Hybrid Functional Maps for Robust Non-Isometric Shape Matching

Kernekoncepter

A novel hybrid basis approach combining intrinsic Laplace-Beltrami and extrinsic elastic eigenfunctions enables robust functional maps for non-isometric shape matching, outperforming state-of-the-art methods.

Resumé

The paper proposes a hybrid functional mapping approach that combines the Laplace-Beltrami (LB) eigenbasis, which is robust to coarse isometric deformations, with the elastic eigenbasis, which better captures high-frequency extrinsic shape details like bending and creases.
The key insights are:

The LB eigenbasis struggles to represent fine extrinsic details, while the elastic basis lacks the robustness of the LB basis for coarse isometric matching.
The authors derive a theoretical framework to effectively integrate the non-orthogonal elastic basis functions into the functional map optimization, generalizing the standard Frobenius norm formulation.
They then construct a hybrid basis by combining the LB and elastic eigenfunctions, resulting in a block-diagonal functional map that can leverage the strengths of both basis types.
Extensive experiments show that the hybrid approach outperforms using either basis individually, achieving significant improvements in non-isometric and topologically noisy settings across supervised, unsupervised, and axiomatic functional mapping frameworks.

Statistik

"Non-isometric shape correspondence remains a fundamental challenge in computer vision."
"Establishing dense correspondences between 3D shapes is a cornerstone for numerous computer vision and graphics tasks such as object recognition, character animation, and texture transfer."
"Many classic correspondence methods leverage that rigid transformations can be represented in six degrees of freedom in R3 and preserve the Euclidean distance between pairs of points."
"For the wider class of isometric deformations (w.r.t. the geodesic distance), the relative embedding of the shape can change significantly, and Euclidean distances between points may not be preserved."

Citater

"A known weakness of the LBO basis, which at the same time comes from its biggest strength, is the reduction to low-frequency information. This leads to efficient optimization and robustness to noise but also inaccuracy in small details."
"To counter this challenge, Hartwig et al. [21] proposed to utilize a basis derived from the spectral decomposition of an elastic thin-shell energy for functional mapping. These bases are particularly suitable for aligning extrinsic features of non-isometric deformations, for example, bending and creases [21]."
"To address the shortcomings of the bases on their own, we propose to estimate functional maps in a hybrid basis representation."

Vigtigste indsigter udtrukket fra

Hybrid Functional Maps for Crease-Aware Non-Isometric Shape Matching

by Lenn... kl. arxiv.org 04-18-2024

https://arxiv.org/pdf/2312.03678.pdf

Dybere Forespørgsler

How can the proposed hybrid basis be extended to handle partial shape matching or noisy point clouds?

The proposed hybrid basis can be extended to handle partial shape matching or noisy point clouds by incorporating additional basis functions that capture specific characteristics of partial shapes or noisy data. For partial shape matching, basis functions that focus on local features or landmarks can be integrated into the hybrid framework to improve alignment and correspondence in regions where data may be missing. These basis functions can help in interpolating and extrapolating information to complete the shape matching process effectively.
In the case of noisy point clouds, basis functions that are robust to noise and outliers can be included in the hybrid basis. These functions can help in filtering out noise and preserving the essential shape features during the matching process. By combining these noise-resistant basis functions with the existing LBO and elastic basis functions, the hybrid framework can adapt to the challenges posed by noisy data and ensure more accurate and reliable shape matching results.

How can the proposed hybrid basis be extended to handle partial shape matching or noisy point clouds?

The proposed hybrid basis can be extended to handle partial shape matching or noisy point clouds by incorporating additional basis functions that capture specific characteristics of partial shapes or noisy data. For partial shape matching, basis functions that focus on local features or landmarks can be integrated into the hybrid framework to improve alignment and correspondence in regions where data may be missing. These basis functions can help in interpolating and extrapolating information to complete the shape matching process effectively.
In the case of noisy point clouds, basis functions that are robust to noise and outliers can be included in the hybrid basis. These functions can help in filtering out noise and preserving the essential shape features during the matching process. By combining these noise-resistant basis functions with the existing LBO and elastic basis functions, the hybrid framework can adapt to the challenges posed by noisy data and ensure more accurate and reliable shape matching results.

How can the proposed hybrid basis be extended to handle partial shape matching or noisy point clouds?

The proposed hybrid basis can be extended to handle partial shape matching or noisy point clouds by incorporating additional basis functions that capture specific characteristics of partial shapes or noisy data. For partial shape matching, basis functions that focus on local features or landmarks can be integrated into the hybrid framework to improve alignment and correspondence in regions where data may be missing. These basis functions can help in interpolating and extrapolating information to complete the shape matching process effectively.
In the case of noisy point clouds, basis functions that are robust to noise and outliers can be included in the hybrid basis. These functions can help in filtering out noise and preserving the essential shape features during the matching process. By combining these noise-resistant basis functions with the existing LBO and elastic basis functions, the hybrid framework can adapt to the challenges posed by noisy data and ensure more accurate and reliable shape matching results.