Surface Comparison with Mass Transportation
For researchers in shape analysis and computer graphics, this provides a novel metric for surface comparison and alignment, but the results are preliminary and lack quantitative benchmarks.
The paper uses mass transportation to compare surfaces by solving a transport problem between conformal densities, requiring Möbius invariance, and defines a metric for simply-connected surfaces with boundary. Numerical experiments on real surfaces demonstrate applications in natural sciences.
We use mass-transportation as a tool to compare surfaces (2-manifolds). In particular, we determine the "similarity" of two given surfaces by solving a mass-transportation problem between their conformal densities. This mass transportation problem differs from the standard case in that we require the solution to be invariant under global Möbius transformations. Our approach provides a constructive way of defining a metric in the abstract space of simply-connected smooth surfaces with boundary (i.e. surfaces of disk-type); this metric can also be used to define meaningful intrinsic distances between pairs of "patches" in the two surfaces, which allows automatic alignment of the surfaces. We provide numerical experiments on "real-life" surfaces to demonstrate possible applications in natural sciences.