A Topological Low-Pass Filter for Quasi-Periodic Signals

Robinson, Michael

doi:10.1109/lsp.2016.2619678

Cited by 8 publications

(9 citation statements)

References 16 publications

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“…Using the algorithm described in [3], we may obtain a homeomorphic copy of C from samples of a CSAS collection u. The algorithm requires that we bundle angle-lagged copies of the domain, forming the vector…”

Section: Theoretical Modelmentioning

confidence: 99%

“…If we instead consider the phase space C for the universal quasiperiodic factorization obtained using the algorithm described in [3], so that u = U • Φ and Φ : S 1 → C, then each "flare" is transformed into a loop via Proposition 2. In this case, the persistent homology can detect all prominent specular reflections, assuming their geometry is favorable.…”

Section: Qualitative Structure Of Sonar Targetsmentioning

confidence: 99%

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The Topology of Circular Synthetic Aperture Sonar Targets

Memon¹

2022

Preprint

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This report presents a connection between the physical acoustics of an object and the topology of the space of echoes resulting from a circular synthetic aperture sonar (CSAS) collection of that object. A simple theoretical model is developed that yields a precise, yet qualitative, description of the space of echoes. This theoretical model is validated in simulation and with experimental data from a laboratory sonar system.

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Section: Theoretical Modelmentioning

confidence: 99%

Section: Qualitative Structure Of Sonar Targetsmentioning

confidence: 99%

The Topology of Circular Synthetic Aperture Sonar Targets

Memon¹

2022

Preprint

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“…We therefore seek a notion of a topological filter [38]. Several kinds of novel topological filters are known [20,40,39], though it is clear that these do not exhaust the possibilities. The results of this article could provide a basis for extending the above works, by supplying the base topological data upon which topological filters are built.…”

Section: Historical Contextmentioning

confidence: 99%

“…Definition 2. [40,39] A quasiperiodic factorization of a smooth map u : M → N consists of a factorization u = U • φ in which φ is a smooth submersion 4 . The category of quasiperiodic factorizations QuasiP(u) is defined analogously to Const(u), with objects consisting of factorizations and morphisms given by diagrams of exactly the same form as in Definition 1.…”

Section: Categories Of Factorizationsmentioning

confidence: 99%

Constant rank factorizations of smooth maps

Robinson

2021

Preprint

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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.

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“…The concept of determining modes and points in fluid mechanics and PDE is related to embedding theory [17,20,21]. Delay coordinates have been used for noise reduction [22,27]. The embedding theory of Sauer et al [23] has been generalized to PDE by Robinson [20,21].…”

Section: Introductionmentioning

confidence: 99%

Delay embedding of periodic orbits using a fixed observation function

Navarrete

Viswanath

2019

Physica D: Nonlinear Phenomena

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Let x j+1 = φ(x j ), x j ∈ R d , be a dynamical system with φ being a diffeomorphism. Although the state vector x j is often unobservable, the dynamics can be recovered from the delay vector (o(x 1 ), . . . , o(x D )), where o is the scalar-valued observation function and D is the embedding dimension. The delay map is an embedding for generic o, and more strongly, the embedding property is prevalent. We consider the situation where the observation function is fixed at o = π 1 , with π 1 being the projection to the first coordinate. However, we allow polynomial perturbations to be applied directly to the diffeomorphism φ, thus mimicking the way dynamical systems are parametrized. We prove that the delay map is an embedding with probability one with respect to the perturbations. Our proof introduces a new technique for proving prevalence using the concept of Lebesgue points.

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A Topological Low-Pass Filter for Quasi-Periodic Signals

Cited by 8 publications

References 16 publications

The Topology of Circular Synthetic Aperture Sonar Targets

The Topology of Circular Synthetic Aperture Sonar Targets

Constant rank factorizations of smooth maps

Delay embedding of periodic orbits using a fixed observation function

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