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?

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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.

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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

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ResultsResults

Jacobian of the age indexed deformation.