Generalized TV--ℓp Structured Priors for Bayesian T1 Mapping

We propose an extended family of structured spatial priors that incorporates the total variation (TV) function with ℓp norms. The prior is proven to be proper and incorporated into a Bayesian regression framework to enable uncertainty quantification in T1 mapping, with posterior inference performed using the No-U-Turn Sampler (NUTS). This TV--ℓp 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 T1 mapping datasets, as well as a real in-vivo breast T1 mapping dataset. The results show that the TV--ℓp 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 ℓp norms in a prior in a Bayesian model improves spatial coherence in T1 maps and enhances uncertainty quantification, offering a robust approach for T1 mapping with uncertainties.