Browsing by Author "Bronstein, M. M."

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    Functional Maps Representation On Product Manifolds
    (© 2019 The Eurographics Association and John Wiley & Sons Ltd., 2019) Rodolà, E.; Lähner, Z.; Bronstein, A. M.; Bronstein, M. M.; Solomon, J.; Chen, Min and Benes, Bedrich
    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.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.
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    Localized Manifold Harmonics for Spectral Shape Analysis
    (© 2018 The Eurographics Association and John Wiley & Sons Ltd., 2018) Melzi, S.; Rodolà, E.; Castellani, U.; Bronstein, M. M.; Chen, Min and Benes, Bedrich
    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.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.
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    Shape Correspondence with Isometric and Non-Isometric Deformations
    (The Eurographics Association, 2019) Dyke, R. M.; Stride, C.; Lai, Y.-K.; Rosin, P. L.; Aubry, M.; Boyarski, A.; Bronstein, A. M.; Bronstein, M. M.; Cremers, D.; Fisher, M.; Groueix, T.; Guo, D.; Kim, V. G.; Kimmel, R.; Lähner, Z.; Li, K.; Litany, O.; Remez, T.; Rodolà, E.; Russell, B. C.; Sahillioglu, Y.; Slossberg, R.; Tam, G. K. L.; Vestner, M.; Wu, Z.; Yang, J.; Biasotti, Silvia and Lavoué, Guillaume and Veltkamp, Remco
    The registration of surfaces with non-rigid deformation, especially non-isometric deformations, is a challenging problem. When applying such techniques to real scans, the problem is compounded by topological and geometric inconsistencies between shapes. In this paper, we capture a benchmark dataset of scanned 3D shapes undergoing various controlled deformations (articulating, bending, stretching and topologically changing), along with ground truth correspondences. With the aid of this tiered benchmark of increasingly challenging real scans, we explore this problem and investigate how robust current state-of- the-art methods perform in different challenging registration and correspondence scenarios. We discover that changes in topology is a challenging problem for some methods and that machine learning-based approaches prove to be more capable of handling non-isometric deformations on shapes that are moderately similar to the training set.