Functional Maps Representation On Product Manifolds

We consider the problem of localizing relevant subsets of non-rigid geometric shapes given only a partial 3D query as the input. Such problems arise in several challenging tasks in 3D vision and graphics, including partial shape similarity, retrieval, and non-rigid correspondence. We phrase the problem as one of alignment between short sequences of eigenvalues of basic differential operators, which are constructed upon a scalar function defined on the 3D surfaces. Our method therefore seeks for a scalar function that entails this alignment. Differently from existing approaches, we do not require solving for a correspondence between the query and the target, therefore greatly simplifying the optimization process; our core technique is also descriptor-free, as it is driven by the geometry of the two objects as encoded in their operator spectra. We further show that our spectral alignment algorithm provides a remarkably simple alternative to the recent shape-from-spectrum reconstruction approaches. For both applications, we demonstrate improvement over the state-of-the-art either in terms of accuracy or computational cost.

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“…with boundary conditions when necessary. Hamiltonian eigenfunctions have been used in shape analysis applications in [10,24,34].…”

Section: Introductionmentioning

confidence: 99%

Correspondence-Free Region Localization for Partial Shape Similarity via Hamiltonian Spectrum Alignment

Rampini

Tallini

Ovsjanikov

et al. 2019

2019 International Conference on 3D Vision (3DV)

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“…Functional maps express maps between surfaces through linear operators transporting functions on one surface to functions on another. The original works have been extended in further research [8,10,20,23,27,30]. These techniques represent the correspondences using the functional map between two manifolds instead of point-to-point matching in Euclidean space.…”

Section: Related Workmentioning

confidence: 99%

“…A probabilistic model for functional maps was introduced in [27] and used soft-maps with analytical distributions. In [23] the isometries challenge of shape matching was tackled by employing functional maps corresponding to a point-wise map with diagonal descriptor matrices.…”

Section: Related Workmentioning

confidence: 99%

“…In this framework, the correspondence for a pair of functions f : X → R and g : Y → R can be written as Cb = a, where a and b are the representation of f and g in the chosen bases. Note that, if the bases are indicator functions, then C ≡ P. Yet, to reduce the representation complexity as well as to better handle the typical near-isometric mappings, it is common to use the first N B Laplace-Beltrami eigenfunctions [24,27] as the bases. Moreover, for two shapes, one can often compute a set of pairs of functions {f k } and {g k }, s.t.…”

Section: Introductionmentioning

confidence: 99%

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Riemannian Functional Map Synchronization for Probabilistic Partial Correspondence in Shape Networks

Huq¹,

Dey²,

Sahra³

et al. 2021

Preprint

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Functional maps are efficient representations of shape correspondences, that provide matching of real-valued functions between pairs of shapes. Functional maps can be modelled as elements of the Lie group SO(n) for nearly isometric shapes. Synchronization can subsequently be employed to enforce cycle consistency between functional maps computed on a set of shapes, hereby enhancing the accuracy of the individual maps. There is an interest in developing synchronization methods that respect the geometric structure of SO(n), while introducing a probabilistic framework to quantify the uncertainty associated with the synchronization results. This paper introduces a Bayesian probabilistic inference framework on SO(n) for Riemannian synchronization of functional maps, performs a maximum-a-posteriori estimation of functional maps through synchronization and further deploys a Riemannian Markov-Chain Monte Carlo sampler for uncertainty quantification. Our experiments demonstrate that constraining the synchronization on the Riemannian manifold SO(n) improves the estimation of the functional maps, while our Riemannian MCMC sampler provides for the first time an uncertainty quantification of the results.

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“…To mitigate this, Yi et al [33] propose a network architecture to synchronize the spectral domains and then perform convolutional operations on it. Rodolà et al [24] design a fully connected network to learn features that can generate functional map matrices; however, fully connected networks may suffer from overfitting, and their method requires point-wise correspondences to train the model which is not required by our method.…”

Section: Related Workmentioning

confidence: 99%

Learning-based Real-time Detection of Intrinsic Reflectional Symmetry

Qiao,

Gao,

Liu

et al. 2019

Preprint

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Reflectional symmetry is ubiquitous in nature. While extrinsic reflectional symmetry can be easily parametrized and detected, intrinsic symmetry is much harder due to the high solution space. Previous works usually solve this problem by voting or sampling, which suffer from high computational cost and randomness. In this paper, we propose a learning-based approach to intrinsic reflectional symmetry detection. Instead of directly finding symmetric point pairs, we parametrize this self-isometry using a functional map matrix, which can be easily computed given the signs of Laplacian eigenfunctions under the symmetric mapping. Therefore, we train a novel deep neural network to predict the sign of each eigenfunction under symmetry, which in addition takes the first few eigenfunctions as intrinsic features to characterize the mesh while avoiding coping with the connectivity explicitly. Our network aims at learning the global property of functions, and consequently converts the problem defined on the manifold to the functional domain. By disentangling the prediction of the matrix into separated basis, our method generalizes well to new shapes and is invariant under perturbation of eigenfunctions. Through extensive experiments, we demonstrate the robustness of our method in challenging cases, including different topology and incomplete shapes with holes. By avoiding random sampling, our learning-based algorithm is over 100 times faster than state-of-the-art methods, and meanwhile, is more robust, achieving higher correspondence accuracy in commonly used metrics.

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Functional Maps Representation On Product Manifolds

Cited by 12 publications

References 45 publications

Correspondence-Free Region Localization for Partial Shape Similarity via Hamiltonian Spectrum Alignment

Correspondence-Free Region Localization for Partial Shape Similarity via Hamiltonian Spectrum Alignment

Riemannian Functional Map Synchronization for Probabilistic Partial Correspondence in Shape Networks

Learning-based Real-time Detection of Intrinsic Reflectional Symmetry

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