LDDMM stochastic interpolants: an application to domain uncertainty quantification in hemodynamics

Abstract

We introduce a novel conditional stochastic interpolant framework for generative modeling of three-dimensional shapes. The method builds on a recent LDDMM-based registration approach to learn the conditional drift between geometries. By leveraging the resulting pull-back and push-forward operators, we extend this formulation beyond standard Cartesian grids to complex shapes and random variables defined on distinct domains. We present an application in the context of cardiovascular simulations, where aortic shapes are generated from an initial cohort of patients. The conditioning variable is a latent geometric representation defined by a set of centerline points and the radii of the corresponding inscribed spheres. This methodology facilitates both data augmentation for three-dimensional biomedical shapes, and the generation of random perturbations of controlled magnitude for a given shape. These capabilities are essential for quantifying the impact of domain uncertainties arising from medical image segmentation on the estimation of relevant biomarkers.

Figures (29)

• Figure 1: Overview of the proposed method and application in forward domain uncertainty quantification for aortic blood flows.
• Figure 2: Left: On a two-dimensional grid, the prescribed velocity field $u^{\text{cond}}_t$\ref{['eq:conditional_drift']} can be defined by computing the difference between the realizations of the random variables $X_0$ and $X_1$. Right: In the case of a three-dimensional shape (described, e.g., as a surface mesh), a definition of $u^{\text{cond}}_t$ based on the difference \ref{['eq:conditional_drift']} is no longer possible, since there is no direct correspondence between points.
• Figure 3: Example of application of the multilevel ResNet-LDDMM romor2025dataassimilationperformedrobust for the registration of computational meshes of aortic shapes. In this example, meshes are refined after 3,000, 4,000, and 5,000 epochs.
• Figure 4: Graphical overview of the proposed conditional LDDMM stochastic interpolant method. The map defining the conditional drift $b^{\theta}$ as a function of the conditioning variable $C$ is learned registering the available shapes onto a common reference ($\mathbf{x}_0$), in order to find a stochastic interpolant between the two distributions.
• Figure 5: The conditional drift $u_t^{\text{cond}}(\mathbf{x}_0, \mathbf{x}_i)$ is defined based on the LDDMM registration. When $\mathbf{x}_0$ and $\mathbf{x}_i$ represent domain coordinates (left), the conditional drift is computed from the registration field ${\phi}^{0,i}$ between the two realizations. When $\mathbf{x}_0$ and $\mathbf{x}_i$ represent physical fields/solutions of PDEs (right), the conditional drift is computed, at each intermediate step $t \in [0,1]$, considering the push-forward of ${\phi}^{0,i}$ (registration of $\mathbf{x}_0$ onto $\mathbf{x}_i$) and the pull-back of ${\psi}^{0,i}$ (registration of $\mathbf{x}_i$ onto $\mathbf{x}_0$).

Theorems & Definitions (14)

• Definition 2.1: Stochastic interpolant
• Remark 2.2: Equivalent ODE flow formulation
• Remark 2.3: Regularizers for stochastic interpolants
• Definition 2.4: Push-forward and pull-back operators
• Remark 2.5: Analogies between LDDMM and diffusion models
• Remark 3.1: General realizations of $X_0$ and $X_1$
• Remark 3.2
• Remark 3.3: Generalization to physical fields on meshes
• Remark 4.1: Generation of the template mesh
• Remark 5.1: Regularity of the perturbation