Simplifying Transformations for a Family of Elastic Metrics on the Space of Surfaces

Abstract

We define a new representation for immersed surfaces in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> by combining the SRNF and the induced surface metric. Using the L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> metric on the space of SRNFs and the DeWitt metric on the space of surface metrics, we obtain a 3-parameter family of metrics that corresponds to the family of "elastic metrics" proposed by Jermyn et al. in [19] on the space of immersed surfaces. Similar to the original SRNF representation this new representation results in an extrinsic distance function on the space of immersed surfaces that is easy to compute as it is given by an explicit formula. In addition to avoiding the degeneracy of the SRNF it allows for a data-driven choice of the parameters of the metric, while still providing for fast and accurate registration of surfaces.