# An Elastic Energy Minimization Framework for Mean Surface Calculation

## TL;DR

This paper introduces a novel elastic energy minimization framework for calculating mean surfaces in 3D image analysis, emphasizing reparameterization equations and centroid calculation within a specific mathematical space.

## Contribution

It presents new equations for surface reparameterization and centroid calculation using RPSN, enhancing surface averaging methods for 3D image delineations.

## Key findings

- RPSN space is invariant under reparameterization and has a meaningful L2 metric.
- Provides reparameterization equations for both RPSN and SRNF representations.
- Addresses theoretical aspects of displacement removal in RPSN for surface analysis.

## Abstract

As the continuation of the contour mean calculation - designed for averaging the manual delineations of 3D layer stack images - in this paper, the most important equations: a) the reparameterization equations to determine the minimizing diffeomorphism and b) the proper centroid calculation for the surface mean calculation are presented. The chosen representation space: escaled Position by Square root Normal (RPSN) is a real valued vector space, invariant under the action of the reparameterization group and the imposed L2 metric (used to define the distance function) has well defined meaning: the sum of the central second moments of the coordinate functions. For comparision purpose, the reparameterization equations for elastic surface matching, using the Square Root Normal Function (SRNF) are also provided. The reparameterization equations for these cases have formal similarity, albeit the targeted applications differ: SRNF representation suitable for shape analysis purpose whereas RPSN is more fit for the cases where all contextual information - including the relative translation between the constituent surfaces - are to be retained (but the sake of theoretical completeness, the possibility of the consistent relative displacement removal in the RPSN case is also addressed).