Frame Field Operators

David R. Palmer; Oded Stein; Justin Solomon

Home/Archive/arXiv

arXiv · 2106.14360

Frame Field Operators

David R. Palmer·Oded Stein·Justin Solomon

View original source ↗Read source document ↗Export metadata ↓

Abstract

Differential operators are widely used in geometry processing for problem domains like spectral shape analysis, data interpolation, parametrization and mapping, and meshing. In addition to the ubiquitous cotangent Laplacian, anisotropic second-order operators, as well as higher-order operators such as the Bilaplacian, have been discretized for specialized applications. In this paper, we study a class of operators that generalizes the fourth-order Bilaplacian to support anisotropic behavior. The anisotropy is parametrized by a symmetric frame field, first studied in connection with quadrilateral and hexahedral meshing, which allows for fine-grained control of local directions of variation. We discretize these operators using a mixed finite element scheme, verify convergence of the discretization, study the behavior of the operator under pullback, and present potential applications.

Explore related subjects

cs.GR

Keep this discovery

BibTeX RIS

David R. Palmer, Oded Stein, Justin Solomon. 2021-06-28. Frame Field Operators. https://arxiv.org/abs/2106.14360

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

ReCHOIR: Contact-guided Human Object Interaction Retargeting to Diverse Characters

We present ReCHOIR, a novel contact-guided motion retargeting method for transferring human object interaction (HOI) motions across diverse humanoid characters. Unlike prior motion retargeting methods that primarily focus on transferring human motion alone, our goal is to preserve not only the semantics of the original body movement but also consistent interaction between the character and the manipulated object, while jointly producing aligned target human and object motions. Given source HOI motion, object geometry, and contact cues extracted from the source interaction, ReCHOIR retargets an HOI sequence to target characters with different skeletal configurations while maintaining both motion semantics and contact-consistent interaction patterns. Our method builds on a Part-Aware Motion Embedding (PAME) autoencoder, which encodes full-body motion into a shared body-part-wise latent space. This representation enables generalization across heterogeneous skeletons while preserving local motion semantics beneficial for part-aware adaptation in HOI retargeting. On top of this representation, we introduce a contact-guided retargeting module and an object motion decoder for HOI retargeting. The contact-guided retargeting module treats the source object interaction as a condition for refining target character motion: object- and contact-related signals are encoded into a body-part-aligned latent representation and injected into decoding through a residual control branch, enabling stronger adaptation in interaction-relevant body regions without discarding the underlying motion prior. In parallel, the object motion decoder predicts a target object motion aligned with the refined target character motion, ensuring that the object trajectory remains consistent with how the interaction is realized by the target character.

cs.GR↗

arXiv2026

Gaussian Light Transport

We present a novel method for computing global illumination by expressing the solution to the light transport equation as a 13D Gaussian mixture model over positions, directions, surface normals, and material properties. We show that including scene properties in the Gaussian representation drastically reduces the number of functions and speeds up evaluation. As opposed to traditional light transport methods based on Neumann series, the parameters of our model are directly estimated by minimizing the residual of the rendering equation. While both optimization and rendering require repeated evaluations of a linear combination of high-dimensional Gaussian functions, we introduce an efficient culling strategy to keep the optimization tractable and produce renderings in real time. Our representation enables to render fast, view-independent solutions to the light transport equation, achieving rendering times on the order of milliseconds, with a fraction of the memory requirements of conventional neural rendering approaches.

cs.GR↗

arXiv2026

Hologram Representation via Quadratic Phase Gaussian Splatting

We introduce Complex-Valued Quadratic Phase Gaussian (CVQPG), a novel hologram representation method that replaces standard 2D Gaussian representations used in 2D Gaussian Splatting with 2D quadratic phase functions. CVQPG incorporates additional learnable parameters to control the curvature of these bases. We evaluate our approach against state-of-the-art methods, exceeding the visual quality by +0.19 dB (RGB) and +0.33 dB (grayscale) on average in holographic reconstructions. Specifically, our equal parameter count evaluations show that modulating the primitive's wavefront is an effective and lightweight enhancement for hologram representations. In addition, our frequency domain analysis illustrates that CVQPG has successfully preserved the mid-to-high frequency band of natural images.

cs.GR↗