Coupled quasi‐harmonic bases

By Artiom Kovnatsky, Michael M. Bronstein, A. Bronstein, Klaus Glashoff, Ron Kimmel. 2012

208 citations recorded by the source corpus.

The use of Laplacian eigenbases has been shown to be fruitful in many computer graphics applications. Today, state‐of‐the‐art approaches to shape analysis, synthesis, and correspondence rely on these natural harmonic bases that allow using classical tools from harmonic analysis on manifolds. However, many applications involving multiple shapes are obstacled by the fact that Laplacian eigenbases computed independently on different shapes are often incompatible with each other. In this paper, we propose the construction of common approximate eigenbases for multiple shapes using approximate joint diagonalization algorithms, taking as input a set of corresponding functions (e.g. indicator functions of stable regions) on the two shapes. We illustrate the benefits of the proposed approach on tasks from shape editing, pose transfer, correspondence, and similarity.

Subfields: Computational Mechanics, Computer Graphics and Computer-Aided Design

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Citations in the corpus

Geometric Deep Learning: Going beyond Euclidean data (2016, background): “It is possible to construct compatible orthogonal bases across different domains resorting to a joint diagonalization procedure [74], [75].”
Dense Non-rigid Shape Correspondence Using Random Forests (2014, background): “Another recent approach [9] attempts to model deviation from isometry in the framework of functional maps, by seeking compatible harmonic bases among two shapes.”
Shape‐from‐Operator: Recovering Shapes from Intrinsic Operators (2015, background): “Kovnatsky et al. [KBB ∗ 13] showed how to ﬁnd a common intrinsic structure of a collection of shapes by ‘averaging’ the eigenspaces of the respective Laplacians by means of a joint diagonalization process.”
Multimodal Manifold Analysis by Simultaneous Diagonalization of Laplacians (2015, unknown): “Other applications heavily relying on the properties of the Laplacian include spectral graph partitioning [5], spectral hashing [6], spectral correspondence , image segmentation [7], and spectral shape analysis [8], [9], [10], [11].”