Combining the Band-Limited Parameterization and Semi-Lagrangian Runge–Kutta Integration for Efficient PDE-Constrained LDDMM

Abstract

The family of PDE-constrained Large Deformation Diffeomorphic Metric Mapping (LDDMM) methods is emerging as a particularly interesting approach for physically meaningful diffeomorphic transformations. The original combination of Gauss–Newton–Krylov optimization and Runge–Kutta integration shows excellent numerical accuracy and fast convergence rate. However, its most significant limitation is the huge computational complexity, hindering its extensive use in Computational Anatomy applied studies. This limitation has been treated independently by the problem formulation in the space of band-limited vector fields and semi-Lagrangian integration. The purpose of this work is to combine both in three variants of band-limited PDE-constrained LDDMM for further increasing their computational efficiency. The accuracy of the resulting methods is evaluated extensively. For all the variants, the proposed combined approach shows a significant increment of the computational efficiency. In addition, the variant based on the deformation state equation is positioned consistently as the best performing method across all the evaluation frameworks in terms of accuracy and efficiency.

Appendix

Appendix gathers the expressions of the gradient and the Hessian for the PDE-LDDMM variants defined in the spatial domain and the method for SL-RK integration.

1.1 A.1 Original PDE-Constrained LDDMM (Variant I)

Let E(v) be the PDE-constrained variational problem given in Eq. 4. Let us define the Lagrange multipliers $\lambda :\varOmega \times [0,1] \rightarrow \mathbb {R}$ and $\eta :\varOmega \rightarrow \mathbb {R}$ associated with the state equation (Eq. 5) and its initial condition. The augmented Lagrangian corresponds with the expression

$$\begin{aligned} E_\text {aug}(v)= & {} E(v) + \int _0^1 \langle \lambda (t), \partial _t m(t) \nonumber \\&+D m(t) \cdot v_t \rangle _{L^2} \mathrm{d}t + \langle \eta , m(0) - I_0 \rangle _{L^2}. \end{aligned}$$

(53)

The first- and second-order optimality conditions are derived from the formal computations of

$$\begin{aligned} \delta E_\text {aug}( v, m, \lambda , \eta ; dv, dm, d\lambda , d\eta ) \end{aligned}$$

(54)

and

$$\begin{aligned} \delta ^2 E_\text {aug}( v, m, \lambda , \eta ; dv, dm, d\lambda , d\eta ). \end{aligned}$$

(55)

The details of the formal derivations can be found in [9].

Since $\delta E_\text {aug}$ needs to vanish for any dv, dm, and $d\eta$, we get the necessary first-order optimality conditions for Variant I. In particular, the expression of the gradient is given by

$$\begin{aligned} (\nabla _v E_\text {aug}(v))_t = L v_t + \lambda (t) \cdot \nabla m(t), \end{aligned}$$

(56)

where m and $\lambda$ are computed from the state and the adjoint equations

$$\begin{aligned} \partial _t m(t) + \nabla m(t) \cdot v_t = 0 \end{aligned}$$

(57)

$$\begin{aligned} -\partial _t \lambda (t) - \nabla \cdot (\lambda (t) \cdot v_t) = 0 \end{aligned}$$

(58)

with their corresponding initial conditions $m(0) = I_0$ and $\lambda (1) = -\frac{2}{\sigma ^2}(m(1)-I_1)$.

The necessary second-order optimality conditions are obtained vanishing $\delta ^2 E_\text {aug}$ for any dv, dm, and $d\eta$. Thus, the Gauss–Newton approximation of the Hessian vector product is given by

$$\begin{aligned} (H_v E_\text {aug}(v))_t \delta v(t) = L \delta v_t + \delta \lambda (t) \cdot \nabla m(t), \end{aligned}$$

(59)

where $\delta \lambda$ is computed from the incremental adjoint equation

$$\begin{aligned} -\partial _t \delta \lambda (t) - \nabla \cdot (\delta \lambda (t) \cdot v_t) = 0 \end{aligned}$$

(60)

with initial condition $\delta \lambda (1) = -\frac{2}{\sigma ^2} \delta m(1)$, where $\delta m(1)$ is computed from the incremental state equation

$$\begin{aligned} \partial _t \delta m(t) + \nabla \delta m(t) \cdot v_t + \nabla m(t) \cdot \delta v(t) = 0 \end{aligned}$$

(61)

with initial condition $\delta m(0) = 0$.

1.2 A.2 PDE-Constrained LDDMM Based on the State Equation (Variant II)

Variant II consists in replacing the computation of the state and the adjoint variables, m and $\lambda$, from the solution of the state and adjoint PDEs to the identities $m(t) = I_0 \circ \phi (t)$ and $\lambda (t) = J(t) \lambda (1) \circ \psi (t)$, where $\phi (t)$ is the direct map, $\psi (t)$ is the inverse map, and J is the Jacobian determinant of $\psi$. As a result, Variants I and II are two theoretically but not numerically equivalent formulations of the original PDE-LDDMM problem.

