Improved functional mappings via product preservation

D. Nogneng, S. Melzi, E. Rodolá, U. Castellani, M. Bronstein, M. Ovsjanikov

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we consider the problem of information transfer across shapes and propose an extension to the widely used functional map representation. Our main observation is that in addition to the vector space structure of the functional spaces, which has been heavily exploited in the functional map framework, the functional algebra (i.e., the ability to take pointwise products of functions) can significantly extend the power of this framework. Equipped with this observation, we show how to improve one of the key applications of functional maps, namely transferring real-valued functions without conversion to point-to-point correspondences. We demonstrate through extensive experiments that by decomposing a given function into a linear combination consisting not only of basis functions but also of their pointwise products, both the representation power and the quality of the function transfer can be improved significantly. Our modification, while computationally simple, allows us to achieve higher transfer accuracy while keeping the size of the basis and the functional map fixed. We also analyze the computational complexity of optimally representing functions through linear combinations of products in a given basis and prove NP-completeness in some general cases. Finally, we argue that the use of function products can have a wide-reaching effect in extending the power of functional maps in a variety of applications, in particular by enabling the transfer of highfrequency functions without changing the representation size or complexity.

Original languageEnglish
Pages (from-to)179-190
Number of pages12
JournalComputer Graphics Forum
Volume37
Issue number2
DOIs
StatePublished - 2018

Funding

FundersFunder number
Ecole Poly-technique
Horizon 2020 Framework Programme758800, 724228
European Research Council
Fonds Unique Interministériel
Institute for Advanced Study, Technische Universität München

    Keywords

    • Computational geometry
    • Shape analysis
    • Theory of computation

    Fingerprint

    Dive into the research topics of 'Improved functional mappings via product preservation'. Together they form a unique fingerprint.

    Cite this