  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.
