Computational Anatomy and PDE

Klas Modin

Outline

  1. Background
     
  2. EPDiff equation
     
  3. New gradient flow PDE

Background

Part 1

MR camera

CT scanner

Radiology or medical imaging is in the middle of a digital or computer revolution that is affecting all of science and its various fields, including medicine

– Nick Bryan, Radiological Society of North America

  • Better understanding of disease
  • Tool to facilitate diagnosis

What is `distance` between images?

Riemannian geometry: distance in curved space

Exemple: great circles (2d)

Exemple: Einstein's space-time (4d)

Exemple: space of diffeomorphisms (∞-dim)

\mathrm{Diff}(\Omega) = \{ \varphi\in C^{\infty}(\Omega,\Omega) \mid \varphi \text{ has smooth inverse}\}
Diff(Ω)={φC(Ω,Ω)φ has smooth inverse}\mathrm{Diff}(\Omega) = \{ \varphi\in C^{\infty}(\Omega,\Omega) \mid \varphi \text{ has smooth inverse}\}

Image dist = Riemannian dist between \(\varphi\) and \(\mathrm{id}\)

What is computational anatomy?

  • Use Riemannian geometry on inf-dim manifold of diffeomorphisms to solve registration problems

(a) Template

(b) Warp

(c) Transformed

(d) Target

History of computational anatomy

EPDiff equation

Part 2

Diffeomorphic image registration

  • Image domain \(\Omega\)
  • Template image \(I_0: \Omega \to \mathbb{R}_+ \)
  • Target image \(I_1:\Omega\to \mathbb{R}_+\)
\displaystyle\min_{\varphi\in\mathrm{Diff}(\Omega)}\lVert I_0\circ\varphi^{-1}-I_1 \rVert^2
minφDiff(Ω)I0φ1I12\displaystyle\min_{\varphi\in\mathrm{Diff}(\Omega)}\lVert I_0\circ\varphi^{-1}-I_1 \rVert^2
\varphi
φ\varphi

Ill-posed with noise!

Large deformation diffeomorphic metric matching (LDDMM)

\displaystyle\min_{\varphi\in\mathrm{Diff}(\Omega)}\lVert I_0\circ\varphi^{-1}-I_1 \rVert^2 + \frac{1}{\sigma}\mathrm{dist}(\mathrm{id},\varphi)^2
minφDiff(Ω)I0φ1I12+1σdist(id,φ)2\displaystyle\min_{\varphi\in\mathrm{Diff}(\Omega)}\lVert I_0\circ\varphi^{-1}-I_1 \rVert^2 + \frac{1}{\sigma}\mathrm{dist}(\mathrm{id},\varphi)^2
\displaystyle\mathrm{dist}(\mathrm{id},\varphi)^2 = \int_0^1 \lVert (1-\alpha \Delta)^k v(t) \rVert^2_{L^2} dt
dist(id,φ)2=01(1αΔ)kv(t)L22dt\displaystyle\mathrm{dist}(\mathrm{id},\varphi)^2 = \int_0^1 \lVert (1-\alpha \Delta)^k v(t) \rVert^2_{L^2} dt
v(t) := \dot\varphi(t)\circ\varphi(t)^{-1} \quad \varphi(0)=\mathrm{id}, \varphi(1)=\varphi
v(t):=φ˙(t)φ(t)1φ(0)=id,φ(1)=φv(t) := \dot\varphi(t)\circ\varphi(t)^{-1} \quad \varphi(0)=\mathrm{id}, \varphi(1)=\varphi
\displaystyle\min_{v\in C^\infty([0,1]\times\Omega,\mathbb{R}^n)}\lVert I_0\circ\varphi(1)^{-1}-I_1 \rVert^2 + \frac{1}{\sigma}\int_0^1 \lVert v(t,\cdot) \rVert^2_\mathcal{A} dt
minvC([0,1]×Ω,Rn)I0φ(1)1I12+1σ01v(t,)A2dt\displaystyle\min_{v\in C^\infty([0,1]\times\Omega,\mathbb{R}^n)}\lVert I_0\circ\varphi(1)^{-1}-I_1 \rVert^2 + \frac{1}{\sigma}\int_0^1 \lVert v(t,\cdot) \rVert^2_\mathcal{A} dt
\frac{\partial}{\partial t}\varphi(t,\cdot) = v(t,\varphi(t,\cdot)), \quad \varphi(0,x) = x
tφ(t,)=v(t,φ(t,)),φ(0,x)=x\frac{\partial}{\partial t}\varphi(t,\cdot) = v(t,\varphi(t,\cdot)), \quad \varphi(0,x) = x

\(\varphi(1)\) related to \(v\) by

\lVert u \rVert_\mathcal{A} = \lVert \mathcal{A} u \rVert_{L^2} = \lVert (1-\alpha \Delta)^k u \rVert_{L^2}
uA=AuL2=(1αΔ)kuL2\lVert u \rVert_\mathcal{A} = \lVert \mathcal{A} u \rVert_{L^2} = \lVert (1-\alpha \Delta)^k u \rVert_{L^2}

Norm \( || \cdot||_\mathcal{A} \) given by

Euler-Lagrange equations

\(\Longrightarrow\) EPDiff equation

  • Velocity \(v:[0,1]\times\Omega\to\mathbb{R}^n\)
  • Momentum \(m:[0,1]\times\Omega\to\mathbb{R}^n \)
\displaystyle\frac{\partial m}{\partial t} + \nabla m\cdot v + (\nabla v)^\top \cdot m + m(\nabla\cdot v) = 0
mt+mv+(v)m+m(v)=0\displaystyle\frac{\partial m}{\partial t} + \nabla m\cdot v + (\nabla v)^\top \cdot m + m(\nabla\cdot v) = 0
\displaystyle m = (1-\alpha\Delta)^k v
m=(1αΔ)kv\displaystyle m = (1-\alpha\Delta)^k v

