Latent Score-Based Bayesian Cram\'er-Rao Bound Estimation for High-Dimensional Imaging Systems

2026-10-02

AI Summary

This study addresses the computational intractability of the Bayesian Cramér-Rao bound (BCRB) arising from complex priors in high-dimensional imaging systems. We propose a BCRB estimation framework based on latent-space dimensionality reduction and sliced score matching in Bochner spaces. Specifically, a variational autoencoder is employed to construct a latent-space mapping, while a Sobolev-norm-constrained sliced score matching objective is designed to suppress high-frequency artifacts. Combined with the change-of-variables formula, this approach enables efficient computation in high-dimensional spaces. Applied to million-parameter quantitative photoacoustic computed tomography (qPACT), the proposed method yields stable, artifact-free BCRB estimates that accurately capture non-Gaussian prior structures. This work establishes a new paradigm for performance evaluation in high-dimensional imaging systems.

Abstract

We propose a data-driven framework for estimating the Bayesian Cram\'er-Rao bound (CRB) in high-dimensional imaging systems with complex, analytically intractable priors. Direct CRB computation is challenging in this setting due to the need to model the prior score and to form and invert the Bayesian Fisher information matrix in very high dimensions. To address these issues, we first reformulate the inverse problem in the latent space of a pre-trained variational autoencoder, thereby dramatically reducing the dimensionality of the bound estimation problem while preserving the spatial structure of the images. The Bayesian CRB is formed in this latent space and mapped back to the native parameter space using a change-of-variables formula. Second, to learn the latent prior score, we introduce a new sliced score matching objective defined in a Bochner space endowed with an $H^1(\Omega)$ Sobolev norm in space. This"Bochner-space sliced score matching"objective is consistent with standard sliced score matching, but penalizes errors in the spatial gradients of the score, suppressing high-frequency artifacts that otherwise contaminate the resulting CRB estimates. We validate the approach on a stylized quantitative photoacoustic computed tomography (qPACT) breast imaging problem with over one million unknown parameters, using a foundation-model autoencoder derived from Stable Diffusion. The proposed method yields stable, artifact-free Bayesian CRB estimates that reflect the highly non-Gaussian structure of the learned prior and reveal the substantial impact of the prior on the relative performance of competing qPACT design schemes.

Problem

• Bayesian Cramér-Rao bound
• high-dimensional imaging systems
• prior score modeling
• Fisher information matrix
• inverse problems

Innovation

• Bayesian Cramér-Rao Bound
• Variational Autoencoder
• Latent Space
• Sliced Score Matching
• Bochner Space