cs.CVAug 20, 2026

Coupled Optimal Transport with Landmark Constraints

Authors: Xiang Gu, Jian Sun, Zongben Xu

Organizations: School of Mathematics and Statistics, Xi’an Jiaotong University, China

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Abstract

Existing optimal transport (OT) models primarily seek an OT map or plan between distributions by minimizing a prescribed transport cost or distortion. However, minimizing transport cost or distortion alone may fail to identify a geometrically meaningful transformation between the two distributions. To address this limitation, this paper proposes a novel coupled OT framework that leverages a small number of annotated landmarks to guide the recovery of an underlying deformation governing the distribution transformation. The coupled OT framework integrates the optimization of the transport plan and the deformation field into a unified model, where the landmark-guided deformation field and the cost-driven transport plan are coupled through a mutual-consistency constraint. As a result, the deformation is jointly determined by the annotated landmarks and cost-driven distribution matching. The proposed framework provides a principled connection between landmark-based registration and transport-based distribution matching, enabling the recovery of transport maps from sparse geometric supervision. We establish the well-definedness of the proposed model in a general variational setting and develop a finite-element-based numerical algorithm for computation whose convergence properties are systematically analyzed. The practical effectiveness of the proposed approach is verified in shape matching.

Figures & tables

Figure 1: Fish-shaped source and target densities, together with the candidate landmarks marked by red crosses.
Figure 2: Left and middle: objective values F(z(k))\mathcal{F}(\mathbf{z}^{(k)}) along the iterations of the KL-proximal (COT (proximal KL)) and entropy-regularized (COT (entropy)) algorithms for coupled OT. Right: relative error ∥z(k+1)−z(k)∥\|\mathbf{z}^{(k+1)}-\mathbf{z}^{(k)}\| between two successive iterates of COT (entropy).
Figure 3: Deformation error (left), held-out landmark error (middle), and density error (right) of four methods under varying numbers of landmarks on the fish-shaped mask. Results are averaged over five replications.
Figure 4: Comparison of the deformation fields estimated by different methods. In each row, the source and target shapes are shown in blue and green, respectively. Two landmark pairs indicated in red and orange are provided for each shape pair, and the estimated deformation fields are visualized by pink arrows.
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