Upload
alexandra-parsons
View
215
Download
2
Tags:
Embed Size (px)
Citation preview
Sarang Joshi #1
Computational Anatomy: Simple Statistics Computational Anatomy: Simple Statistics on Interesting Spaceson Interesting Spaces
Sarang Joshi, Tom FletcherScientific Computing and Imaging Institute
Department of Bioengineering, University of UtahNIH Grant R01EB007688-01A1
Brad Davis, Peter Lorenzen University of North Carolina at Chapel Hill
Joan Glaunes and Alain TruouveENS de Cachan, Paris
Sarang Joshi #2
Motivation: A Natural QuestionMotivation: A Natural Question
Given a collection of Anatomical Images what is the Image of the “Average Anatomy”.
Sarang Joshi #3
Motivation: A Natural QuestionMotivation: A Natural Question
Given a set of Surfaces what is the “Average Surface”
Given a set of unlabeled Landmarks points what is the “Average Landmark Configuration”
Sarang Joshi #5
Regression Regression Given an age index
population what are the “average” anatomical changes?
Sarang Joshi #6
OutlineOutline
Mathematical Framework– Capturing Geometrical variability via Diffeomorphic transformations.
“Average” estimation via metric minimization: Fréchet Mean. “Regression” of age indexed anatomical imagery
Sarang Joshi #7
Motivation: A Natural QuestionMotivation: A Natural Question
What is the Average?
Consider two simple images of circles:
Sarang Joshi #8
Motivation: A Natural QuestionMotivation: A Natural Question
What is the Average?
Consider two simple images of circles:
Sarang Joshi #9
Motivation: A Natural QuestionMotivation: A Natural Question
What is the Average?
Sarang Joshi #10
Motivation: A Natural QuestionMotivation: A Natural Question
Average considering “Geometric Structure”
A circle with “average radius”
Sarang Joshi #11
Mathematical Foundations of Computational Mathematical Foundations of Computational AnatomyAnatomy
Structural variation with in a population represented by transformation groups: – For circles simple multiplicative group of positive reals
(R+)– Scale and Orientation: Finite dimensional Lie Groups
such as Rotations, Similarity and Affine Transforms.– High dimensional anatomical structural variation:
Infinite dimensional Group of Diffeomorphisms.
Sarang Joshi #12
•G. E. Christensen, S. C. Joshi and M. I. Miller, "Volumetric Transformation of Brain Anatomy," IEEE Transactions on Medical Imaging, volume 16, pp. 864-877, DECEMBER 1997. •S. C. Joshi and M. I. Miller, “Landmark Matching Via Large Deformation Diffeomorphisms”, IEEE Transactions on Image Processing, Volume 9 no 8,PP.1357-1370, August 2000.
Sarang Joshi #13
Mathematical Foundations of Mathematical Foundations of Computational AnatomyComputational Anatomy
transformations constructed from the group of diffeomorphisms of the underlying coordinate system– Diffeomorphisms: one-to-one onto (invertible) and differential
transformations. Preserve topology.
Anatomical variability understood via transformations – Traditional approach: Given a family of images
construct “registration” transformations that map all the images to a single template image or the Atlas.
How can we define an “Average anatomy” in this framework: The template estimation problem!!
Sarang Joshi #14
Large deformation diffeomorphismsLarge deformation diffeomorphisms
Space of all Diffeomorphisms forms a group under composition:
Space of diffeomorphisms not a vector space.
Small deformations, or “Linear Elastic” registration approaches ignore this.
)(Diff
)(:)(, 2121 DiffhhhDiffhh
)(:)(, 2121 DiffhhhDiffhh
Sarang Joshi #15
Large deformation diffeomorphisms.Large deformation diffeomorphisms.
infinite dimensional “Lie Group”. Tangent space: The space of smooth vector valued
velocity fields on . Construct deformations by integrating flows of velocity
fields.
Sarang Joshi #16
Relationship to Fluid DeformationsRelationship to Fluid Deformations
Newtonian fluid flows generate diffeomorphisms: John P. Heller "An Unmixing Demonstration," American Journal of Physics, 28, 348-353 (1960).
For a complete mathematical treatment see:
– Mathematical methods of classical mechanics, by Vladimir Arnold (Springer)
Sarang Joshi #17
Metric on the Group of Diffeomorphisms:Metric on the Group of Diffeomorphisms:
Induce a metric via a sobolev norm on the velocity fields. Distance defined as the length of geodesics under this norm.
Distance between e, the identity and any diffeomorphis is defined via the geodesic equation:
Right invariant distance between any two diffeomorphisms is defined as:
Sarang Joshi #18
Simple Statistics on Interesting Spaces: Simple Statistics on Interesting Spaces: ‘Average Anatomical Image’‘Average Anatomical Image’
Given N images use the notion of Fréchet mean to define the “Average Anatomical” image.
The “Average Anatomical” image: The image that minimizes the mean squared metric on the semi-direct product space.
Sarang Joshi #19
Simple Statistics on Interesting Spaces: Simple Statistics on Interesting Spaces: ‘Averaging Anatomies’‘Averaging Anatomies’
The average anatomical image is the Image that requires “Least Energy for each of the Images to deform and match to it”:
•Can be implemented by a relatively efficient alternating algorithm.
Sarang Joshi #20
Simple Statistics on Interesting Spaces: Simple Statistics on Interesting Spaces: ‘Averaging Anatomies’‘Averaging Anatomies’
Sarang Joshi #22
Initial Images
Initial Absolute Error
Deformed Images
Final Absolute Error Final Average
Initial Average
Averaging Brain ImagesAveraging Brain Images
Sarang Joshi #23
Regression Regression Given an age index
population what are the “average” anatomical changes?
Sarang Joshi #24
Regression analysis on ManifoldsRegression analysis on Manifolds
Given a set of observation where
Estimate function
An estimator is defined as the conditional expectation.– Nadaraya-Watson estimator: Moving weighted average, weighted
by a kernel.
Replace simple moving weighted average by weighted Fréchet mean!
Sarang Joshi #25
Kernel regression on Riemannien manifoldsKernel regression on Riemannien manifolds
B. C. Davis, P. T. Fletcher, E. Bullitt and S. Joshi, "Population Shape Regression From Random Design Data", IEEE International Conference on Computer Vision, ICCV, 2007. (Winner of David Marr Prize for Best Paper)
Sarang Joshi #26
Results
Regressed Image at Age 35
Regressed Image at Age 55
Sarang Joshi #27
ResultsResults
Jacobian of the age indexed deformation.