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Topological and geometric characterization of synthetic aperture sonar collections

Michael Robinson, Zander Memon, Maxwell Gualtieri

TL;DR

The paper addresses classifying objects from synthetic aperture sonar by leveraging the topology and geometry of the echo signature space, rather than conventional images. It develops a trajectory-invariant framework in which the signature space $v(M)$ is analyzed via topological data analysis, proving that under generic conditions it decomposes into a wedge of spheres whose dimension matches the configuration space $M$, with topology governed by the number and placement of prominent echos and geometry by their cross sections. The authors establish theoretical results (including a key 'echoes bouquet spheres' theorem), specializations to circular SAS, and implications for persistence diagrams, then validate them through both simulation and laboratory AirSAS data, showing phase spaces as wedge-like loop complexes and persistence bounds linked to sonar cross sections. Collectively, this work provides a principled, trajectory-robust basis for sonar classification and suggests practical classifiers based on persistent homology that are sensitive to the presence of artificial targets while remaining resilient to trajectory variations. The practical impact lies in enabling robust, geometry-aware, trajectory-invariant sonar interpretation and potential new discriminators for target origin.

Abstract

This article explores the theoretical underpinnings of -- and practical results for -- topological and geometric features of data collected by synthetic aperture sonar systems. We prove a strong theoretical guarantee about the structure of the space of echos (the signature space) that is relevant for classification, and that this structure is measurable in laboratory conditions. The guarantee makes minimal assumptions about the sonar platform's trajectory, and establishes that the signature space divides neatly into topological features based upon the number of prominent echos and their placement, and geometric features that capture their corresponding sonar cross section.

Topological and geometric characterization of synthetic aperture sonar collections

TL;DR

The paper addresses classifying objects from synthetic aperture sonar by leveraging the topology and geometry of the echo signature space, rather than conventional images. It develops a trajectory-invariant framework in which the signature space is analyzed via topological data analysis, proving that under generic conditions it decomposes into a wedge of spheres whose dimension matches the configuration space , with topology governed by the number and placement of prominent echos and geometry by their cross sections. The authors establish theoretical results (including a key 'echoes bouquet spheres' theorem), specializations to circular SAS, and implications for persistence diagrams, then validate them through both simulation and laboratory AirSAS data, showing phase spaces as wedge-like loop complexes and persistence bounds linked to sonar cross sections. Collectively, this work provides a principled, trajectory-robust basis for sonar classification and suggests practical classifiers based on persistent homology that are sensitive to the presence of artificial targets while remaining resilient to trajectory variations. The practical impact lies in enabling robust, geometry-aware, trajectory-invariant sonar interpretation and potential new discriminators for target origin.

Abstract

This article explores the theoretical underpinnings of -- and practical results for -- topological and geometric features of data collected by synthetic aperture sonar systems. We prove a strong theoretical guarantee about the structure of the space of echos (the signature space) that is relevant for classification, and that this structure is measurable in laboratory conditions. The guarantee makes minimal assumptions about the sonar platform's trajectory, and establishes that the signature space divides neatly into topological features based upon the number of prominent echos and their placement, and geometric features that capture their corresponding sonar cross section.
Paper Structure (22 sections, 13 theorems, 16 equations, 16 figures)

This paper contains 22 sections, 13 theorems, 16 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Historical context
  4. Signal model
  5. Prominent echos
  6. Limitations
  7. The structure of signature space
  8. The general case
  9. Special results for circular SAS
  10. Superposition of point scatterers
  11. Persistent homology
  12. Experimental verification
  13. Simulation
  14. Laboratory dataset description
  15. Laboratory results
  16. ...and 7 more sections

Key Result

Proposition 1

Suppose that $\mathcal{D}_1$ and $\mathcal{D}_2$ are two finite collections of pairwise disjoint disks in a manifold $M$ such that the union $\cup \mathcal{D}_1$ is a proper subset of the union $\cup \mathcal{D}_2$. Under this condition, we have that if $N = \mathbb{R}^n$ for some $n$,

Figures (16)

  • Figure 1: Specular reflections arising from broadside illumination of a copper pipe form prominent echoes.
  • Figure 2: Mapping the boundary of a disk to a point results in a sphere.
  • Figure 3: Components of the tangent map of the signal map in Example \ref{['eg:knotted_sphere_2scatter']}. Brighter pixels correspond to higher received signal strength.
  • Figure 4: A visualization of the phase space in Example \ref{['eg:knotted_sphere_2scatter']}, with prominent echoes indicated.
  • Figure 5: Components of the tangent map of the signature space in Example \ref{['eg:knotted_sphere_3scatter']}. Brighter pixels correspond to higher received signal strength.
  • ...and 11 more figures

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Example 1
  • Example 2
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 22 more