Local versus Global in Quasi-Conformal Mapping for Medical Imaging

A method and algorithm of flattening folded surfaces, for two-dimensional representation and analysis of medical images, are presented. The method is based on an application to triangular meshes of classical results of Gehring and Väisälä regarding the existence of quasi-conformal and quasi-isometric mappings.The proposed algorithm is basically local and, therefore, suitable for extensively folded surfaces encountered in medical imaging. The theory and algorithm guarantee minimal distance, angle and area distortion. Yet, the algorithm is relatively simple, robust and computationally efficient, since it does not require computational derivatives. Both random-starting-point and curvature-based versions of the algorithm are presented.We demonstrate the algorithm using medical data obtained from real CT images of the colon and MRI scans of the human cortex. Further applications of the algorithm, for image processing in general are also considered. The globality of this algorithm is also studied, via extreme length methods for which we develop a technique of computing straightest geodesics on polyhedral surfaces.