# Inexact elastic shape matching in the square root normal field framework

## TL;DR

This paper introduces a new elastic shape matching method for unparametrized curves and surfaces that leverages square root normal fields and varifold metrics, enabling flexible, efficient, and topology-agnostic matching with texture incorporation.

## Contribution

It presents a novel variational framework that minimizes over reparametrizations without discretizing the reparametrization group, allowing for efficient and flexible shape matching.

## Key findings

- Effective matching of curves and surfaces with different samplings and topologies.
- Incorporation of texture as additional matching information.
- Efficient numerical implementation of the objective and gradient.

## Abstract

This paper puts forth a new formulation and algorithm for the elastic matching problem on unparametrized curves and surfaces. Our approach combines the frameworks of square root normal fields and varifold fidelity metrics into a novel framework, which has several potential advantages over previous works. First, our variational formulation allows us to minimize over reparametrizations without discretizing the reparametrization group. Second, the objective function and gradient are easy to implement and efficient to evaluate numerically. Third, the initial and target surface may have different samplings and even different topologies. Fourth, texture can be incorporated as additional information in the matching term similarly to the fshape framework. We demonstrate the usefulness of this approach with several numerical examples of curves and surfaces.