  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.
