Equi-affine invariant geometry for shape analysis

Dan Raviv*, Alexander M. Bronstein, Michael M. Bronstein, Dan Waisman, Nir Sochen, Ron Kimmel

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


Traditional models of bendable surfaces are based on the exact or approximate invariance to deformations that do not tear or stretch the shape, leaving intact an intrinsic geometry associated with it. These geometries are typically defined using either the shortest path length (geodesic distance), or properties of heat diffusion (diffusion distance) on the surface. Both measures are implicitly derived from the metric induced by the ambient Euclidean space. In this paper, we depart from this restrictive assumption by observing that a different choice of the metric results in a richer set of geometric invariants. We apply equi-affine geometry for analyzing arbitrary shapes with positive Gaussian curvature. The potential of the proposed framework is explored in a range of applications such as shape matching and retrieval, symmetry detection, and computation of Voroni tes- sellation. We show that in some shape analysis tasks, equiaffine- invariant intrinsic geometries often outperform their Euclidean-based counterparts.We further explore the potential of this metric in facial anthropometry of newborns.

Original languageEnglish
Pages (from-to)144-163
Number of pages20
JournalJournal of Mathematical Imaging and Vision
Issue number1
StatePublished - 2014


FundersFunder number
European Community’s FP7-ERC267414


    • Affine
    • Equi-affine
    • Intrinsic geometry
    • Metric invariant
    • Shape analysis


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