Gustavo K Rohde Email gustavorcmuedu URL http wwwandrewcmuedu user gustavor Center for Bioimage Informatics Department of Biomedical Engineering Department of Electrical and Computer Engineering ID: 931687
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Slide1
Computational methods for modeling and quantifying shape information of biological forms
Gustavo K. Rohde
Email:
gustavor@cmu.edu
URL: http://www.andrew.cmu.edu/user/gustavor/
Center for Bioimage Informatics, Department of Biomedical Engineering Department of Electrical and Computer Engineering Lane Center for Computational Biology
Slide2Soheil
Kolouri, BME, CMUDejan
Slepcev, Math, CMU
Robert Murphy, CompBio, MLD, BME, CMUMMBioSCenter for Biomage Informatics, CMUNational Institutes of Health P41, GM103712Acknowledgements
Slide3TR&D 3: Topic
Tools for understanding morphological/dynamic information in cells
Today:
focus on cell shapesYin et al, BioEssays, 2014.
Slide4Modeling information in cell images
Method 1:
descriptive
Sailem et al, Open Biology, 2014Method 2: generative models
Rohde et al,
Cytometry, 2008
Slide5Generative models
Parametric
Zhao, Murphy,
Cytometry, 2007Non-parametricRohde,…, Murphy, Cytometry, 2008nuclear shapecell shape
s
hape space
Based on work of Miller et al., Quart. Appl. Math. 1998
Slide6Shapes can be deformed onto one another.
Quantifying the difference between shapes using these deformations:
Goal:
Find the mapping which causes least amount of “bending.”
Deformation-based shape distances
Slide7Previous work:
based on the “large deformation diffeomorphic
metric mapping” (LDDMM) work of Miller et al, JHU.
Improvements described here:Robustness, more difficult shapesMake the method faster Provide options regarding: types of differences to measureallowing for different regions have different penaltiesdistance
Shape distance definition:
Slide8Outline of the method
Preprocessing:
Morphological analysisCenteringRotation Initial mapping:Multi-resolution MSE basedSmooth diffeomorphic mappingEnergy minimization:Symmetric energy functionPhysically meaningfulGradient descent
Energy
Gradient descent iteration
Slide9Robust smooth invertible initial mapping:Penalized MSE:
Solved by a multi-resolution gradient descent approach.
Interpolate &
Multiply by 2
Smooth initial invertible mapping
Slide10Elapsed time= 3.96 sec
Smooth initial invertible mapping
Slide11Physically meaningful energy terms
From continuum mechanics, the strain rate tensor, E, is defined as:
Based on f and E, we propose the following energy terms:
Viscous friction:Volume change:Total mass transport: Compression:
Slide12The symmetric energy terms are defined as,
We avoid the calculation of the inverse map, g, by rewriting the inverse energy terms as a function of the forward mapping. For instance,
Symmetrizing the energy terms
where, n=2,3 is the dimension of the problem and is the
k’th
order symmetric polynomial of the eigenvalues of the Jacobian matrix.
Slide13Symmetric energy terms
Slide14For two shape images, the symmetrized similarity measure is defined as
Where are the mixing coefficients of energy terms.
Measurement calibration:The mixing coefficients are tuned to incorporate the relative importance of each term:
User inputNormalization with respect to dataSymmetrized similarity measure
Slide15With a
non symmetric
energy, computations are order dependent.
ForwardBackwardWhy symmetry?
Slide16With a
symmetric
energy, same answer either way.
ForwardBackwardWhy symmetry?
Slide17Sample result
Slide18Sample result
Slide19Comparison with LDDMM
Large errors
Slide20Current & Future work
Finish 3D implementation
Finish testing with sample applications
Integrate into our CellOrganizerMove on to transport-based distances for densities (e.g. proteins)Microscopy images3D cell model
CellOrganizer.org
Slide21Thank you