  Recent developments in elastic shape analysis (ESA) are motivated by the fact
that it provides comprehensive frameworks for simultaneous registration,
deformation, and comparison of shapes. These methods achieve computational
efficiency using certain square-root representations that transform invariant
elastic metrics into Euclidean metrics, allowing for applications of standard
algorithms and statistical tools. For analyzing shapes of embeddings of
$\mathbb{S}^2$ in $\mathbb{R}^3$, Jermyn et al. introduced square-root normal
fields (SRNFs) that transformed an elastic metric, with desirable invariant
properties, into the $\mathbb{L}^2$ metric. These SRNFs are essentially surface
normals scaled by square-roots of infinitesimal area elements. A critical need
in shape analysis is to invert solutions (deformations, averages, modes of
variations, etc) computed in the SRNF space, back to the original surface space
for visualizations and inferences. Due to the lack of theory for understanding
SRNFs maps and their inverses, we take a numerical approach and derive an
efficient multiresolution algorithm, based on solving an optimization problem
in the surface space, that estimates surfaces corresponding to given SRNFs.
This solution is found effective, even for complex shapes, e.g. human bodies
and animals, that undergo significant deformations including bending and
stretching. Specifically, we use this inversion for computing elastic shape
deformations, transferring deformations, summarizing shapes, and for finding
modes of variability in a given collection, while simultaneously registering
the surfaces. We demonstrate the proposed algorithms using a statistical
analysis of human body shapes, classification of generic surfaces and analysis
of brain structures.
