Harmonic Beltrami Signature: A Novel 2D Shape Representation for Object Classification

TL;DR

This paper introduces the Harmonic Beltrami signature (HBS), a novel, unique, and robust 2D shape representation derived from conformal welding and Beltrami coefficients, enabling effective shape classification.

Contribution

The paper presents the HBS, a new shape signature that uniquely characterizes 2D shapes up to similarity transformations, eliminating conformal ambiguity and improving shape classification.

Findings

HBS provides a one-to-one correspondence with 2D shapes.

HBS is robust and can be reconstructed from shape signatures.

Shape classification experiments show high accuracy with HBS.

Abstract

There is a growing interest in shape analysis in recent years. We present a novel shape signature for 2D bounded simply-connected domains, named the Harmonic Beltrami signature (HBS). The proposed signature is based on the harmonic extension of the conformal welding map of a unit circle and its Beltrami coefficient. We show that there is a one-to-one correspondence between the quotient space of HBS and the space of 2D simply-connected shapes up to a translation, rotation and scaling. With a suitable normalization, each equivalence class in the quotient space of HBS is associated to a unique representative. It gets rid of the conformal ambiguity. As such, each shape is associated to a unique HBS. Conversely, the associated shape of a HBS can be reconstructed based on quasiconformal Teichmuller theories, which is uniquely determined up to a translation, rotation and scaling. The HBS is thus an effective fingerprint to represent a 2D shape. The robustness of HBS is studied both theoretically and experimentally. With the HBS, simple metric, such as L2, can be used to measure geometric dissimilarity between shapes. Experiments have been carried out to classify shapes in different classes using HBS. Results show good classification performance, which demonstrate the efficacy of our proposed shape signature.