Presentation on theme: "A Fast Diffeomorphic Image Registration Algorithm"— Presentation transcript:
1A Fast Diffeomorphic Image Registration Algorithm John Ashburner2008
2Overview Parameterization Objective Function Optimization Scaling and SquaringObjective FunctionOptimizationGroup-wise RegistrationComparison with LDDMM
3Julian Huxley (1932). PROBLEMS OF RELATIVE GROWTH 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.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, …
4exp(u) = φ(1) = ∫ u φ(t) dt, where φ(0) = 1 AllometryJulian 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) = 11t=0
5Parameterization Diffeomorphic Anatomical Registration Through ExponentiatedLie AlgebraDeformations 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.
6Euler Integration φ(0)(x) = x φ(1)(x) = ∫ u(φ(t)(x))dt Parameterising the deformationφ(0)(x) = xφ(1)(x) = ∫ u(φ(t)(x))dtu is a flow field to be estimatedScaling and squaring is used to generate deformations.c.f. matrix exponentiation1t=0
8For (e.g) 8 time steps Simple integration φ(1/8) = x + u/8 7 compositionsScaling and squaringφ(1/8) = x + u/8φ(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 compositionsSimilar procedure used for the inverse.Starts withφ(-1/8) = x - u/8
13See 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 LNCS vol 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 LNCS 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.
14Overview Parameterization Objective Function Optimization Group-wise RegistrationFuture directions
15Original objective function Simultaneously minimize the sum ofLikelihood componentSum of squares difference½ ∑i∑k(tk(xi) – μk(φ(1)(xi)))2φ(1) parameterized by uPrior componentA measure of deformation roughness½uTHu
16Likelihood TermImages assumed to be partitioned into different tissue classes.E.g., a 3 class registration simultaneously matches:Grey matter with grey matterWhite matter wit white matterBackground (1 – GM – WM) with background
17Prior Term ½uTHu DARTEL has three different models for H Membrane energyLinear elasticityBending energyH is very sparseAn example H for 2D registration of 6x6 images (linear elasticity)
18“Membrane Energy” Penalises first derivatives. Sum of squares of the elements of the Jacobian (matrices) of the flow field.Sparse Matrix RepresentationConvolution Kernel
19“Bending Energy” Penalises second derivatives. Sparse Matrix RepresentationConvolution Kernel
20“Linear Elasticity” Decompose the Jacobian of the flow field into Symmetric component½(J+JT)Encodes non-rigid part.Anti-symmetric component½(J-JT)Encodes rigid-body part.Penalise sum of squares of symmetric part.Trace of Jacobian encodes volume changes. Also penalised.
21Regularization models “Membrane energy”Images registered using a small deformation approximation“Bending energy”
22Overview Parameterization Objective Function Optimization Gauss-NewtonMulti-gridGroup-wise RegistrationComparison with LDDMM
23Optimization Uses Gauss-Newton Requires a matrix solution to a very large set of equations at each iterationu(k+1) = u(k) - (H+A)-1 bb are the first derivatives of objective functionA is a sparse matrix of second derivativesComputed efficiently, making use of scaling and squaring
24Computing 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
25Relaxation To solve Mx = c Split M into E and F, whereE is easy to invertF is more difficultIf 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|
26M = H+A = E+F 2nd derivs of prior term 2nd derivs of likelihood term Easy to invertDifficult to invert
27Relaxation Strategies Jacobi’s method if not done in place, such that updates are not used immediately.Easier to implement in MATLABGauss-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.
30A 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.
31B 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.
32C Restrict high resolution residuals to current resolution. Perform a few iterations of relaxation.Restrict residuals down to lower resolution.
33E Restrict higher resolution residuals to current resolution. Obtain exact solution by matrix inversion.Prolongation of solution to higher resolution.
34See also…W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery. Numerical Recipes in C (Second Edition). Cambridge University Press, Cambridge, UKChapter 15, Section 5 explains Gauss-Newton optimization (Levenberg-Marquardt without the regularisation).Chapter 19, Section 6 explains the basics of multi-grid methods.
35Overview Principles Objective Function Optimization Group-wise RegistrationMultinomial distributionsSimultaneous registration of GM & WMTissue probability map creationComparison with LDDMM
36“Average Shaped” Template For CA, work in the tangent space of the manifold, using linear approximations.Average-shaped templates give less bias, as the tangent-space at this point is a closer approximation.For spatial normalisation of fMRI, warping to a more average shaped template is less likely to cause signal to disappear.If a structure is very small in the template, then it will be very small in the spatially normalised individuals.Smaller deformations are needed to match with an average-shaped template.Smaller errors.
37Average shaped templates Average on Riemannian manifoldLinear Average(Not on Riemannian manifold)
38Template Generation Initial Average Iteratively generated from 471 subjects.Began with rigidly aligned tissue probability maps.Regularization lighter for later iterations.After a few iterationsFinal template
39Multinomial Model log p(t|μ,ϕ) = ΣjΣk tjk log(μk(ϕj)) The extended model is multinomial for matching tissue class images.log p(t|μ,ϕ) = ΣjΣk tjk log(μk(ϕj))t – individual GM, WM and backgroundμ – template GM, WM and backgroundϕ – deformationA general purpose template should not have regions where log(μ) is –Inf.
42μk(x) = exp(ak(x))/(Σj exp(aj(x))) Template modelled as softmax of a Gaussian processμk(x) = exp(ak(x))/(Σj exp(aj(x)))MAP solution determined for a, by Gauss-Newton optimisation, using multi-grid.
43Determining amount of regularisation Matrices too big for Bayesian variance component estimation.Used cross-validation.Smooth an image by different amounts, see how well it predicts other images:Nonlinear registeredRigidly alignedlog p(t|μ) = ΣjΣk tjk log(μjk)
44ML and MAP templates from 6 subjects Nonlinearly RegisteredRigidly registeredMLMAPlog
54Overview Principles Objective Function Optimization Group-wise RegistrationComparison with LDDMM
55Slight ProblemThe DARTEL framework is a parameterisation over the background space through which the deformation evolves.Each part of the flow field is not uniquely associated with one part of the brain.Not an ideal representation for morphometry.It is not able to represent the full group of diffeomorphic deformations.
58Variable velocity framework (as in LDDMM) “Initial momentum”
59Variable velocity framework (as in LDDMM) “Initial momentum”
60Variable VelocityThe variable velocity framework of LDDMM actually does associate its parameters with unique points in the brain.The entire trajectory of an evolving LDDMM deformation is (in theory) uniquely specified by the starting conditions.These initial conditions provide a more meaningful measure of shape.
61Initial Momentum MapsThe assumptions behind allometry fail for overlapping or adjacent structures.Initial momentum maps may be the way to go….
62See also…M. F. Beg, M. I. Miller, A. Trouvé and L. Younes. “Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms”. International Journal of Computer Vision 61(2):139–157 (2005).M. Vaillant, M. I. Miller, L. Younes and A. Trouvé. “Statistics on diffeomorphisms via tangent space representations”. NeuroImage 23:S161–S169 (2004).L. Younes, “Jacobi fields in groups of diffeomorphisms and applications”. Quart. Appl. Math. 65:113–134 (2007).
64Rigid Rotation x1(t) = x2(t) x2(t) = -x1(t) x(t) = Ax(t), where Consider a 2D rotation y=Rx, whereThis can be formulated as the solution of a differential equation at time t=1x1(t) = x2(t)x2(t) = -x1(t)orx(t) = Ax(t), where
65More on rotationsA rigid-body transform can be parameterised by the matrix exponential of an anti-symmetric (skew-symmetric) matrix.