Constant rank factorizations of smooth maps

TL;DR

This paper introduces new topological invariants based on constant rank factorizations of smooth functions, enabling object classification in synthetic aperture sonar independently of trajectory estimation.

Contribution

It develops theoretical tools for factorizations of smooth maps that decouple classification from trajectory knowledge, with broad applicability beyond sonar.

Findings

New topological invariants for smooth functions

Decoupling of classification from trajectory estimation

Applicability to non-small perturbations in trajectories

Abstract

Sonar systems are frequently used to classify objects at a distance by using the structure of the echoes of acoustic waves as a proxy for the object's shape and composition. Traditional synthetic aperture processing is highly effective in solving classification problems when the conditions are favorable but relies on accurate knowledge of the sensor's trajectory relative to the object being measured. This article provides several new theoretical tools that decouple object classification performance from trajectory estimation in synthetic aperture sonar processing. The key insight is that decoupling the trajectory from classification-relevant information involves factoring a function into the composition of two functions. The article presents several new general topological invariants for smooth functions based upon their factorizations over function composition. These invariants specialize to the case when a sonar platform trajectory is deformed by a non-small perturbation. The mathematical results exhibited in this article apply well beyond sonar classification problems. This article is written in a way that supports full mathematical generality.