Parallelised Differentiable Straightest Geodesics for 3D Meshes

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• The paper provides a parallel GPU implementation of straightest geodesics for 3D meshes.technical 95%
  Evidencepartial

  We provide a parallel GPU implementation and derive two different methods for differentiating through the straightest geodesics

  Implicationpartial

  Explicitly stated in abstract as a core contribution

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• The method enables differentiable exponential map computation on meshes, which can improve learning and optimization pipelines.method 90%
  Evidencepartial

  demonstrate how our differentiable exponential map can improve learning and optimisation pipelines on general geometries

  Implicationpartial

  Directly stated in abstract with demonstration promise

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• The paper introduces a new geodesic convolutional layer for learning on meshes.method 90%
  Evidencepartial

  we propose a new geodesic convolutional layer, a new flow matching method for learning on meshes, and a second-order optimiser

  Implicationpartial

  Explicitly stated as one of three demonstration applications

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• The method includes two different approaches for differentiating through straightest geodesics: one using an extrinsic proxy function and one based on geodesic finite differences.technical 95%
  Evidencepartial

  derive two different methods for differentiating through the straightest geodesics, one leveraging an extrinsic proxy function and one based upon a geodesic finite differences scheme

  Implicationpartial

  Explicitly stated in abstract with specific technical details

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• The paper demonstrates application of the method to centroidal Voronoi tessellation using a second-order optimizer.result 90%
  Evidencepartial

  a second-order optimiser that we apply to centroidal Voronoi tessellation

  Implicationpartial

  Directly stated as one of the demonstration applications

  Verificationpartial

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• The implementation is available as a pip-installable library called digeo.technical 95%
  Evidencepartial

  Our code, models, and pip-installable library (digeo) are available at: circle-group.github.io/research/DSG

  Implicationpartial

  Explicitly stated in abstract with specific library name

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• The method addresses limitations of previous approaches including lack of closed-form Riemannian operators and poor parallelization capabilities.limitation 85%
  Evidencepartial

  The lack of closed-form Riemannian operators, the non-differentiability of their discrete counterparts, and poor parallelisation capabilities have been the main obstacles to the development of the field on meshes

  Implicationpartial

  Strongly implied in abstract as motivation for the work

  Verificationpartial

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• Straightest geodesics provide a principled framework for computing exponential maps on Riemannian surfaces discretized as meshes.method 90%
  Evidencepartial

  A principled framework to compute the exponential map on Riemannian surfaces discretised as meshes is straightest geodesics

  Implicationpartial

  Directly stated in abstract as theoretical foundation

  Verificationpartial

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