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We propose a novel way to incorporate morphology operators through structured regularization of machine learning models. Specifically, we introduce a feature map in the models that performs structured variable selection. The feature map is automatically processed by approximate morphology operators and is learned together with the model coefficients. Experiments were conducted with linear regression on both synthetic data, demonstrating that the proposed methods are effective in selecting groups of parameters with much less noise than baseline models, and on three-dimensional T1-weighted brain magnetic resonance images (MRI) for age prediction, demonstrating that the proposed methods enforce sparsity and select homogeneous regions of non-zero and relevant regression coefficients. The proposed methods improve interpretability in pattern analysis. The minimum size of features in the structured variable selection can be controlled by adjusting the structuring element in the approximate morphology operator, tailored to the specific study of interest. With these added benefits, the proposed methods still perform on par with commonly used variable selection and structured variable selection methods in terms of the coefficient of determination and the Pearson correlation coefficient.

Structured regularization allows machine learning models to consider spatial relationships among parameters, leading to results that generalize better and are more interpretable compared to norm penalties. In this study, we evaluated a novel structured regularization method that incorporates approximate morphology operators defined using harmonic mean-based fW-filters. We extended this method to multiclass classification and conducted experiments aimed at classifying magnetic resonance images (MRI) of subjects into four stages of Alzheimer's disease progression. The experimental results demonstrate that the novel structured regularization method not only performs better than standard sparse and structured regularization methods in terms of prediction accuracy (ACC), F1 scores, and the area under the receiver operating characteristic curve (AUC), but also produces interpretable coefficient maps.

This work proposes a structured prior integrated within the Bayesian framework for variable flip angle T1 mapping. The proposed structured prior combines total variation (TV) and L1 norm functions, and is proven to be a proper prior. The TV–L1 prior promotes sparsity in the spatial gradients of the parametric maps, resulting in smooth and coherent image reconstructions. Embedding the prior within the Bayesian framework enables uncertainty quantification for both T1 and M0 estimates. Posterior inference was performed using the No-U-Turn Sampler (NUTS). The proposed method is compared to maximum likelihood estimation and to alternative Bayesian models that employ uniform, Laplace, and bounded TV priors. The results show that the proposed method yields narrower probability density functions, indicating reduced uncertainty. The proposed method also achieves lower variance and exhibits a smaller negative bias, reflecting more stable estimates. Overall, the integration of TV and L1 functions in a prior within the Bayesian framework enhances spatial coherence in T1 mapping and delivers improved uncertainty quantification, making it a promising tool for robust quantitative MRI parameter estimation.

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.