A Framework for Wasserstein-1-Type Metrics

Optimal transport and Wasserstein distances have become foundational in image processing, shape analysis, and machine learning, with classical models like Monge-Kantorovich providing rigorous mathematical tools. Recent developments introduced unbalanced models such as the Hellinger-Kantorovich and sliced-Wasserstein distances, addressing issues of mass variation and noise robustness. However, these models often lack a unified theoretical framework and computational efficiency for large-scale problems. The need for a flexible, convex, and computationally feasible extension of W1 remains pressing, especially in applications like biomedical imaging and video analysis where mass can fluctuate due to biological activity or measurement noise.

Existing Wasserstein-1 metrics are limited to measures with equal total mass, which restricts their use in practical scenarios involving mass creation or destruction. Nonlinear models like entropy-regularized OT improve computational aspects but often sacrifice theoretical guarantees or interpretability. The challenge is to develop a convex, efficient, and unified framework that can handle unbalanced measures, incorporate local discrepancies, and be scalable for high-dimensional data. Addressing this gap is crucial for advancing applications in dynamic imaging, shape deformation, and complex data analysis where mass change is intrinsic.

The paper introduces two novel classes of Wasserstein-1 type distances: (1) a convex discrepancy model based on concave functions \(h_0, h_1\) and a convex set \(B\), which generalizes the Kantorovich-Rubinstein formula; (2) an infimal convolution-based model that combines W1 with local discrepancy functionals \(D\). These models are shown to be equivalent via convex duality, unifying existing non-balanced transport models like flat metrics and Hellinger-Kantorovich. Algorithmically, the authors develop a variable splitting scheme and a Primal-Dual iterative solver, enabling scalable and robust computation. Theoretically, they prove properties like existence, semi-continuity, and bounds, ensuring stability and interpretability.

• �� Define the discrepancy \(Wh_{0,h_1,B}\) via a supremum over functions \(\alpha, eta\) constrained by Lipschitz and set \(B\).• Use convex duality to relate this to an infimal convolution model \(WD_{0,D_{01},D_1}\), involving local discrepancy functionals \(D\).• Prove the equivalence of these models, ensuring a flexible yet rigorous framework. • Develop a numerical scheme based on variable splitting: splitting variables \(\alpha, eta\), and auxiliary variables for discrepancy terms. • Implement Proximal operators for each component, ensuring convergence. • Discretize continuous models via finite differences or finite elements, enabling implementation on digital data. • Validate through experiments on biological images, videos, and texture tasks, comparing with classical W1 and other unbalanced models.

Experiments utilize datasets from microscopy (biological cells), synthetic videos, and texture images. Baselines include classical W1, entropy-regularized OT, and recent unbalanced models. Metrics include mean absolute error, structural similarity, and flow accuracy. Hyperparameters like regularization weights are tuned via cross-validation. Ablation studies analyze the impact of different discrepancy functions and regularization terms. Results show the proposed models outperform baselines in accuracy, robustness, and computational efficiency, especially in scenarios with mass variation and noise. Visualizations include flow fields, reconstructed images, and texture decompositions, demonstrating the models’ practical effectiveness.

In biological imaging, the unbalanced W1 reduced flow estimation error by 15%, with a 20% error decrease in noisy conditions. Video analysis showed a 10% improvement in tracking cellular migration under noise. Texture decomposition experiments achieved clearer separation of noise and structures, with quantitative metrics surpassing existing methods. Ablation results confirmed the importance of the convex function choice and discrepancy parameters, guiding optimal model configurations. Overall, the models demonstrated superior robustness and scalability across diverse tasks.

The framework applies directly to biomedical imaging, video motion tracking, texture analysis, and shape deformation. It is suitable for scenarios with incomplete or noisy data, biological mass changes, and dynamic environments. The models can be integrated into existing pipelines for image registration, segmentation, and flow estimation, providing more accurate and stable results. Long-term, the approach can be combined with deep learning for end-to-end training, enabling real-time processing in medical diagnostics, autonomous navigation, and multimedia editing.

Parameter tuning remains critical, as inappropriate regularization can lead to suboptimal results. Computational costs, especially in high dimensions, are still significant, requiring further optimization. In extreme cases of rapid mass change or severe noise, convergence may slow or results may degrade. Future work should focus on adaptive schemes, acceleration techniques, and broader validation across application domains.