PRISM: A 3D Probabilistic Neural Representation for Interpretable Shape Modeling

TL;DR

PRISM addresses the need for interpretable, uncertainty-aware shape modeling by learning a conditional, spatially varying Gaussian field of displacements $p(\boldsymbol{d}\mid \boldsymbol{p}, t)=\mathcal{N}(\mu(\boldsymbol{p}, t), \Sigma(\boldsymbol{p}, t))$ and by deriving a closed-form Fisher Information metric to quantify local temporal uncertainty. It couples this probabilistic representation with an amortized inverse encoder to estimate intrinsic developmental time $\tau$ directly from local shapes, enabling dense temporal localization and personalized trajectory forecasting. The framework supports continuous shape trajectories, intrinsic time estimation, longitudinal prediction, and OOD detection, validated on synthetic datasets with known ground-truth timing and on pediatric airway data, showing accurate mean trajectories and spatially varying uncertainty that align with clinical expectations. Findings indicate that spatially resolved uncertainty improves anomaly detection and personalized predictions, with competitive performance against state-of-the-art covariate-conditioned implicit methods while offering analytic tractable uncertainty in anatomically meaningful coordinates.

Abstract

Understanding how anatomical shapes evolve in response to developmental covariates and quantifying their spatially varying uncertainties is critical in healthcare research. Existing approaches typically rely on global time-warping formulations that ignore spatially heterogeneous dynamics. We introduce PRISM, a novel framework that bridges implicit neural representations with uncertainty-aware statistical shape analysis. PRISM models the conditional distribution of shapes given covariates, providing spatially continuous estimates of both the population mean and covariate-dependent uncertainty at arbitrary locations. A key theoretical contribution is a closed-form Fisher Information metric that enables efficient, analytically tractable local temporal uncertainty quantification via automatic differentiation. Experiments on three synthetic datasets and one clinical dataset demonstrate PRISM's strong performance across diverse tasks within a unified framework, while providing interpretable and clinically meaningful uncertainty estimates.

Figures (7)

• Figure 1: Overview of the PRISM framework. (a) illustrates the population distribution of developmental stages (e.g., physiological ages) at a fixed covariate (e.g., chronological age), indicating that individuals can be developmentally delayed, expected, or fast. (b) visualizes the probabilistic shape deformation trajectory $p(\boldsymbol{d} \mid \boldsymbol{p}, t)$, differentiating between variations in developmental progression (indicated by blue, purple, pink dots) and intrinsic shape attributes independent of the time $t$ (represented by the orange plate). (c) details the shape modeling component, where a neural network (NN) estimates the mean trajectory $\mu(\boldsymbol{p}, t)$ and total covariance $\Sigma(\boldsymbol{p}, t)$. These learned parameters enable the estimation of temporal uncertainty through the Fisher information (Eq. \ref{['eq.fisher_info_full']}). (d) presents the Inverse Encoder, which facilitates downstream analysis by inferring the developmental time $\hat{\tau}$ from a template query point $\boldsymbol{p}$ and deformation $\boldsymbol{d}$, serving as the foundation for tasks in (e). (e) shows various applications of PRISM in shape analysis, including generating population-level shape trajectories, inferring individual developmental stage (time inference), predicting future shapes (shape transfer), and detecting abnormal development (OOD detection) by evaluating the likelihood of an observed shape within the population distribution.
• Figure 2: Visual comparison of uncertainty-aware shape reconstruction on the Starman dataset. We decode shapes using time points at the mean and $\pm 2\sigma$ of the predicted intrinsic time distribution. The red contours represent the shapes reconstructed by PRISM at these time points, while the blue contours represent the ground truth shapes. The high degree of overlap verifies that the learned temporal uncertainty correctly translates into valid geometrical deformations.
• Figure 3: Qualitative validation of uncertainty estimation on the simulated Starman datasets. (a) Results on the Starman(G) dataset and (b) results on the Starman(L) dataset. The plot compares the uncertainty estimates from PRISM at different locations with the ground truth. The blue shaded regions represent the ground truth distribution of the conditional distribution $p(\tau \mid \boldsymbol{p}, t)$, while the red shaded regions show the distribution estimated by our method. The tight alignment between the two demonstrates PRISM's ability to accurately recover the true underlying uncertainty profile of the data-generating process.
• Figure 4: Spatially-varying uncertainty quantification across anatomical landmarks in pediatric airways. Each subplot displays the relationship between chronological age $t$ (x-axis) and the predicted intrinsic developmental age $\hat{\tau} = g(\boldsymbol{p}, \boldsymbol{d})$ (y-axis) at a specific anatomical landmark, where $g(\cdot)$ is the learned inverse encoder. Landmarks progress from the nose tip (top-left) to the carina (bottom-right). The shaded region indicates $\tau \pm 2\sigma$ estimated by PRISM; the diagonal line represents $\mathbb{E}[\tau|t] = t$. Blue points denote normal subjects; red points denote abnormal cases.
• Figure 5: Starman dataset generation. Left: Template shape with four control points governing arm and leg deformations. Middle: Shape evolution across physical time $t \in [0, 1]$. Right: Temporal uncertainty functions $\sigma_\tau(t)$ for Starman(L), where arms (red) exhibit early growth and legs (blue) exhibit late growth.

Theorems & Definitions (2)

• Theorem 1.1: Cramér--Rao Lower Bound
• proof