Complex Functional Maps : a Conformal Link Between Tangent Bundles

• URL: http://arxiv.org/abs/2112.09546v1
• Date: Fri, 17 Dec 2021 14:54:01 GMT
• Title: Complex Functional Maps : a Conformal Link Between Tangent Bundles
• Authors: Nicolas Donati (LIX), Etienne Corman (LORIA, CNRS, PIXEL), Simone Melzi (Sapienza University of Rome), Maks Ovsjanikov (LIX)
• Abstract summary: We introduce complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces. A key property of these maps is their orientation awareness. We show that functional maps and their complex counterparts can be estimated jointly to promote orientation preservation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we introduce complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces. A key property of these maps is their orientation awareness. More specifically, we demonstrate that unlike regular functional maps that link functional spaces of two manifolds, our complex functional maps establish a link between oriented tangent bundles, thus permitting robust and efficient transfer of tangent vector fields. By first endowing and then exploiting the tangent bundle of each shape with a complex structure, the resulting operations become naturally orientationaware, thus favoring orientation and angle preserving correspondence across shapes, without relying on descriptors or extra regularization. Finally, and perhaps more importantly, we demonstrate how these objects enable several practical applications within the functional map framework. We show that functional maps and their complex counterparts can be estimated jointly to promote orientation preservation, regularizing pipelines that previously suffered from orientation-reversing symmetry errors.

Related papers

• Zero-Shot Image Feature Consensus with Deep Functional Maps [20.988872402347756]
  We show that a better correspondence strategy is available, which directly imposes structure on the correspondence field: the functional map. We demonstrate that our technique yields correspondences that are not only smoother but also more accurate, with the possibility of better reflecting the knowledge embedded in the large-scale vision models that we are studying.
  arXiv  Detail & Related papers  (2024-03-18T17:59:47Z)
• Spectral Meets Spatial: Harmonising 3D Shape Matching and Interpolation [50.376243444909136]
  We present a unified framework to predict both point-wise correspondences and shape between 3D shapes. We combine the deep functional map framework with classical surface deformation models to map shapes in both spectral and spatial domains.
  arXiv  Detail & Related papers  (2024-02-29T07:26:23Z)
• Revisiting Map Relations for Unsupervised Non-Rigid Shape Matching [18.957179015912402]
  We propose a novel unsupervised learning approach for non-rigid 3D shape matching. We show that our method substantially outperforms previous state-of-the-art methods.
  arXiv  Detail & Related papers  (2023-10-17T17:28:03Z)
• Understanding and Improving Features Learned in Deep Functional Maps [31.61255365182462]
  We show that features learned within deep functional map approaches can be used as point-wise descriptors across different shapes. We propose effective modifications to the standard deep functional map pipeline, which promote structural properties of learned features.
  arXiv  Detail & Related papers  (2023-03-29T08:32:16Z)
• Smooth Non-Rigid Shape Matching via Effective Dirichlet Energy Optimization [46.30376601157526]
  We introduce pointwise map smoothness via the Dirichlet energy into the functional map pipeline. We propose an algorithm for optimizing it efficiently, which leads to high-quality results in challenging settings.
  arXiv  Detail & Related papers  (2022-10-05T14:07:17Z)
• Learning Implicit Feature Alignment Function for Semantic Segmentation [51.36809814890326]
  Implicit Feature Alignment function (IFA) is inspired by the rapidly expanding topic of implicit neural representations. We show that IFA implicitly aligns the feature maps at different levels and is capable of producing segmentation maps in arbitrary resolutions. Our method can be combined with improvement on various architectures, and it achieves state-of-the-art accuracy trade-off on common benchmarks.
  arXiv  Detail & Related papers  (2022-06-17T09:40:14Z)
• Entangled Residual Mappings [59.02488598557491]
  We introduce entangled residual mappings to generalize the structure of the residual connections. An entangled residual mapping replaces the identity skip connections with specialized entangled mappings. We show that while entangled mappings can preserve the iterative refinement of features across various deep models, they influence the representation learning process in convolutional networks.
  arXiv  Detail & Related papers  (2022-06-02T19:36:03Z)
• Deep Orientation-Aware Functional Maps: Tackling Symmetry Issues in Shape Matching [32.03608983026839]
  We propose a new deep learning approach to learn orientation-aware features in a fully unsupervised setting. Our architecture is built on top of DiffusionNet, making it robust to discretization changes.
  arXiv  Detail & Related papers  (2022-04-28T12:36:09Z)
• Multiway Non-rigid Point Cloud Registration via Learned Functional Map Synchronization [105.14877281665011]
  We present SyNoRiM, a novel way to register multiple non-rigid shapes by synchronizing the maps relating learned functions defined on the point clouds. We demonstrate via extensive experiments that our method achieves a state-of-the-art performance in registration accuracy.
  arXiv  Detail & Related papers  (2021-11-25T02:37:59Z)
• Learning Lane Graph Representations for Motion Forecasting [92.88572392790623]
  We construct a lane graph from raw map data to preserve the map structure. We exploit a fusion network consisting of four types of interactions, actor-to-lane, lane-to-lane, lane-to-actor and actor-to-actor. Our approach significantly outperforms the state-of-the-art on the large scale Argoverse motion forecasting benchmark.
  arXiv  Detail & Related papers  (2020-07-27T17:59:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.