**Project Description:**

We address the problem of surface inpainting, which aims to
fill in holes or missing regions on a Riemann surface based on its surface
geometry. In practical situation, surfaces obtained from range scanners
often have holes or missing regions where the 3D models are incomplete. In
order to analyze the 3D shapes effectively, restoring the incomplete shape
by filling in the surface holes is necessary. We propose a novel conformal
approach to inpaint surface holes on a Riemann surface based on its surface
geometry. The basic idea is to represent the Riemann surface using its conformal
factor and mean curvature. According to Riemann surface theory, a Riemann
surface can be uniquely determined by its conformal factor and mean curvature
up to a rigid motion. Given a Riemann surface S, its mean curvature H and
conformal factor \lambda can be computed easily through its conformal parameterization.
Conversely, given \lambda and H, a Riemann surface can be uniquely reconstructed
by solving the Gauss-Codazzi equation on the conformal parameter domain.
Hence, the conformal factor and the mean curvature are two geometric quantities
fully describing the surface. With this \lambda-H representation of the surface,
the problem of surface inpainting can be reduced to the problem of image
inpainting of and H on the conformal parameter domain. The inpainting of
\lambda and H can be done by conventional image inpainting models. Once \lambda
and H are inpainted, a Riemann surface can be reconstructed which effectively
restores the 3D surface with missing holes. Since the inpainting model is
based on the geometric quantities \lambda and H, the restored surface follows
the surface geometric pattern as much as possible. We test the proposed algorithm
on synthetic data, 3D human face data and MRI-derived brain surfaces. Experimental
results show that our proposed method is an effective surface inpainting
algorithm to fill in surface holes on an incomplete 3D models based their
surface geometry.

**Publication:
**

- L.M. Lui, C.F. Wen, X.F. Gu, A conformal approach for surface inpainting, accepted, Journal of Inverse Problems and Imaging (2013)