFinslerianstructure
• thisFinslerianstructurecanbeinferredfromdiffusionMRImeasurements
• theFinsleriandualmetriccanbeinterpretedasa‘5-dimensional(3x3)DTI’tensor
• FinslergeometryencompassesRiemanniangeometryasaspecialcase
• theFinslermetricadmits∞d.o.f.’sperspatialpointasopposedto6d.o.f.’sfortheRiemannianlimit
• theFinsler-DTIparadigmadmitsversatiledimensionalityreductionintrade-offwithacquisitiontime
*TomDelaHaije,PhDthesis,May162017,Eindhoven
Finsler-DTIparadigmoperationalization*
![Page 57: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/57.jpg)
FinslerfuncPon:
F (x,�y) = |�|F (x, y)
F
⇤(x,�q) = |�|F ⇤(x, q)
Finsler-DTIparadigmsummary
![Page 58: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/58.jpg)
FinslerfuncPon:
F (x,�y) = |�|F (x, y)
F
⇤(x,�q) = |�|F ⇤(x, q)
Finsler-DTIparadigmsummary
![Page 59: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/59.jpg)
FLAIR DTI
© Neda Sepasian, Joep Kamps, Andrea Fuster
applications&outlook
![Page 60: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/60.jpg)
applications&outlook
StephanMeestersetal.,electronicposter3476,ISMRM2017,Hawaii
![Page 61: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/61.jpg)
preoperative postoperative
© Stephan Meesters
applications&outlook
![Page 62: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/62.jpg)
preoperative postoperative
© Stephan Meesters
applications&outlook
![Page 63: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/63.jpg)
©DaichiKominami
applications&outlook
![Page 64: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/64.jpg)
Bernhard Riemann Sophus Lie Elie Cartan Albert Einstein Paul Finsler
conclusion
• FinslergeometryisagenericandpotentiallypowerfulframeworkfordiffusionMRIbeyondclassicalDTI
• thisframeworkallowsustoexploitarichbodyofknowledgegainedovermorethanacenturybygreatscientists
…
![Page 65: Diffusion MRI - Australian National Universityusers.cecs.anu.edu.au/~koniusz/tensors-cvpr17/present/...physics in a nutshell: nuclear spin quantization → Zeeman splitting → Boltzmann](https://reader034.fdocuments.net/reader034/viewer/2022042710/5f5b52334671936ede4f5052/html5/thumbnails/65.jpg)