# Methods and Applications of Analysis

## Volume 15 (2008)

### Variational Method on Riemann Surfaces using Conformal Parameterization and its Applications to Image Processing

Pages: 513 – 538

DOI: http://dx.doi.org/10.4310/MAA.2008.v15.n4.a7

Tony F. Chan

Xianfeng Gu

Lok Ming Lui

Shing-Tung Yau

#### Abstract

Variational method is a useful mathematical tool in various areas of research, espe- cially 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 com- putation, 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 para- meterization, 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 segmen- tation, 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.

#### Keywords

Variational method; conformal parameterization; holomorphic one form; conformal factor; covariant derivative

#### 2010 Mathematics Subject Classification

35Q93, 49M25, 58C05, 65K10

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