The paper starts by introducing Finsler manifolds as a generalization of Riemannian manifolds, where the metric not only depends on the location but also on the tangential direction of motion. This allows for more flexibility in capturing anisotropic features on manifolds.
The authors then revisit the heat diffusion on Finsler manifolds and derive a tractable Finsler heat equation. From this, they exhibit a simplified equation that behaves like a regular heat equation with an external heat source. The solution to this new problem is explicit and governed by an anisotropic heat diffusion with diffusivity involving the Randers metric, a specific type of Finsler metric.
This heat kernel allows the construction of a new Finsler-Laplace-Beltrami operator (FLBO) that generalizes traditional heuristic anisotropic LBOs (ALBOs) while being theoretically motivated. The FLBO can directly replace the ALBOs in surface processing tasks like shape correspondence, notably by constructing shape-dependent anisotropic convolutions in their spectral domain.
The authors provide a discretization of the FLBO and evaluate it on shape correspondence tasks, demonstrating that the proposed FLBO is a valuable alternative to the traditional Riemannian-based LBO and ALBOs. They show that the FLBO can complement state-of-the-art geometric deep learning methods using ALBOs to boost performance.
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