Regularized Discrete Optimal Transport (1307.5551v1)

Abstract: This article introduces a generalization of the discrete optimal transport, with applications to color image manipulations. This new formulation includes a relaxation of the mass conservation constraint and a regularization term. These two features are crucial for image processing tasks, which necessitate to take into account families of multimodal histograms, with large mass variation across modes. The corresponding relaxed and regularized transportation problem is the solution of a convex optimization problem. Depending on the regularization used, this minimization can be solved using standard linear programming methods or first order proximal splitting schemes. The resulting transportation plan can be used as a color transfer map, which is robust to mass variation across images color palettes. Furthermore, the regularization of the transport plan helps to remove colorization artifacts due to noise amplification. We also extend this framework to the computation of barycenters of distributions. The barycenter is the solution of an optimization problem, which is separately convex with respect to the barycenter and the transportation plans, but not jointly convex. A block coordinate descent scheme converges to a stationary point of the energy. We show that the resulting algorithm can be used for color normalization across several images. The relaxed and regularized barycenter defines a common color palette for those images. Applying color transfer toward this average palette performs a color normalization of the input images.

• The paper introduces a relaxed optimal transport model with an added regularization term to enable flexible mass mapping and improve visual outputs.
• It applies convex optimization and proximal splitting techniques to efficiently solve color image manipulation tasks.
• Empirical results demonstrate enhanced color coherence and texture preservation, outperforming traditional methods in barycenter computation.

Overview of "Regularized Discrete Optimal Transport"

The research paper titled "Regularized Discrete Optimal Transport" by Sira Ferradans et al. explores an advanced optimization approach for discrete optimal transport (OT) with specific applications in color image manipulation. The core contributions of this work are the relaxation of the mass conservation constraint and the integration of a regularization term within the OT framework. These modifications aim to address significant challenges in image processing, particularly those involving multimodal histograms with large variations in mass across different modes.

Key Contributions and Methodology

The paper introduces a modified OT formulation designed to generate more visually appealing transformations for tasks like color transfer between images. The main components of the proposed method include:

1. Relaxation and Regularization: Traditional OT demands strict mass conservation, which can be problematic when the source and target distributions differ significantly in scale. The introduced relaxation allows flexibility in mapping, accommodating mass variation. Regularization, on the other hand, smoothens the transport map, thereby mitigating artifacts commonly induced by irregular mappings.
2. Optimization Techniques: The transportation problem is reformulated as a convex optimization problem which can be tackled through linear programming or proximal splitting techniques depending on the type of regularization applied. These approaches facilitate efficient computation while maintaining the desired relaxations and regularizations in the transport map.
3. Barycenter Computation: Extending the OT framework, the paper explores the computation of barycenters, effectively averaging multiple distributions to find a central palette. This has implications for tasks such as color normalization across a series of images.

Numerical Results and Claims

The authors provide convincing empirical evidence supporting their claims. The methodology is applied to color transfer tasks, demonstrating robust results even with significant variations in image palettes. The regularization and relaxation lead to improved color coherence and reduced artifact visibility, compared to traditional OT methods or existing alternatives like Papadakis's and Pitié's methods. Specifically, the approach preserves texture details better and avoids undesirable noise amplification.

Implications and Future Directions

The proposed approach holds significant promise in broader image processing applications. By allowing for relaxed matching and smooth transformations, this method can benefit areas such as image registration, enhancement, and synthetic texture creation. Future research can explore extending these techniques to higher-dimensional data or integrating learning-based methods to further optimize computational efficiency and result fidelity.

Conclusion

The authors have successfully showcased a regularized and relaxed discrete OT framework that significantly advances the state of color manipulation in digital imaging. The balance between theoretical grounding and practical application places this research as a valuable contribution to computational imaging, with potential expansions into various domains requiring efficient transport solutions.