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Photorealistic Learned Landscapes for Augmented Reality

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Symmetric projection attractor reconstruction: Embedding in higher dimensions.

J V Lyle¹, P J Aston¹

¹Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom.

Chaos (Woodbury, N.Y.)

|December 9, 2021

Summary

Symmetric Projection Attractor Reconstruction (SPAR) visualizes periodic signals. This study extends SPAR to higher dimensions, offering a generalized method for enhanced waveform analysis and understanding subtle morphological changes.

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Area of Science:

Signal Processing
Dynamical Systems Theory
Nonlinear Dynamics

Background:

Periodic signals are ubiquitous in science and engineering.
Symmetric Projection Attractor Reconstruction (SPAR) offers a 2D visualization of 3D phase space reconstructions for periodic signals.
Existing SPAR methods are limited to 3D embeddings.

Purpose of the Study:

To generalize SPAR for delay coordinate embeddings in any dimension N≥3.
To develop a method for creating meaningful 2D visualizations from higher-dimensional attractors.
To analyze the morphology and variability of approximately periodic signals in higher dimensions.

Main Methods:

Generalized delay coordinate embedding for N≥3 dimensions.
Identification of invariant subspaces for 2D projection with rotational symmetry.
Equivalence to trigonometric interpolating polynomial coefficients.
Derivation of bounds on mean and frequency response.

Main Results:

A generalized SPAR method producing 2D visualizations with rotational symmetry from N-dimensional embeddings.
Demonstration of the method's equivalence to analyzing trigonometric interpolating polynomial coefficients.
Derived bounds on the mean and frequency response of the new coordinates.
Successful application to real, approximately periodic signals.

Conclusions:

The generalized SPAR method effectively visualizes higher-dimensional attractors for periodic signals.
This approach enhances the understanding of waveform morphology and variability.
The method reveals subtle waveform changes in higher dimensions, improving signal analysis.