A Fast Diffeomorphic Image Registration Algorithm John Ashburner 2008.

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A Fast Diffeomorphic Image Registration Algorithm

John Ashburner

Overview

• Parameterization– Scaling and Squaring

• Objective Function

• Optimization

• Group-wise Registration

• Comparison with LDDMM

Julian Huxley (1932). PROBLEMS OF RELATIVE GROWTH

A constant partition of growth intensity between different regions implies constant differences in their rates of growth. Thus any genes controlling relative size of parts will have to exert their action by influencing the rates of processes, …

One essential fact about growth is that it is a process of self-multiplication of living substance – i.e. that the rate of growth of an organism growing equally in all its parts is at any moment proportional to the size of the organism.

Allometry

• Julian Huxley proposed that simpler relationships among lengths, volumes or areas of anatomical structures could be found by working with the logarithms of the measures.

exp(u) = φ(1) = ∫ u φ(t) dt,where φ(0) = 1

Parameterization

DiffeomorphicAnatomicalRegistrationThroughExponentiatedLie Algebra

Deformations parameterized by a single flow field, which is considered to be constant in time.

Not really a proper Lie Group.Often referred to as a one parameter subgroup.

Euler Integration• Parameterising the deformation

• φ(0)(x) = x• φ(1)(x) = ∫ u(φ(t)(x))dt• u is a flow field to be estimated

• Scaling and squaring is used to generate deformations.– c.f. matrix exponentiation

Euler integration

• The differential equation is

dφ(x)/dt = u(φ(t)(x))• By Euler integration

φ(t+h) = φ(t) + hu(φ(t))• Equivalent to

φ(t+h) = (x + hu) o φ(t)

For (e.g) 8 time steps

Simple integration• φ(1/8) = x + u/8• φ(2/8) = φ(1/8) o φ(1/8) • φ(3/8) = φ(1/8) o φ(2/8) • φ(4/8) = φ(1/8) o φ(3/8) • φ(5/8) = φ(1/8) o φ(4/8) • φ(6/8) = φ(1/8) o φ(5/8) • φ(7/8) = φ(1/8) o φ(6/8) • φ(8/8) = φ(1/8) o φ(7/8)

7 compositions

Scaling and squaring• φ(1/8) = x + u/8• φ(2/8) = φ(1/8) o φ(1/8)

• φ(4/8) = φ(2/8) o φ(2/8)

• φ(8/8) = φ(4/8) o φ(4/8)

3 compositions

• Similar procedure used for the inverse.Starts withφ(-1/8) = x - u/8

Scaling and squaring example

Deformations at different times

Jacobians

• Jacobian fields can also be obtained by scaling and squaring.

• If warps are composed by:ϕC=ϕB○ϕA

then Jacobian matrices are obtained by the following matrix multiplication:JϕC=(JϕB○ϕA) JϕA

Jacobian determinants remain positive (almost)

See also…• C. Moler and C. van Loan. “Nineteen Dubious Ways to Compute the

Exponential of a Matrix, Twenty-Five Years Later”. SIAM Review 45(1):3-49 (2003).

• V. Arsigny, O. Commowick, X. Pennec and N. Ayache. “A Log-Euclidean Polyaffine Framework for Locally Rigid or Affine Registration”. Proc. Of the 3rd International Workshop on Biomedical Image Registration (WBIR'06), 2006, pp. 120-127. LNCS vol 4057. Springer-Verlag, Utrecht, NL.

• V. Arsigny, O. Commowick, X. Pennec and N. Ayache. “A Log-Euclidean Framework for Statistics on Diffeomorphisms”. Proc. of the 9th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI'06), 2006, pp. 924-931. LNCS 4190. Springer-Verlag, Berlin, Germany.

• M. Hernandez, M. N. Bossa, and S. Olmos. “Registration of anatomical images using geodesic paths of diffeomorphisms parameterized with stationary vector fields”. IEEE workshop on Math. Meth. in Biom. Image Anal. (MMBIA’07), 2007.

