Functional correspondence by matrix completion

Castellani

et al. 2017

The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer graphics and geometry processing applications. In particular, Laplacian eigenbases allow generalizing the classical Fourier analysis to manifolds. A key drawback of such bases is their inherently global nature, as the Laplacian eigenfunctions carry geometric and topological structure of the entire manifold. In this paper, we introduce a new framework for local spectral shape analysis. We show how to efficiently construct localized orthogonal bases by solving an optimization problem that in turn can be posed as the eigendecomposition of a new operator obtained by a modification of the standard Laplacian. We study the theoretical and computational aspects of the proposed framework and showcase our new construction on the classical problems of shape approximation and correspondence. We obtain significant improvement compared to classical Laplacian eigenbases as well as other alternatives for constructing localized bases.

“…By choosing the Laplacian eigenfunctions on

scriptX

and

scriptY

as the bases

{φ_{i}^{scriptX}}

and

{φ_{j}^{scriptY}}

, one can truncate the series to the first

k^{'}

terms—hence obtaining a compact representation which can be interpreted as a band‐limited approximation of the full map. Correspondence problems can then be phrased as searching for a

k^{'} \times k^{'}

matrix

boldC

minimizing simple data fidelity criteria [OBCS*12, NO17] or exhibiting a particular structure depending on the correspondence setting [PBB*13, KBBV15, RCB*17].…”

Section: Applicationsmentioning

confidence: 99%

Localized Manifold Harmonics for Spectral Shape Analysis

Melzi

Castellani

et al. 2017

“…One of the key innovations of this framework is allowing bringing a new set of algebraic methods into the domain of shape correspondence. Several follow‐up works tried to improve the framework by employing sparsity‐based priors [PBB*13], manifold optimization [KBB*13, KGB16], non‐orthogonal [KBBV15] or localized [CSBK17, MRCB18] bases, coupled optimization over the forward and inverse maps [ERGB16, EBC17, HO17], combination of functional maps with metric‐based approaches [ADK16, SK17] and kernelization [WGBS18]. Recent works of [NO17, NMR*18] considered functional algebra (function point‐wise multiplications together with addition).…”

Section: Introductionmentioning

confidence: 99%

Functional Maps Representation On Product Manifolds

Lähner²,

Bronstein

et al. 2019

We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.

“…Extensions of the functional map framework have been proposed by several authors, covering the problem of non‐isometric deformations [PBB*13, KBB*13, RBW*14, KBBV15], partial similarity [RCB*16, LRB*16], clutter [CRM*16], shape exploration [ROA*13, HWG14] and image segmentation [WHG13] among others.…”

Section: Introductionmentioning

confidence: 99%

Regularized Pointwise Map Recovery from Functional Correspondence

Moeller

Cremers³

2017

The concept of using functional maps for representing dense correspondences between deformable shapes has proven to be extremely effective in many applications. However, despite the impact of this framework, the problem of recovering the point‐to‐point correspondence from a given functional map has received surprisingly little interest. In this paper, we analyse the aforementioned problem and propose a novel method for reconstructing pointwise correspondences from a given functional map. The proposed algorithm phrases the matching problem as a regularized alignment problem of the spectral embeddings of the two shapes. Opposed to established methods, our approach does not require the input shapes to be nearly‐isometric, and easily extends to recovering the point‐to‐point correspondence in part‐to‐whole shape matching problems. Our numerical experiments demonstrate that the proposed approach leads to a significant improvement in accuracy in several challenging cases.