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**Variational method on Riemann surfaces using conformal parameterization and its applications to image processing.**
*(English)*
Zbl 1185.68807

Summary: Variational method is a useful mathematical tool in various areas of research, especially in image processing. Recently, solving image processing problems on general manifolds using variational techniques has become an important research topic. In this paper, we solve several image processing problems on general Riemann surfaces using variational models defined on surfaces. We propose an explicit method to solve variational problems on general Riemann surfaces, using the conformal parameterization and covariant derivatives defined on the surface. To simplify the computation, the surface is firstly mapped conformally to the two dimensional rectangular domains, by computing the holomorphic 1-form on the surface.

It is well known that the Jacobian of a conformal map is simply the scalar multiplication of the conformal factor. Therefore with the conformal parameterization, the covariant derivatives on the parameter domain are similar to the usual Euclidean differential operators, except for the scalar multiplication. As a result, any variational problem on the surface can be formulated to a 2D problem with a simple formula and efficiently solved by well developed numerical scheme on the 2D domain. With the proposed method, we solve various image processing problems on surfaces using different variational models, which include image segmentation, surface denoising, surface inpainting, texture extraction and automatic landmark tracking. Experimental results show that our method can effectively solve variational problems and tackle image processing tasks on general Riemann surfaces.

It is well known that the Jacobian of a conformal map is simply the scalar multiplication of the conformal factor. Therefore with the conformal parameterization, the covariant derivatives on the parameter domain are similar to the usual Euclidean differential operators, except for the scalar multiplication. As a result, any variational problem on the surface can be formulated to a 2D problem with a simple formula and efficiently solved by well developed numerical scheme on the 2D domain. With the proposed method, we solve various image processing problems on surfaces using different variational models, which include image segmentation, surface denoising, surface inpainting, texture extraction and automatic landmark tracking. Experimental results show that our method can effectively solve variational problems and tackle image processing tasks on general Riemann surfaces.

### MSC:

68U10 | Computing methodologies for image processing |

35Q93 | PDEs in connection with control and optimization |

49M25 | Discrete approximations in optimal control |

58C05 | Real-valued functions on manifolds |

65K10 | Numerical optimization and variational techniques |