Overview

• Parameterization

• Optimization

• Future directions

Original objective function

Simultaneously minimize the sum of – Likelihood component

• Sum of squares difference

• ½ ∑i∑k(tk(xi) – μk(φ(1)(xi)))2

• φ(1) parameterized by u

– Prior component• A measure of deformation roughness

• ½uTHu

Likelihood Term

• Images assumed to be partitioned into different tissue classes.– E.g., a 3 class registration simultaneously

matches:• Grey matter with grey matter• White matter wit white matter• Background (1 – GM – WM) with background

Prior Term

• ½uTHu• DARTEL has three different models for H

– Membrane energy– Linear elasticity– Bending energy

• H is very sparse

An example H for 2D registration of 6x6 images (linear elasticity)

“Membrane Energy”

Convolution KernelSparse Matrix Representation

Penalises first derivatives.Sum of squares of the elements of the Jacobian (matrices) of the flow field.

“Bending Energy”

Sparse Matrix Representation Convolution Kernel

Penalises second derivatives.

“Linear Elasticity”• Decompose the Jacobian of the flow field into

– Symmetric component• ½(J+JT)• Encodes non-rigid part.

– Anti-symmetric component• ½(J-JT)• Encodes rigid-body part.

• Penalise sum of squaresof symmetric part.

• Trace of Jacobianencodes volume changes.Also penalised.

Regularization models“Membrane energy”

“Bending energy”Images registered using a small deformation approximation

Overview

• Parameterization

• Optimization– Gauss-Newton– Multi-grid

Optimization

• Uses Gauss-Newton– Requires a matrix solution to a very large set

of equations at each iteration

u(k+1) = u(k) - (H+A)-1 b

– b are the first derivatives of objective function– A is a sparse matrix of second derivatives– Computed efficiently, making use of scaling

and squaring

Computing Derivatives

• First derivatives w.r.t. flow field are considered as a vector field when being computed.

• Second derivatives are treated as a field of positive definite symmetric matrices when computed.

• Approximations involved, as a result of multiple interpolations used for the computations.– Derivatives resulting from evolution from time 0 to time 1 are

simply the sum of the derivatives from time 0 to time 0.5 and those from time 0.5 to time 1.

– Derivatives from 0.5 to 1 can be simply computed from those from time 0 to time 0.5.

– Similarly derivatives from time 0 to 0.5 are the sum of derivatives from 0 to 0.25 and 0.25 to 0.5.

– etc

Relaxation

• To solve Mx = cSplit M into E and F, where

• E is easy to invert• F is more difficult

• If M is diagonally dominant (membrane energy):

x(k+1) = E-1(c – F x(k))• Otherwise regularize (bending or linear elastic

energy):

x(k+1) = x(k) + (E+sI)-1(c – M x(k))– Diagonal dominance is when |mii| > Σi≠j |mij|

M = H+A = E+F

2nd derivs of prior term

2nd derivs of likelihood term

Easy to invert

Difficult to invert

Relaxation Strategies• Jacobi’s method if not done in place, such that updates are not

used immediately.– Easier to implement in MATLAB

• Gauss-Siedel when done in place.– Faster convergence and uses less memory.– “Red-black” alternating update scheme is used with membrane energy.

– Sweeps in alternating directions are used for updates when regularization is linear-elastic or bending energy.

• Both methods fit high frequencies quickly, but low frequencies slowly.

Relaxation 128x128

Highest resolution

Lowest resolution

Full Multi-Grid

•Prolongation of low resolution solution to current resolution.•Add this to existing solution.•Perform a few iterations of relaxation.•Restrict residuals down to lower resolution.

•Prolongation of low resolution solution to current resolution.•Add this to existing solution at current resolution.•Perform a few iterations of relaxation.•Prolongation of solution to higher resolution.

•Restrict high resolution residuals to current resolution.•Perform a few iterations of relaxation.•Restrict residuals down to lower resolution.

•Restrict higher resolution residuals to current resolution.•Obtain exact solution by matrix inversion.•Prolongation of solution to higher resolution.

A Fast Diffeomorphic Image Registration Algorithm John Ashburner 2008.

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