Optimal Transport on the Lie Group of Roto-translations

TL;DR

It is observed that the framework of lifting images to SE2 and optimal transport with left-invariant anisotropic metrics leads to equivariant transport along dominant contours and salient line structures in the image, which yields sharper and more meaningful interpolations compared to their counterparts on R^2.

Abstract

The roto-translation group SE2 has been of active interest in image analysis due to methods that lift the image data to multi-orientation representations defined on this Lie group. This has led to impactful applications of crossing-preserving flows for image de-noising, geodesic tracking, and roto-translation equivariant deep learning. In this paper, we develop a computational framework for optimal transportation over Lie groups, with a special focus on SE2. We make several theoretical contributions (generalizable to matrix Lie groups) such as the non-optimality of group actions as transport maps, invariance and equivariance of optimal transport, and the quality of the entropic-regularized optimal transport plan using geodesic distance approximations. We develop a Sinkhorn like algorithm that can be efficiently implemented using fast and accurate distance approximations of the Lie group and GPU-friendly group convolutions. We report valuable advancements in the experiments on 1) image barycentric interpolation, 2) interpolation of planar orientation fields, and 3) Wasserstein gradient flows on SE2. We observe that our framework of lifting images to SE2 and optimal transport with left-invariant anisotropic metrics leads to equivariant transport along dominant contours and salient line structures in the image. This yields sharper and more meaningful interpolations compared to their counterparts on R^2

Figures (14)

• Figure 1: An orientation score transform (see \ref{['eq:lift']}) disentangles line structures due to lifting onto the Lie group $\text{SE(2)}$. This is a useful property in many applications like image de-noising, geodesic tracking, equivariant deep learning, etc. In this paper, we use it for the barycentric interpolation of images using optimal transport as seen in the schematic in \ref{['fig:intro']}
• Figure 1:
• Figure 1: Tracing U-Curve Geodesics in $\text{SE(2)}$: We interpolate between two Dirac masses using a anisotropic metric $(w_1,w_2,w_3) = (1, 5, 2.5)$ in $\text{SE(2)}$. The overlaid red curve is the exact sub-Riemannian geodesic between the two endpoints computed via analytic solutions from duits2014association. 1st Row: Interpolation with Heat Diffusion in $\text{SE(2)}$. 2nd Row: Interpolation with the proposed half-angle distance approximation $\rho_b$ from equation \ref{['eq:rhob']}. We can observe that the transport from our distance approximation is more accurate to the analytic solution in comparison to the analogous solution with Heat Diffusion. Columns 1-7: Spatially projected $\text{SE(2)}$ interpolations on ${\mathbb{R}}^2$. Column 8: A cumulation of columns 1-7 showing the path that was traced. See \ref{['fig:s_curve']} for an additional example.
• Figure 1: Average barycenters of ($\sim 6000$ images per class) images from the QuickDraw dataset. 1st Row: $L^2$ barycenter of all images per class - L2R . 2nd Row: 2-Wasserstein barycenter in $\mathbb{R}^2$ - WR , 3rd Row: 2-Wasserstein barycenter in $\text{SE(2)}$ - WG . WG shows sharper interpolations in comparison with its counterparts on $\mathbb{R}^2$.
• Figure 1: Comparing image barycenters in ${\mathbb{R}}^2$ and $\text{SE(2)}$: Another example similar to \ref{['fig:umbrella']}, with MNIST digits. The Images at the corners are interpolated using linear weights. Again, one observes a more meaningful interpolation with left-invariant anisotropic optimal transport in $\text{SE(2)}$

Theorems & Definitions (16)

• Remark 2.1
• Lemma 3.1
• Proof 1
• Lemma 3.2
• Proof 2
• Theorem 3.3
• Proof 3
• Corollary 3.4
• Proof 4
• Corollary 4.1