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Sarang Joshi #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

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Sarang Joshi #2 Motivation: A Natural Question Given a collection of Anatomical Images what is the Image of the Average Anatomy.

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Sarang Joshi #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

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Sarang Joshi #4Results

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Sarang Joshi #5Regression Given an age index population what are the average anatomical changes?

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Sarang Joshi #6Outline Mathematical Framework –Capturing Geometrical variability via Diffeomorphic transformations. Average estimation via metric minimization: Fréchet Mean. Regression of age indexed anatomical imagery

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

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Sarang Joshi #8 Motivation: A Natural Question What is the Average? Consider two simple images of circles:

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Sarang Joshi #9 Motivation: A Natural Question What is the Average?

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Sarang Joshi #10 Motivation: A Natural Question Average considering Geometric Structure A circle with average radius

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Sarang Joshi #11 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.

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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.

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Sarang Joshi #13 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!!

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Sarang Joshi #14 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.

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Sarang Joshi #15 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.

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Sarang Joshi #16 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)

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Sarang Joshi #17 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:

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Sarang Joshi #18 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.

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Sarang Joshi #19 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.

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Sarang Joshi #20 Simple Statistics on Interesting Spaces: Averaging Anatomies

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Sarang Joshi #21 Averaging of 16 Bulls eye images

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Sarang Joshi #22 Initial Images Initial Absolute Error Deformed Images Final Absolute Error Final Average Initial Average Averaging Brain Images

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Sarang Joshi #23Regression Given an age index population what are the average anatomical changes?

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Sarang Joshi #24 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!

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Sarang Joshi #25 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)

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Sarang Joshi #26 Results Regressed Image at Age 35 Regressed Image at Age 55

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Sarang Joshi #27Results Jacobian of the age indexed deformation.

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Sarang Joshi #28 Next: Bandwidth Selection and Significance of Trend Results of 500, 000 random permutations. (d) The probability of the null hypothesis (x and y are independent) as a function of kernel bandwidth.

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