Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors

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Principal Geodesic Analysis on Symmetric Spaces
Statistics of Diffusion Tensors

• P. Thomas Fletcher and Sarang Joshi
• In proceedings of Computer Vision Approaches of
  Medical Image Analysis 2004 (CVAMIA)

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Outline

• DT-MRI
• Geometry of Diffusion Tensors
• Lie Group Actions
• Invariant Metrics
• Computing Geodesics
• Statistics of Diffusion Tensors
• Averages of Diffusion Tensors
• Principal Geodesic Analysis

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Diffusion Tensor Magnetic Resonance Imaging
(DT-MRI)

• Within biological systems water molecules follow
  stochastic Brownian motion.
• In some tissues the rate is anisotropic (faster
  in some directions than others).
• A 2D diffusion tensor image (DTI) and an
  associated anatomical scalar field, created
  during the tensor calculation, define seven
  values at each voxel.

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Diffusion Tensor Magnetic Resonance Imaging
(DT-MRI)

• 3x3 matrix
• Symmetric, AAT
• Positive-definite matrix, AMMT, where M is
  non-singular (M?0),
• AMMTM2gt0
• aiigt0, for all i
• We define as P(n), the space of all nxn
  symmetric, positive-definite matrices
• DT P(3)

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Diffusion Tensor Magnetic Resonance Imaging
(DT-MRI)
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Diffusion Tensor Magnetic Resonance Imaging
(DT-MRI)
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DT geometry

• P(n) forms a convex subset or Rnxn
• i.e. if A,B ? P(n), CA?(B-A) ? P(n), 0lt?lt1
• .
• Linear averages do not interpolate natural
  properties
• Straight lines do not stay within P(n)
• If A ? P(n) gt -A is not positive-definite

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DT geometry
A ? P(2)
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Useful Definitions

• A manifold is a topological space that is locally
  Euclidean.
• Riemannian manifold is a manifold possessing a
  symmetric and positive-definite metric. For a
  complete Riemannian manifold, the metric d(x,y)
  is defined as the length of the shortest curve
  (geodesic) between x and y.

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Riemannian symmetric space

• A Riemannian symmetric space is a connected
  Riemannian manifold M such that for each x ? M
  there is an isometry (map that preserves
  distances) sx which
• is involutive, i.e. sx(sxx)x gt sx2 id
• and has x as an isolated fixed point, that is,
  there is a neighborhood U of x where sx leaves
  only x fixed.

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Lie Group Actions

• It is an algebraic group that also forms a smooth
  manifold, where multiplication and inversion are
  smooth mappings.
• Smooth mapping
• G Lie Group Action, M Manifold
• f(e,x)x and f(g,f(h,x))f(gh,x)
• g,h ? G and x ? M

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Lie Group Actions

• GL(n) group of all nxn real matrices with
  positive determinant.
• Orbit
• P(n) is homogeneous space single orbit

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Lie Group Actions

• Isotropy subgroup of x
• For P(n) isotropy group of In is
• connected compact

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Invariant Metric

• We need a metric
• for P(n) that it is an isometry for each g ? G
• (called G-invariant)
• The space of DT, P(n) has a metric that is
  invariant under the GL(n) action.
• A Riemannian metric on a manifold M assigns to
  each point p ? M, an inner product lt,gtx on TxM

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Invariant Metric

• The tangent space of P(n) at In, TInPn is the
  space of n x n symetric matrices, Sym(n)
• The tangent space at any point p ? P(n) is also
  Sym(n)

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Invariant Metric

• If X, Y ? Sym(n) tangent vectors
• at p ? P(n), where pggT , g ? GL(n),
• the Riemannian metric at p is

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Computing Geodesics

• If p ? P(n) and X a tangent vector at p there is
  unique geodesic ?,
• ?(0)p , ?(0)X
• If pIn and X diagonal then

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Computing Geodesics

• If arbitrary p ? P(n) and X ? Sym(n)
• pggT , g ? GL(n)
• We can write Y?S?T, ? rotation matrix
• S diagonal

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Computing Geodesics

• So generally the geodesic is obtained by
• For t1 we get the Riemannian exponential at p

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Averages of DT
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Averages of DT

• Input p1,,pN ? P(n)
• Output µ ? P(n), the intrinsic mean
• µ0I
• Do
• While Xigt e

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Variance of DT

• Following similar reasoning we define

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Geodesic submanifolds

• Line Geodesic
• Submanifold H of M is geodesic at x ? H if all
  geodesics of H passing through x are also
  geodesics of M.
• Submanifolds geodesic at x preserves distances to
  x

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Projections

• Projection of x ? M onto a geodesic sub-manifold
  H of M
• If v1, , vk is an orthonormal basis for TµH

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Principal Geodesic Analysis
PCA
PGA
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Principal Geodesic Analysis

• Properties of PGA on P(n)
• Positive-definiteness
• Determinant
• and Orientation

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Summary

• DT-MRI
• Geometry of Diffusion Tensors
• Lie Group Actions
• Invariant Metrics
• Computing Geodesics
• Statistics of Diffusion Tensors
• Averages of Diffusion Tensors
• Principal Geodesic Analysis