Methods and Applications of Analysis

Volume 15 (2008)

Number 4

Shape Analysis by Conformal Modules

Pages: 539 – 556



Xianfeng Gu

Lok Ming Lui

Shing-Tung Yau

Wei Zeng


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 algo- rithms 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 compari- son. Experiments on surfaces acquired from real world demonstrate the efficiency and efficacy of the conformal module method.


Conformal module; holomorphic 1-form; curvature flow; shape classification; shape analysis

2010 Mathematics Subject Classification

30F20, 68R99

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