On Visualizing Branched Surface: an Angle/Area Preserving Approach

Abstract

The techniques of surface deformation and mapping are useful tools for the visualization of medical surfaces, especially for highly undulated or branched surfaces. In this thesis, two algorithms are presented for flattened visualizations of multi-branched medical surfaces, such as vessels. The first algorithm is an angle preserving approach, which is based on conformal analysis. The mapping function is obtained by minimizing two Dirichlet functionals. On a triangulated representation of vessel surfaces, this algorithm can be implemented efficiently using a finite element method. The second algorithm adjusts the result from conformal mapping to produce a flattened representation of the original surface while preserving areas. It employs the theory of optimal mass transport via a gradient descent approach. A new class of image morphing algorithms is also considered based on the theory of optimal mass transport. The mass moving energy functional is revised by adding an intensity penalizing term, in order to reduce the undesired "fading" effects. It is a parameter free approach. This technique has been applied on several natural and medical images to generate in-between image sequences.Ph.D