Abstract: A coordinate system describing the interior of organs is a powerful tool for a systematic localization of injured tissue. If the same coordinate values are assigned to specific anatomical landmarks, the coordinate system allows integration of data across different medical image modalities. Harmonic mappings have been used to produce parametric coordinate systems over the surface of anatomical shapes, given their flexibility to set values
at specific locations through boundary conditions. However, most of the existing implementations in medical imaging restrict to either anatomical surfaces, or the depth coordinate with boundary conditions is given at sites
of limited geometric diversity. In this paper we present a method for anatomical volumetric parameterization that extends current harmonic parameterizations to the interior anatomy using information provided by the
volume medial surface. We have applied the methodology to define a common reference system for the liver shape and functional anatomy. This reference system sets a solid base for creating anatomical models of the patient’s liver, and allows comparing livers from several patients in a common framework of reference.

Abstract: Medial surfaces are powerful tools for shape description, but their use has been limited due to the sensibility existing methods to branching artifacts. Medial branching artifacts are associated to perturbations of the object boundary rather than to geometric features. Such instability is a main obstacle for a condent application in shape recognition and description. Medial branches correspond to singularities of the medial surface and, thus, they are problematic for existing morphological and energy-based algorithms. In this paper, we use algebraic geometry concepts in an energy-based approach to compute a medial surface presenting a stable branching topology. We also present an ecient GPU-CPU implementation using standard image processing tools. We show the method computational eciency and quality on a custom made synthetic database. Finally, we present some results on a medical imaging application for localization of abdominal pathologies.