Computational Anatomy & Statistical Shape Models John Ashburner firstname.lastname@example.org Functional Imaging Lab, 12 Queen Square, London, UK. Why?. The Wellcome Trust is keen that there is a translational component to the work in the FIL. E.g. develop some potentially useful diagnostic stuff.
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Mathematics in Brain Imaging
Edited by P.M. Thompson, M.I. Miller, T. Ratnanather, R.A. Poldrack and T.E. Nichols
Computational anatomy: shape, growth, and atrophy comparison via diffeomorphismsMichael I. Miller
a is a weighted linear combination of the support vectors
y = f(Si wi(xi,x))
Small deformation: displacements are linear within the Eulerian framework
Small def. approx. to backward transform
Small def. approx. to forward transform
C = log (R-1S)
and then computing the RMS of C.
(12+ 22 + 32)1/2
A = (A+AT)/2 + (A-AT)/2
Diffeomorphisms have curved trajectories (variable velocity) if followed in the Eulerian reference frame (fixed).
If followed within the Lagrangian frame (moves over time), they appear to have constant velocity.
Model one image as it deforms to match another.
x(t) = u(x(t))
x(1) = eu (x(0))
x(1) = eUx(0)
x(0) = e-Ux(1)
For large k
eU ≈ (I+U/k)k
Large deformations generated from compositions of small deformations
S1 = S1/8oS1/8oS1/8oS1/8oS1/8oS1/8oS1/8oS1/8
S1 = S1/2oS1/2,S1/2 = S1/4oS1/4,S1/4 = S1/8oS1/8
S1/8 ≈ I + U/8
Register to a mean shaped image
Totally impractical for lots of scans
Problem: How can the distance between e.g. A and B be computed? Inverse exponentiating is iterative and slow.
Sometimes unstable. Looks like proper nonlinear methods would be impractical.
Average of 452 images
2D average of 471 images
Registration of each 2D image takes about 3 seconds per iteration, and about 16 iterations. I see no problems scaling it to 3D.
Only affine registered
A simpler model can often do better...
Use half the data for training.
and the other half for testing.
Then swap around the training and test data.
Use all data except one point for training.
The one that was left out is used for testing.
Then leave another point out.
And so on...
AoB remains in the same group.
(AoB)oC = Ao(BoC)
Identity transform I exists.
A-1 exists, and A-1oA=AoA-1 = I
It is a Lie Group.
The group of diffeomorphisms constitute a smooth manifold.
The operations are differentiable.Group Theory
E.g. SO(2) : Special Orthogonal 2D (rigid-body rotation in 2D).
Manifold is a circle
Lie Algebra is exponentiated to give Lie group. For square matrices, this involves a matrix exponential.Lie Groups
Velocities are the Lie Algebra. These are exponentiated to a deformation by recursive application of tiny displacements, over a period of time=0..1.
A(1) = A(1/2)oA(1/2)
A(1/2) = A(1/4) oA(1/4)
Don’t actually use matrices.
For tiny deformations, things are almost linear.
x(1/1024)x(0) + vx/1024
y(1/1024)y(0) + vy/1024
z(1/1024)z(0) + vz/1024
Recursive application by
x(1/2) = x(1/4) (x(1/4), y(1/4),z(1/4))
y(1/2) = y(1/4) (x(1/4), y(1/4),z(1/4))
z(1/2) = z(1/4) (x(1/4), y(1/4),z(1/4))Relevance to Diffeomorphisms
Distances on the manifold are given by geodesics.
Average of a number of deformations is a point on the manifold with the shortest sum of squared geodesic distances.
E.g. average position of London, Sydney and Honolulu.
Negate the velocities, and exponentiate.
x(1/1024)x(0) - vx/1024
y(1/1024)y(0) - vy/1024
z(1/1024)z(0) - vz/1024
Priors for registration
Based on smoothness of the velocities.
Velocities relate to distances from origin.Working with Diffeomorphisms