Discrete Lie flow: A measure controllable parameterization method

Computing measure controllable parameterizations for general surface is a fundamental task for medical imaging and computer graphics, which is designed to control the measures of the regions of interest in the parameterization domain for more accurate and thorough detection and examination of data. Previous works usually handle just some certain kind of topology and boundary shapes, or are computationally complex. In this paper, a modified approach based on the technique of lie advection is presented for the measure controllable parameterization of geometry objects in the general context of 2-manifold surfaces. Given a general surface with arbitrary initial parameterization without flips but usually with great area distortion, the Lie derivative is introduced to eliminate the difference between the initial parameterization and the prescribed measure. The vertices flow in the directions derived through the Lie derivative and finally converge to the ideal measure, and by its geometric meaning, this method will be called as DLF (Discrete Lie Flow) intuitively. Compared with previous methods based on Lie derivative, two key modifications were made: an adaptive step-length scheme resulting in a substantive acceleration and robustness and a measure controllable function. Area preserving mapping can be generated easily through our DLF algorithm as a special case for measure controllable parameterization. With various algorithms developed for mesh parameterization based on energy optimization approaches in recent years, our DLF is the minority that is supported by a solid differential geometric theory. We tested our method on plenty of cases, including disk models with convex and non-convex boundaries, and spherical models. Experimental results demonstrate the efficiency of the proposed algorithm.