  Spectral analysis of open surfaces is gaining momentum for studying surface
morphology in engineering, computer graphics, and medical domains. This
analysis is enabled using proper parameterization approaches on the target
analysis domain. In this paper, we propose the usage of customizable
parameterization coordinates that allow mapping open surfaces into oblate or
prolate hemispheroidal surfaces. For this, we proposed the usage of Tutte,
conformal, area-preserving, and balanced mappings for parameterizing any given
simply connected open surface onto an optimal hemispheroid. The hemispheroidal
harmonic bases were introduced to spectrally expand these parametric surfaces
by generalizing the known hemispherical ones. This approach uses the radius of
the hemispheroid as a degree of freedom to control the size of the
parameterization domain of the open surfaces while providing numerically stable
basis functions. Several open surfaces have been tested using different mapping
combinations. We also propose optimization-based mappings to serve various
applications on the reconstruction problem. Altogether, our work provides an
effective way to represent and analyze simply connected open surfaces.
