Hierarchical Functional Maps between Subdivision Surfaces

Shoham, Moshe; Vaxman, Amir; Ben‐Chen, Mirela

doi:10.1111/cgf.13789

Cited by 15 publications

(8 citation statements)

References 66 publications

(78 reference statements)

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“…While computing a functional map reduces to solving a least squares system, the conversion from a functional map to a point-wise map is not trivial and can lead to errors and noise [Ezuz and Ben-Chen 2017;Rodolà et al 2015]. To improve accuracy, several desirable map attributes have been promoted via regularizers for the functional map estimation rst using geometric insights [Burghard et al 2017;Eynard et al 2016;Litany et al 2017b;Nogneng and Ovsjanikov 2017;Ren et al 2018;Shoham et al 2019;Wang et al 2018a,b], and more recently using learning-based techniques [Halimi et al 2019;Roufosse et al 2019]. Nevertheless, despite signicant progress, the reliance on descriptors and decoupling of continuous optimization and pointwise map conversion remains common to all existing methods.…”

Section: Shape Matching With Functional Mapsmentioning

confidence: 99%

MapTree

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In this paper we propose an approach for computing multiple high-quality near-isometric dense correspondences between a pair of 3D shapes. Our method is fully automatic and does not rely on user-provided landmarks or descriptors. This allows us to analyze the full space of maps and extract multiple diverse and accurate solutions, rather than optimizing for a single optimal correspondence as done in most previous approaches. To achieve this, we propose a compact tree structure based on the spectral map representation for encoding and enumerating possible rough initializations, and a novel efficient approach for refining them to dense pointwise maps. This leads to a new method capable of both producing multiple high-quality correspondences across shapes and revealing the symmetry structure of a shape without a priori information. In addition, we demonstrate through extensive experiments that our method is robust and results in more accurate correspondences than state-of-the-art for shape matching and symmetry detection.

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Section: Shape Matching With Functional Mapsmentioning

confidence: 99%

MapTree

et al. 2020

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“…While computing a functional map reduces to solving a least squares system, the conversion from a functional map to a point-wise map is not trivial and can lead to inaccuracy and noise [Ezuz and Ben-Chen 2017;Rodolà et al 2015]. To improve accuracy, several desirable map attributes have been promoted via regularizers for the functional map estimation first using geometric insights [Burghard et al 2017;Eynard et al 2016;Litany et al 2017b;Nogneng and Ovsjanikov 2017;Ren et al 2018;Shoham et al 2019;Wang et al 2018a,b], and more recently using learning-based techniques [Halimi et al 2019;Roufosse et al 2019]. Nevertheless, despite significant progress, the reliance on descriptors and decoupling of continuous optimization and pointwise map conversion remains common to all existing methods.…”

Section: Shape Matching With Functional Mapsmentioning

confidence: 99%

MapTree: Recovering Multiple Solutions in the Space of Maps

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Melzi,

Ovsjanikov

et al. 2020

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“…We also note other commonly-used relaxations for matching problems based on optimal transport, e.g. [Mandad et al 2017;Solomon et al 2016], which are often solved through iterative refinement. Other techniques that exploit a similar formalism for solving optimal assignment include the Product Manifold Filter and its variants [Vestner et al 2017a,b].…”

Section: Contributions To Summarizementioning

confidence: 99%

“…Spatial refinement methods such as PMF [Vestner et al 2017a,b] operate via an alternating diffusion process based on solving a sequence of linear assignment problems; this approach demonstrates high accuracy in challenging cases, but is severely limited by mesh resolution. Other approaches formulate shape correspondence by seeking for optimal transport plans iteratively via Sinkhorn projections, but they either scale poorly [Solomon et al 2016] or can have issues with non-isotropic meshes [Mandad et al 2017]. Interestingly, although fundamentally different, a link exists between ZoomOut and PMF that we describe in the supplementary materials.…”

Section: Relation To Other Techniquesmentioning

confidence: 99%

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ZoomOut

et al. 2019

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We present a simple and efficient method for refining maps or correspondences by iterative upsampling in the spectral domain that can be implemented in a few lines of code. Our main observation is that high quality maps can be obtained even if the input correspondences are noisy or are encoded by a small number of coefficients in a spectral basis. We show how this approach can be used in conjunction with existing initialization techniques across a range of application scenarios, including symmetry detection, map refinement across complete shapes, non-rigid partial shape matching and function transfer. In each application we demonstrate an improvement with respect to both the quality of the results and the computational speed compared to the best competing methods, with up to two orders of magnitude speed-up in some applications. We also demonstrate that our method is both robust to noisy input and is scalable with respect to shape complexity. Finally, we present a theoretical justification for our approach, shedding light on structural properties of functional maps.155:2 • Melzi. et al the same or better quality at a fraction of the cost compared to the current top performing methods.(2) We demonstrate how higher-frequency information can be extracted from low-frequency spectral map representations. (3) We introduce a novel variational optimization problem and develop a theoretical justification of our method, shedding light on structural properties of functional maps. RELATED WORKShape matching is a very well-studied area of computer graphics. Below we review the methods most closely related to ours, concentrating on spectral techniques for finding correspondences between non-rigid shapes. We refer the interested readers to recent surveys including [Biasotti et al. 2016;Tam et al. 2013;Van Kaick et al. 2011] for a more in-depth treatment of the area.Point-based Spectral Methods. Early spectral methods for shape correspondence were based on directly optimizing pointwise maps between spectral shape embeddings based on either adjacency or Laplacian matrices of graphs and triangle meshes [Jain and Zhang 2006;Jain et al. 2007;Mateus et al. 2008;Ovsjanikov et al. 2010;Scott and Longuet-Higgins 1991;Umeyama 1988]. Such approaches suffer from the requirement of a good initialization, and rely on restricting assumptions about the type of transformation relating the shapes. An initialization algorithm with optimality guarantees, although limited to few tens of points, was introduced in [Maron et al. 2016] and later extended to deal with intrinsic symmetries in [Dym and Lipman 2017]. Spectral quantities (namely, sequences of Laplacian eigenfunctions) have also been used to define pointwise descriptors, and employed within variants of the quadratic assignment problem in Kimmel 2010, 2011]. These approaches have been recently generalized by spectral generalized multidimensional scaling [Aflalo et al. 2016], which explicitly formulates minimumdistortion shape correspondence in the spectral domain.

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Hierarchical Functional Maps between Subdivision Surfaces

Cited by 15 publications

References 66 publications

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