For Variant II, the derivation of the gradient and the Hessian vector product proceeds as for Variant I. However, the computation of the state and adjoint variables is performed using their identities, transferring PDE resolution to the deformation state equation for $\phi$ and $\psi$

$$\begin{aligned}&\partial _t \phi (t) + D \phi (t) \cdot v_t = 0 \end{aligned}$$

(62)

$$\begin{aligned}&-\partial _t \psi (t) - D \psi (t) \cdot v_t = 0 \end{aligned}$$

(63)

with initial condition $\phi (0) = id$ and $\psi (1) = id$, and the Jacobian equation for J

$$\begin{aligned} -\partial _t J(t) - v_t \cdot \nabla J(t) = -J(t) \nabla \cdot v_t \end{aligned}$$

(64)

with initial condition $J(1)=1$. The incremental state and adjoint variables are computed from the incremental expression of the identities

$$\begin{aligned}&\delta m(t) = \nabla I_0 \circ \phi (t) \cdot \delta \phi (t) \end{aligned}$$

(65)

$$\begin{aligned}&\delta \lambda (t) = J(t) \nabla \lambda (1) \circ \psi (t) \cdot \delta \psi (t), \end{aligned}$$

(66)

and, again, the PDE resolution is transferred to the incremental deformation state equations for $\delta \phi$ and $\delta \psi$

$$\begin{aligned}&\partial _t \delta \phi (t) + D \delta \phi (t) \cdot v_t + D \phi (t) \cdot \delta v(t)= 0 \end{aligned}$$

(67)

$$\begin{aligned}&-\partial _t \delta \psi (t) - D \delta \psi (t) \cdot v_t - D \psi (t) \cdot \delta v(t) = 0. \end{aligned}$$

(68)

1.3 A.3 PDE-Constrained LDDMM Based on the Deformation State Equation (Variant III)

For Variant III, the Lagrange multipliers are $\rho :\varOmega \times [0,1] \rightarrow \mathbb {R}^d$, associated with the deformation state equation (Eq. 7), and $\mu :\varOmega \rightarrow \mathbb {R}^d$, associated with its initial condition. The augmented Lagrangian corresponds with

$$\begin{aligned} E_\text {aug}(v)= & {} E(v) + \int _0^1 \langle \rho (t), \partial _t \phi (t) \nonumber \\&+\,D \phi (t) \cdot v_t \rangle _{L^2} \mathrm{d}t + \langle \mu , \phi (0) - id \rangle _{L^2}. \end{aligned}$$

(69)

The first- and second-order optimality conditions are derived from the formal computations of

$$\begin{aligned} \delta E_\text {aug}( v, \phi , \rho , \mu ; dv, d\phi , d\rho , d\mu ) \end{aligned}$$

(70)

and

$$\begin{aligned} \delta ^2 E_\text {aug}( v, \phi , \rho , \mu ; dv, d\phi , d\rho , d\mu ). \end{aligned}$$

(71)

Table 8 Original PDEs involved in PDE-LDDMM and corresponding PDEs written in SL form

The necessary first- and second-order optimality conditions are obtained from the need to vanish $\delta E_\text {aug}$ and $\delta ^2 E_\text {aug}$ for any dv, $d\phi$, $d\rho$, and $d\mu$, yielding

$$\begin{aligned}&(\nabla _v E_\text {aug}(v))_t = L v_t + D \phi (t) \cdot \rho (t) \nonumber \\&(H_v E_\text {aug}(v))_t \delta v(t) = L \delta v_t + D \phi (t) \cdot \delta \rho (t). \end{aligned}$$

(72)

where $\phi$ is computed from the deformation state equation, $\rho$ from the deformation adjoint equation, and $\delta \rho$ from the incremental deformation adjoint equation

$$\begin{aligned} \partial _t \phi (t) + D \phi (t) \cdot v_t= & {} 0 \end{aligned}$$

(73)

$$\begin{aligned} -\partial _t \rho (t) - \nabla \cdot (\rho (t) \cdot v_t)= & {} 0 \end{aligned}$$

(74)

$$\begin{aligned} -\partial _t \delta \rho (t) - \nabla \cdot ( \delta \rho (t) \cdot v_t)= & {} 0 \end{aligned}$$

(75)

with initial conditions $\phi (0) = id$, $\rho (1) = \lambda (1) \nabla m(1)$, $\delta \rho (1) = \delta \lambda (1) \nabla m(1)$. It should be noticed that the divergence operator acting on tensors operates row-wise.

1.4 A.4 Semi-Lagrangian Runge–Kutta Integration

As we mentioned in Sect. 3, to be able to apply SL integration, the differential equations for different spatial variants need to be written in the shape of Eq. 46. The state equations, the deformation state equations, and their incremental counterparts (Eqs. 57, 62, 61, 67) are already in the shape of Eq. 46 by just moving to the right-hand side of the equation a remaining term. For the adjoint and the incremental adjoint equations (Eqs. 58, 74, 60, 75), we use the identity

$$\begin{aligned} \nabla \cdot (u \cdot v) = u \nabla \cdot v + v \nabla u \end{aligned}$$

(76)

and move the divergence term to the right-hand side the transformed equation. Table 8 gathers the expressions of the resulting differential equations, needed for the implementation of PDE-LDDMM methods in SL form. For SL-RK, the right-hand side expressions can be directly plugged into an RK differential solver. Algorithm A1 shows the pseudocode for SL-RK integration.