Convex Potential Learning for Optimal Transport Between Medical Images
Authors: Hongyu Kan, Ivan Medri, Natasha Ironside, Kristofor Pas, Richard Corp, Shinjini Kundu, Gustavo Kunde Rohde
Organizations: George Washington University, Washington, DC 20052, USA · University of Illinois Chicago, Chicago, IL 60607, USA · University of Virginia, Charlottesville, VA 22903, USA · Washington University in St. Louis, St. Louis, MO 63110, USA
Computing optimal transport maps between medical images plays a fundamental role in morphological analysis. However, existing methods are either difficult to scale to large 3D medical images or rely on regularized formulations that provide only approximate transport maps. To improve the accuracy of downstream medical image analysis, we propose a novel approach for estimating optimal transport maps by parameterizing the convex potential function. By construction, the resulting transport map is guaranteed to be curl-free, consistent with Brenier's theorem. We evaluate the proposed method on both 2D and 3D medical imaging datasets. Experimental results demonstrate that our approach achieves lower transport estimation errors and produces higher-quality image reconstructions than existing methods.
Figures & tables
Figure 1: An example of optimal transport between two Gaussian distributions with different centers. I0 (left) and I1 (second from left) denote the source and target distributions, respectively. The estimated optimal transport map pushes the mass of I0 onto I1 . The reconstructed source image (second from the right) obtained via formula ( 1 ) and the estimated transport map f , closely matches I0 , demonstrating accurate push-forward consistency. The rightmost panel shows the estimated transport vector field, where the red and blue contours represent I0 and I1 , respectively, and the white arrows indicate the direction of the optimal transport map computed by the proposed method.
Cell dataset reconstruction error
ICH dataset recontruction error
Method
L1
L2
L1
L2
S. Kundu et al. [ 10 ]
0.334 ± 0.187
(21.04 ± 16.06) ∗10−5
1.098 ± 0.398
(7.89 ± 6.24) ∗10−5
Convex Potential
0.239 ± 0.068
(13.74 ± 7.88) ∗10−5
0.287 ± 0.139
(1.31 ± 1.72) ∗10−5
Table 1: L1 and L2 distances comparison
Figure 2: Visual comparison of image reconstruction using the estimated transport maps. The reconstructed images are compared with the reference image I0 according to det(Df(x))I1(f(x)) . The left panel shows representative samples from the cell dataset, while the right panel presents the middle slice of representative samples from the ICH dataset.
Figure 3: Comparison of the evolution of the L1 error, L2 error, transport cost, and curl over optimization iterations in cell dataset. Curves of the same color correspond to the same image pair.
Figure 4: Comparison of the evolution of the L1 error, L2 error, transport cost, and curl over optimization iterations in ICH dataset. Curves of the same color correspond to the same image pair.