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ShapeModelinginCellOrganizerRobertF.Murphy
Ray & Stephanie Lane Professor of Computational Biology and Professor of Biological Sciences, Biomedical Engineering and Machine Learning
External Senior Fellow, Freiburg Institute for Advanced Studies Honorary Professor, Faculty of Biology, University of Freiburg, Germany
AnNIHBiomedicalTechnologyResearchCenter
March9,2018
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Methodsformodelingcellshape• Parametric
– Outline– Ratio-metricrelativetonuclearshape
• Nonparametric– Diffeomorphic– Autoencoder
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Parametricshapeoutlinemodels
Kerenetal.
2008
Imagessh
owing
realsh
apes
Gene
rativ
e
shapemod
el
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Shapespace
Kerenetal.2008
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Limitationsofcommonoutlinemodel
Srivastavaet
al.2005
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Cell shape: eigenshape ratio model
• Conditionedonnuclearshape:– Sampleevenlyaroundacircletorepresenttheshapebyradiusratios
– Parameterization
– Keep10principalcomponents(2D)– Keep25principalcomponents(3D)
1 2/r d d=
∑ =+≈
k
i iib1 φrr d1 d2
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However…• Cellsdon’talwayssatisfyassumptionsofparametricmodels.
SegmentedPC12cell Star-polygonratiomodelrepresentation
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LDDMM-LargeDeformationDiffeomorphicMetricMapping
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Whatisadiffeomorphism?• Essentiallyasmoothandinvertiblemappingfromonecoordinatespacetoanother
Adiffeomorphicmappingfromaregularrectangulargrid.
https://en.wikipedia.org/wiki/Diffeomorphism
Diffeomorphicmappingsofcontinentstoa2Dprojectionofaglobe
http://wwwx.cs.unc.edu/~mn/classes/comp875/doc/diffeomorphisms.pdf
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Adiffeomorphicmappingfromoneimagetoanother.http://wwwx.cs.unc.edu/~mn/classes/comp875/doc/diffeomorphisms.pdf
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Nonparametricshapeimage-basedmodels
Pengetal.2009
Real2Dnuclearshapes
http://alumni.media.mit.edu/~maov/classes/comp_photo_vision08f/
Cannotjustinterpolateimagesasiftheywerevectors
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Morphingtointerpolateimages
12http://alumni.media.mit.edu/~maov/classes/comp_photo_vision08f/
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ShapeBShapeAWorksofar
0.0191
Distancebetweentwoshapes
0.01650 0.0194 0.0195
Distance
Pengeta
l.2009
0.01650 0.0191 0.0194 0.0195
Distance
IterativereductionindifferencebetweendeformedshapeAandB
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LDDMM-LargeDeformationDiffeomorphicMetricMapping
• Minimalenergytransformationwithrespecttothegradientofthedeformationfieldi.e.Geodesicdistance
• Deformationfieldisanonlinearmanifoldthatcontainstheinformationoftheimage,includinggradient,secondorderderivationetc.Theupdateisageodesicpathonthemanifold.
Shadel1974 http://wwwx.cs.unc.edu/~mn/classes/comp875/doc/diffeomorphisms.pdf
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f1.png
0 0.2 0.4 0.6 0.8 1
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Constructingacellshapespace• Finddistancesofeverycelltoeveryothercell
• Trytofinda“map”thatputseachcellthecorrectdistancefromtheothers(i.e.,putscellswithshortdistancesneareachother)
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Distancematrix…BOS CHI DC DEN LA MIA NY SEA SF
BOS 0 963 429 1949 2979 1504 206 2976 3095CHI 963 0 671 996 2054 1329 802 2013 2142DC 429 671 0 1616 2631 1075 233 2684 2799DEN 1949 996 1616 0 1059 2037 1771 1307 1235LA 2979 2054 2631 1059 0 2687 2786 1131 379MIA 1504 1329 1075 2037 2687 0 1308 3273 3053NY 206 802 233 1771 2786 1308 0 2815 2934SEA 2976 2013 2684 1307 1131 3273 2815 0 808SF 3095 2142 2799 1235 379 3053 2934 808 0
http://personality-project.org/r/mds.html
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…tocoordinates
http://personality-project.org/r/mds.html
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Shapespacesmodeljointdistributionacrossmorphologicalfeatures
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Shapespace
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DiffeomorphicTraining
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ShapestoSpace
MDS
Butthistakesalotoftime
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PartialDistanceMatrixLearning
• Mostcompleteshapespace
MDS
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ShapeSpaceEmbedding• Thisembeddingmethodrequiresallpairsofdistances• Let’ssaywehave250cellsandittakes~30sectoregisterapair• (30sec*2502)/2≈10days.Waytoolong…..• canweinferembeddingwithmissingdata?• MDSwithmissingdata
wherewi,jistheweightofobservationDi,j
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PartialDistanceMatrixLearning• LandmarkMDS
?
ApproximateMDS
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DiffeomorphicSynthesis
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SpacetoShapes
?
Synthesisstrategyfornewpoints
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Modelingthedistributionofshapes• Theshapespacedefinesanimplicitprobabilitydensity.
x
Nonparametricdensityestimation
p(x)=1/vin
Modelingdistributionofshapes–p(x)
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Modelingthedistributionofshapes• Theshapespacedefinesanimplicitprobabilitydensity.
x
Nonparametricdensityestimation
p(x)=1/vin
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ParametricRepresentationGaussianmixturemodel2components
Modelingdistributionofshapes–p(x)n=1 n=2
n=3 n=4
ShapespacemodeledasaGaussianMixtureModel
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Diffeomorphicspace• Newfeaturespace
– Positionsinspacecorrespondtoarealimage– Featuredimensionscorrespondtodimensionswithhighesteigenvalues
– Canbetreatedexactlylikeanormalfeaturespace
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HeLashapespacewithDNAintensity
Component1(R2=0.04)
Compo
nent2(R
2 =0.08)
Component3(R2=0.57)
DNAintensity
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Docellandnuclearshapedependoneachother?
• Buildshapespacesfor– nuclearshapesonly– cellshapesonly
• Foreachnuclearshape,predictapositionincellshapespaceforitbyinterpolatingatthesamerelativedistancesfromthecellshapesofitsneighborsinnuclearshape(andviceversa)
• Measurepredictionerrorasthedistanceintheshapespacebetweenthepredictedpositionandtheactualposition
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Predictionofcellandnuclearshapedependency
John
sonetal.
2015
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H1299shapespace,coloredbyproteinlabel
Johnsonetal.2015