Two-dimensional sampling and representation of folded surfaces embedded in higher dimensional manifolds

The general problem of sampling and flattening of folded surfaces for the purpose of their two-dimensional representation and analysis as images is addressed. We present a method and algorithm based on extension of the classical results of Gehring and Väaisäaläa regarding the existence of quasi-conformal and quasi-isometric mappings between Riemannian manifolds. Proper surface sampling, based on maximal curvature is first discussed. We then develop the algorithm for mapping of this surface triangulation into the corresponding flat triangulated representation. The proposed algorithm is basically local and, therefore, suitable for extensively folded surfaces such as encountered in medical imaging. The theory and algorithm guarantee minimal metric, angular and area distortion. Yet, it is relatively simple, robust and computationally efficient, since it does not require computational derivatives. In this paper we present the sampling and flattening only, without complementing them by proper interpolation. We demonstrate the algorithm using medical and synthetic data.