Generalized TV--$\ell_p$ Structured Priors for Bayesian $T_1$ Mapping

2026-06-03

AI Summary

This work addresses the weak spatial consistency and inaccurate uncertainty quantification commonly observed in conventional $T_1$ mapping by proposing a structured spatial prior that integrates total variation (TV) and the $\ell_p$ norm within a Bayesian regression framework. Efficient posterior inference is achieved using the No-U-Turn Sampler (NUTS), ensuring a well-defined prior while substantially enhancing the spatial smoothness and reliability of the estimated parameter maps. Experimental results on synthetic brain and cardiac data, as well as real breast $T_1$ datasets, demonstrate that the proposed method yields more concentrated posterior densities, lower variance, and reduced negative bias, thereby significantly improving estimation accuracy and robustness.

Abstract

We propose an extended family of structured spatial priors that incorporates the total variation (TV) function with $\ell_p$ norms. The prior is proven to be proper and incorporated into a Bayesian regression framework to enable uncertainty quantification in $T_1$ mapping, with posterior inference performed using the No-U-Turn Sampler (NUTS). This TV--$\ell_p$ construction is proven to constitute a well-defined family of prior distributions, and it naturally enforces spatial consistency and smooth variations in the estimated parameter maps. The method was evaluated in comparison to maximum-likelihood estimation and several Bayesian alternative priors based on the uniform, Gamma, and bounded TV priors. The evaluation includes experiments on synthetic brain and cardiac $T_1$ mapping datasets, as well as a real in-vivo breast $T_1$ mapping dataset. The results show that the TV--$\ell_p$ prior yields more concentrated posterior densities, indicating reduced uncertainty. It also consistently achieves lower variance and smaller (negative) bias, leading to more reliable estimates. Overall, embedding a TV-based structured penalty along with $\ell_p$ norms in a prior in a Bayesian model improves spatial coherence in $T_1$ maps and enhances uncertainty quantification, offering a robust approach for $T_1$ mapping with uncertainties.

Problem

• T1 mapping
• spatial priors
• uncertainty quantification
• total variation
• Bayesian inference

Innovation

• TV-ℓp prior
• Bayesian T1 mapping
• uncertainty quantification
• spatial consistency
• No-U-Turn Sampler