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

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.

💡 Research Summary

PRISM (Probabilistic Implicit Shape Modeling) is a novel framework that unifies neural implicit representations with rigorous uncertainty quantification for statistical shape analysis. The authors start from a template‑based formulation: each observed anatomy is expressed as a displacement field ϕi(p) that maps a canonical template point p to its location on subject i. By focusing on the displacement d = ϕ(p) rather than the absolute coordinates, the conditional distribution of a point given its template coordinate and a covariate (e.g., chronological age t) can be written as a simple 3‑D Gaussian: p(d | p, t) = N(μ(p, t), Σ(p, t)). This model captures both a mean developmental trajectory μ(p, t) and a spatially heteroscedastic covariance Σ(p, t), allowing the variance to differ across anatomical regions.

Both μ and Σ are parameterized by a coordinate‑based multilayer perceptron fθ(p, t). Training minimizes the negative log‑likelihood over all observed displacement‑template pairs, which reduces to a sum of Mahalanobis distances plus a log‑determinant term. The network thus learns a continuous, resolution‑agnostic field that can be queried at any point in ℝ³ and for any covariate value.

A central theoretical contribution is a closed‑form Fisher Information metric for the conditional Gaussian field. The score function U(d; p, t) = ∂/∂t log p(d | p, t) leads to the Fisher Information I(p, t) = E