Numerical approach for LDDMM

  • Steepest descent over time-dependent vector fields
     
  • EPDiff IVP solver using particle methods (solitons)
     
  • Use shooting over initial velocity \(v_0\) to minimize
\displaystyle\lVert I_0\circ \varphi(1)^{-1}-I_1\rVert^2 + \frac{1}{\sigma}\lVert v_0 \rVert^2_\mathcal{A}
I0φ(1)1I12+1σv0A2\displaystyle\lVert I_0\circ \varphi(1)^{-1}-I_1\rVert^2 + \frac{1}{\sigma}\lVert v_0 \rVert^2_\mathcal{A}

Result: excellent! (but computationally expensive)

Open problems

  • Develop FE discretization for EPDiff IVP
  • Develop FE discretization for LDDMM
  • Adaptive FE
  • Numerical convergence analysis

*images by Matteo Molteni

New gradient flow PDE

Part 3

Deformation gradient flows

Right-invariant metric on \(\mathrm{Diff}(\Omega)\) defined by \(\mathcal A\)

Minimization problem

\displaystyle\min_{\varphi\in \mathrm{Diff}(\Omega)} \lVert(I_0\circ\varphi^{-1}-I_1\rVert^2
minφDiff(Ω)(I0φ1I12\displaystyle\min_{\varphi\in \mathrm{Diff}(\Omega)} \lVert(I_0\circ\varphi^{-1}-I_1\rVert^2
\displaystyle + \frac{1}{\sigma} d_{\text{FR}}(dx,\varphi_*dx)^2
+1σdFR(dx,φdx)2\displaystyle + \frac{1}{\sigma} d_{\text{FR}}(dx,\varphi_*dx)^2

Origin of expensiveness: no formula for \(d_\mathrm{Diff}\)

\displaystyle + \frac{1}{\sigma} d_{\mathrm{Diff}}(\mathrm{id},\varphi)^2
+1σdDiff(id,φ)2\displaystyle + \frac{1}{\sigma} d_{\mathrm{Diff}}(\mathrm{id},\varphi)^2

Regularization penalizing change of volume

\(\mathrm{Diff}(\Omega)\) acts on \(Q = C^{\infty}(\Omega)\times \mathrm{Dens}(\Omega) \)

\( E(\varphi) = F_{q_1}(\varphi\cdot q_0)\) where \(q_i = (I_i,dx)\)

\dot \varphi = - \nabla_\mathcal{A} E(\varphi)
φ˙=AE(φ)\dot \varphi = - \nabla_\mathcal{A} E(\varphi)
\mathrm{Diff}(\Omega)
Diff(Ω)\mathrm{Diff}(\Omega)
Q
QQ
\mathrm{id}
id\mathrm{id}
q_0
q0q_0
\varphi(t)
φ(t)\varphi(t)
q_1
q1q_1
q(t) = \varphi(t)\cdot q_0
q(t)=φ(t)q0q(t) = \varphi(t)\cdot q_0
Q = C^{\infty}(\Omega)\times \mathrm{Dens}(\Omega)
Q=C(Ω)×Dens(Ω)Q = C^{\infty}(\Omega)\times \mathrm{Dens}(\Omega)
q_i = (I_i,dx)
qi=(Ii,dx)q_i = (I_i,dx)

Gradient flow is

 

 

 

 

 

 

\dot \varphi = - v(\varphi\cdot q_0)\circ \varphi
φ˙=v(φq0)φ\dot \varphi = - v(\varphi\cdot q_0)\circ \varphi
(1-\alpha\Delta)^s v = \frac{1}{\sigma}\rho \nabla \rho + \nabla(I-I_\infty)
(1αΔ)sv=1σρρ+(II)(1-\alpha\Delta)^s v = \frac{1}{\sigma}\rho \nabla \rho + \nabla(I-I_\infty)
\rho = \det(\varphi^{-1})
ρ=det(φ1)\rho = \det(\varphi^{-1})
I = I_0\circ\varphi^{-1}
I=I0φ1I = I_0\circ\varphi^{-1}

Lie-Euler method

 

 

 

 

 

\rho_k = \det(\varphi_k^{-1}) \quad I_k = I_0\circ\varphi_k^{-1}
ρk=det(φk1)Ik=I0φk1\rho_k = \det(\varphi_k^{-1}) \quad I_k = I_0\circ\varphi_k^{-1}
\varphi_{k+1} = \exp(-h v_k)\circ\varphi_k
φk+1=exp(hvk)φk\varphi_{k+1} = \exp(-h v_k)\circ\varphi_k
(1-\alpha\Delta)^s v_k = \frac{1}{\sigma}\rho_k \nabla \rho_k + \nabla(I_k-I_\infty)
(1αΔ)svk=1σρkρk+(IkI)(1-\alpha\Delta)^s v_k = \frac{1}{\sigma}\rho_k \nabla \rho_k + \nabla(I_k-I_\infty)

Toy example (2D)

Open problem

  • Develop moving mesh FE discretization for gradient flow PDE

Course project

slides available at slides.com/kmodin/cmm-kristineberg

Computational Anatomy and PDE

By Klas Modin

Computational Anatomy and PDE

Guest seminar in the Nordic Graduate Course on Computational Mathematical Modeling at Kristineberg on the west-coast of Sweden.

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