Computational Anatomy: Simple Statistics on Interesting Spaces

1. Computational Anatomy: Simple Statistics on Interesting Spaces Sarang Joshi, Tom Fletcher Scientific Computing and Imaging Institute Department of Bioengineering, University of Utah NIH Grant R01EB007688-01A1 Brad Davis, Peter Lorenzen University of North Carolina at Chapel Hill Joan Glaunes and Alain Truouve ENS de Cachan, Paris

2. Motivation: A Natural Question • Given a collection of Anatomical Images what is the Image of the “Average Anatomy”.

3. Motivation: 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”

4. Regression • Given an age index population what are the “average” anatomical changes?

5. Outline • Mathematical Framework • Capturing Geometrical variability via Diffeomorphic transformations. • “Average” estimation via metric minimization: Fréchet Mean. • “Regression” of age indexed anatomical imagery

6. Motivation: A Natural Question Consider two simple images of circles: What is the Average?

7. Motivation: A Natural Question Consider two simple images of circles: What is the Average?

8. Motivation: A Natural Question What is the Average?

9. Motivation: A Natural Question Average considering “Geometric Structure” A circle with “average radius”

10. Mathematical Foundations of Computational Anatomy • 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.

11. 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, DECEMBER1997. • 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.

12. Mathematical Foundations of Computational 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!!

13. Large 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.

14. 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.

15. Relationship 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)

16. 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:

17. Simple Statistics on Interesting Spaces: ‘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.

18. Simple Statistics on Interesting Spaces: ‘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.

19. Simple Statistics on Interesting Spaces: ‘Averaging Anatomies’

20. Averaging Brain Images Initial Images Initial Average Initial Absolute Error Deformed Images Final Average Final Absolute Error

21. Regression • Given an age index population what are the “average” anatomical changes?

22. Regression 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!

23. Kernel 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)

24. Results Regressed Image at Age 35 Regressed Image at Age 55

25. Results • Jacobian of the age indexed deformation.