TPFM-DIR: Two-Parameter Flow Map Diffeomorphic Image Registration

TPFMDIR in a Nutshell

TPFMDIR Methodology

To model a diffeomorphism $\phi$ warping a moving image $I_m$ to a fixed image $I_f$, TPFMDIR learns the solution of a non-autonomous ODE

$$ \begin{cases} \frac{d\phi_t}{dt} &= v(\phi_t, t) \\ \phi_0 &= \mathrm{id} \end{cases} $$

TPFMDIR takes the dynamical systems perspective of the ODE solutions:

$$ \phi_0 = \mathrm{Id},\quad\underbrace{\phi_{r, t}\circ\phi_{s, r}=\phi_{s, t}}_\text{Cocycle Property} \iff \begin{cases}\frac{d\phi_{s, t}}{dt} &= v(\phi_{s, t}, t) \\\phi_0 &= \mathrm{id}\end{cases}\quad\text{(for some velocity field }v\text{)} $$

Note that, for the non-autonomous ODE, the solution depends on both the initial time $s$ and the end time $t$, whereas for the stationary case, the solution only depends on the elapsed time. This is the reason the solutions are called two-parameter family.

Usually, to find the solution of such non-autonomous ODEs, the velocity field $v(., t)$ is modeled and the solution is obtained by expensive numerical integrations such as RK4 method. The cost of such methods for the task of medical image registration which usually deals with large images, has caused this type of modeling to be largely overlooked.

TPFMDIR mitigates this issue and directly models the two-parameter flow map solution of the ODE and learns the diffeomorphism without velocity parameterization, without any numerical integration, and withour multiple regularization terms.

To this end, TPFMDIR introduces a anchored cocycle regularization as the only regularization term and employs a time-dependent loss function, which provably drives the model towards learning the ODE solution, hence inheriting their smoothness and diffeomorphic properties.

Importantly, we show that the full cocycle property is deducible from the anchored cocycle regularization. Practically, we don't need independent time poinst $r$, $s$ and $t$ to impose the full cocycle; rather, only using time points $s$ and $t$ suffices for a guaranteed learning of the ODE solution:

$$ \underbrace{\phi_{s, t}\circ\phi_{0, s}=\phi_{0, t}, \phi_{t, s}\circ\phi_{1, t}=\phi_{1, s}}_\text{Anchored Cocycle Property} \implies \underbrace{\phi_{r, t}\circ\phi_{s, r}=\phi_{s,t}}_\text{Full Cocycle Property} $$

Moreover, we show that without any numerical differentiation, one could extract the learned underlying instataneous velocity field governing the dynamics of the ODE.

Results Summary