Computational Anatomy: Simple Statistics on Interesting Spaces

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Computational Anatomy Simple Statistics on
Interesting Spaces

• Sarang Joshi,
• Scientific Computing and Imaging Institute
• Department of Bioengineering, University of Utah
• Brad Davis, Peter Lorenzen
• University of North Carolina at Chapel Hill
• Joan Glaunes and Alain Truouve
• ENS de Cachan, Paris

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Motivation A Natural Question

• Given a collection of Anatomical Images what is
  the Image of the Average Anatomy.

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Motivation A Natural Question

• Given a set of unlabeled Landmarks points what is
  the Average Landmark Configuration

• Given a set of Surfaces what is the Average
  Surface

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Regression

• Given an age index population what are the
  average anatomical changes?

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Outline

• Mathematical Framework
• Capturing Geometrical variability via
  Diffeomorphic transformations.
• Average estimation via metric minimization
  Fréchet Mean.
• Averaging Images.
• Averaging Point sets
• Weak norm on signed dirac measures.
• Averaging Curves and Surfaces
• Weak norm on dirac currents
• Regression of age indexed anatomical imagery

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Motivation A Natural Question
Consider two simple images of circles
What is the Average?
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Motivation A Natural Question
Consider two simple images of circles
What is the Average?
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Motivation A Natural Question
What is the Average?
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Motivation A Natural Question
Average considering Geometric Structure
A circle with average radius
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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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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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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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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.
• Induce a metric via a differential norm on
  velocity fields.

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Space of Images and Anatomical Structure

• Images as function of a underlying coordinate
  space
• Image intensities
• Space of structural transformations
  diffeomorphisms of the underlying coordinate
  space
• Space of Images and Transformations a semi-direct
  product of the two spaces.

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Simple Statistics on Interesting Spaces Average
Anatomical Image

• 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.
• Metamorphoses Through Lie Group Action. Alain
  Trouvé, Laurent Younes Foundations of
  Computational Mathematics 5(2) 173-198 (2005) .

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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
  diffeomorphism j is defined via the geodesic
  equation
• Right invariant distance between any two
  diffeomorphisms is defined as

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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

• Warning Not Consistent for large noise.
• For reasonable noise in practice very stable.
  (Mode approximation holds).

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Simple Statistics on Interesting Spaces
Averaging Anatomies
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Alternating Algorithm

• If the transformations are fixed than the average
  image is simply the average of the deformed
  images!!
• Alternate until convergence between estimating
  the average and the transformations.

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Results Sample of 16 Bulls eye Images
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Averaging of 16 Bulls eye images
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Works for reasonable noise
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Averaging of 16 Bulls eye images
Voxel Averaging
LDMM Averaging
Numerical geometric average of the radii
of the individual circles forming the bulls eye
sample.
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Averaging Brain Images
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Averaging Unlabeled Point Sets

• Given a set of unlabeled Landmarks points what is
  the Average Landmark Configuration

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Averaging Unlabeled Point Sets

• Given a set of Unlabelled weighted point sets
• we define an Average point set in the Large
  Deformation Fréchet sense by
• Defining a metric between two Unlabelled point
  sets Diffeomorphic matching of distributions A
  new approach for unlabelled point-sets and
  sub-manifolds matching Glaunes, et. al.
• Analogues to Image averaging, using the metric
  defined previously on the space to diffeomorphic
  transformations to define the Average via a
  minimization problem.

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Unlabelled Point sets as signed measures
(Glaunes et al.)

• Treat a set of points in as collection of
  signed dirac delta measures.
• Space of signed measures is a vector space.
• Let be the measure associated with Point
  set
• Action of diffeomorphism defined via
  integration of measurable functions
• is the measure associated with the
  transformed point set

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Weak RKHS norm on signed measures.

• Induce a week norm on signed measures via a
  reproducing kernel Hilbert space structure with a
  kernel k on the dual space The space of bounded
  continuous functions
• If than after
  a simple calculation

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Averaging Unlabeled Point Sets

• Give a collection of unlabeled point sets
  ,j 1,,N
• Let be the measure associated with point
  set
• Let be the action of the diffeomorphic
  transformation on
• The average point set estimation problem
  becomes
• If the transformations are fixed than the optimal
  measure is given by

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Results
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Results
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Representing Curves and Surfaces via Currents
(Glaunes et al.)

• The space of currents is the dual space of
  differential forms with compact support.
• Generalization of Schwarz distributions (0
  dimensional currents).
• Treat discretized curves and surfaces as Dirac
  currents.

Electrical Engineering perspective If the
dimension or the co-dimension is 1 then Vector
weighted Dirac deltas
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Averaging Curves and Surfaces via Currents
(Glaunes et al.)

• Action of diffeomorphism on the current is
  the current of the transformed surface

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Representing Curves and Surfaces via Currents
(Glaunes et al.)

• Space of all currents forms a vector space
• Let be a current associated with a surface
  then
• is the current associated with the same
  surface with the opposite orientation.
• Induce a week norm on currents via a Reproducing
  Kernel Hilbert Space structure, with a kernel K,
  on the dual space.
• If than after
  a simple calculation
• K a matrix kernel. If
  ,then

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Averaging Curves and Surfaces

• Give a collection of curves or surfaces ,j
  1,,N
• Let be the current associated with
• Let be the action of the diffeomorphic
  transformation on
• The Average estimation problem becomes
• If the transformations are fixed than the optimal
  current is given by

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Results
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Regression analysis (Review of Kernel Regression)

• Given a set of observation where
• Estimate function
• An estimator defined as a conditional
  expectation
• Nadaraya-Watson kernel regression replaces the
  unknown densities via a kernel densities of
  bandwidths g and h

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Regression analysis (Review of Kernel Regression)

• Finally, assuming that the kernels is symmetric
  about the origin, integration of the numerator
  leads to
• Weighted average weighted by a kernel.

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Kernel regression on Riemannien manifolds

• Replace conditional expectation by Fréchet mean!

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Results
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Diffeomorphic growth model

• Now given a dense regressed image estimate a
  real time indexed deformation to quantify the
  shape changes

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Results

• Jacobian of the age indexed deformation.