Latent Score-Based Bayesian Cramér-Rao Bound Estimation for High-Dimensional Imaging Systems
Authors: Evan Scope Crafts, Thomas Wynn, Seonyeong Park, Mark Anastasio, Umberto Villa
Organizations: Oden Institute for Computational Engineering and Sciences, the University of Texas at Austin, Austin, TX, 78712 · Graduate School of Biomedical Sciences at UT Southwestern Medical Center · Mallinckrodt Institute of Radiology at Washington University in St. Louis · Department of Biomedical Engineering and the Oden Institute for Computational Engineering and Sciences, the University of Texas at Austin, Austin, TX, 78712
We propose a data-driven framework for estimating the Bayesian Cramér-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 H1(Ω) 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.
Figures & tables
Fig. 1: Fluence maps simulated under the interlaced (top) and contiguous (bottom) illumination schemes used in the numerical studies. These maps correspond to the diagonals of the weighting matrices {Di}i=1I in ( 21 ), with I=4 . The white regions indicate locations that contained only water across the dataset, i.e., regions outside of the breast phantoms; the fluence was set to zero in these regions. Although the two illumination schemes produce different spatial fluence patterns, both exhibit significant light attenuation near the center of the domain due to the highly scattering nature of biological tissue.
Fig. 2: Uncurated μa maps from our dataset (top), the corresponding reconstructions obtained using the autoencoder (middle), and the associated error maps (bottom). Despite not being trained or fine-tuned on our dataset, the autoencoder produced accurate reconstructions without noticeable visual distortions.
Fig. 3: (a) Diagonals of the estimates of VmP in ( 22 ) obtained using the original (left) and weighted (right) SSM objectives. (b) Pointwise variance of the samples {mi}i=1N . In both panels, regions containing only water across the dataset were masked out. Both VmP estimates exhibit similar global structure, consistent with that of the pointwise variance. However, the estimate obtained using the original SSM objective contains high-frequency artifacts that are absent from the estimate obtained using the proposed objective.
Fig. 4: Estimates of the diagonals of the Bayesian CRB Vm (bottom) and the Bayesian CRB under an uninformative prior VmD (top; see ( 23 )) for the contiguous and interlaced illumination schemes. The final column shows the pointwise ratio of the two bounds, with red indicating locations where the interlaced scheme performs better and blue indicating locations where the contiguous scheme is superior. The spatial structure of the bound under the uninformative prior is consistent with that of the diagonal weighting matrices shown in Fig. 1 . Incorporating the data-informed prior changes both the spatial structure of the bounds and their relative strengths because the prior strongly informs the parameters near the center of the domain; cf. Fig. 3 (b).
Design Scheme
Uninformative Prior
Data-Informed Prior
Trace VzD
Log Det. VzD
Trace VmD
Trace Vz
Log Det. Vz
Trace Vm
Interlaced
217819.2410
−33163.6517
3065.9900
11.2323
−131923.8601
147.4793
Contiguous
167677.041
−35779.3460
3844.3887
10.9382
−132472.7493
134.2040
TABLE I: Trace and log-determinant of the estimated Bayesian CRBs (lower is better). With the uninformative prior, the ranking of the two illumination schemes depends on the chosen metric: the latent-space bound favors the contiguous scheme, whereas the parameter-space bound favors the interlaced scheme. The incorporation of the learned prior resolves this discrepancy, with the contiguous scheme favored by all reported metrics.
Bayesian imaging inverse problems often require sampling from high-dimensional posterior distributions. While recent score-based and diffusion models provide expressive Bayesian priors, their sampling procedures remain inherently sequential and computationally expensive for large-scale imaging applications. We propose PiX-MC, a time-parallel posterior sampling framework based on proximal Langevin dynamics and Picard iteration. The proximal-likelihood formulation exploits the fact that many imaging likelihoods admit efficient, problem-specific proximal operators, while Picard refinement exposes parallelism across discretization nodes and naturally supports multi-GPU implementation. To further improve practical scalability and sampling performance, we develop multi-block and annealed variants of the proposed framework. We establish convergence guarantees under transparent assumptions, accommodating non-log-concave posteriors, imperfect learned score models, multi-block implementations, and annealing schedules. Experiments on a diverse collection of imaging inverse problems demonstrate that PiX-MC substantially reduces wall-clock time while preserving reconstruction quality. On a 512×512×80 sparse-view computed tomography (CT) problem, annealed multi-block PiX-MC achieves up to a 50× runtime speedup over the standard Langevin sampler using eight GPUs.
Deliang Wei, Evan Bell, Wenhan Guo +2
Johns Hopkins University · University of California, Los Angeles
We propose a novel Bayesian framework for joint image reconstruction and uncertainty quantification from compressed sensing magnetic resonance imaging data. The problem is formulated as a linear inverse problem, where prior distributions are assigned to the unknown image parameters. Specifically, the image is assumed to be sparse in a given transform domain. We develop a general framework applicable to any sparsifying transform and demonstrate its performance using (1) a total variation transform based on image spatial gradients and (2) a wavelet-domain transform. Bayesian inference is performed using a split-and-augmented Gibbs sampler, while the resulting non-differentiable conditional distributions are efficiently sampled using a proximal Markov chain Monte Carlo method. The proposed algorithms are validated on both single-coil and multi-coil datasets using various k-space sampling patterns and acceleration factors. The results demonstrate that the proposed Bayesian methods consistently outperform their optimisation-based counterparts in image reconstruction while providing uncertainty estimates for the reconstructed images. Furthermore, the estimated uncertainty maps show a strong correlation with the true reconstruction errors and substantially outperformed deep learning-based uncertainty estimation methods.
Ahmed Karam Eldaly, Matteo Figini, Daniel C. Alexander
Department of Computer Science, University of Exeter, Exeter EX4 4QF, U.K. · UCL Hawkes Institute, Department of Computer Science, University College London, London, U.K.
This paper reviews how a diverse set of popular data-driven priors commonly used in Bayesian inverse problems can be unified through their respective score functions. By framing these priors under this common perspective, we show that they can benefit from their straightfoward and effective integration into a recently proposed sampling algorithm. The applicability of this common framework is illustrated by considering several data-driven priors, namely regularization-by-denoising, normalizing flow-based priors, score-based generative models, and convex-ridge regularizers. For these four particular priors, the performance of the method is evaluated when conducting image inpainting and single image super-resolution. These results, as well as those obtained when restoring real images acquired in a geological context, demonstrate the efficiency of the method. This unified framework proves versatile enough to handle any posterior distribution defined by a broad class of score function-based priors, beyond the specific cases considered in this paper.