All the surfaces in real life are Riemann surfaces, therefore with conformal structures. Two surfaces share the same conformal structure, if there exists a conformal (angle-preserving) mapping between them. Conformal modules are the complete invariants of conformal structures, which can be treated as shape descriptors for shape analysis applications.
This work focuses on the computational methods of conformal modules for genus zero surfaces with boundaries, including topological quadrilaterals, annuli, multiply connected annuli. The algorithms are based on both holomorphic 1-forms and discrete curvature flows, which are rigorous and practical. The conformal module shape descriptors are applied for shape classification and comparison. Experiments on surfaces acquired from real world demonstrate the efficiency and efficacy of the conformal module method.