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Lorenz Attractor Quaternion Filtering

Overview

This application demonstrates advanced quaternion filtering using the chaotic Lorenz attractor system. It showcases how quaternion linear algebra can be applied to signal processing and filtering problems with multi-dimensional chaotic data.

Mathematical Foundation

Quaternion Signal Encoding

A three-dimensional signal is encoded as a quaternion-valued time series: 

x(t) = x_r(t)𝐢 + x_g(t)𝐣 + x_b(t)𝐤
where x_r, x_g, x_b represent real-valued channels (e.g., RGB components).

Quaternion Filter Design

We seek a finite-length quaternion filter {w(s)} such that: 

w(s) = w⁽⁰⁾(s) + w⁽ʳ⁾(s)𝐢 + w⁽ᵍ⁾(s)𝐣 + w⁽ᵇ⁾(s)𝐤

The filtering operation uses right quaternion multiplication: 

y(t) = Σ x(t-s) ∗ w(s)

Lorenz System Generation

The target channels (y_r, y_g, y_b) are generated by numerically integrating the Lorenz system: 

dx/dt = α(y - x)
dy/dt = x(ρ - z) - y
dz/dt = xy - βz

Parameters: - α = 10 (coupling strength) - β = 8/3 (damping parameter) - ρ = 28 (driving parameter) - Initial condition: (x₀, y₀, z₀) = (1, 1, 1)

Linear System Formulation

The filtering problem becomes a quaternion linear system: 

X ∗ w = y
where: - X: Toeplitz-like quaternion data matrix - w: Unknown quaternion filter coefficients - y: Target quaternion signal from Lorenz system

Available Scripts

Q-GMRES Lorenz Application

python applications/signal_processing/lorenz_attractor_qgmres.py
Demonstrates quaternion GMRES solver applied to Lorenz-based filtering problems.

Method Comparison Benchmark

python applications/signal_processing/benchmark_lorenz_methods.py
Compares Q-GMRES vs Newton-Schulz methods for Lorenz attractor filtering.

Key Features

Signal Processing Capabilities

  • Multi-channel filtering with quaternion coherence
  • Chaotic signal handling using Lorenz attractor dynamics
  • Noise robustness with quaternion-aware processing
  • Real-time filtering potential with optimized algorithms

Solver Comparisons

  • Q-GMRES: Native quaternion iterative solver
  • Block GMRES: Real-valued block formulation
  • Newton-Schulz: Quaternion pseudoinverse methods
  • Performance metrics: Convergence rates, residual analysis

Applications

Signal Processing

  • Multi-channel audio filtering with spatial coherence
  • Color image processing preserving chrominance relationships
  • Sensor fusion for multi-dimensional data streams
  • Chaotic system identification and control

Research Applications

  • Quaternion linear algebra method validation
  • Iterative solver benchmarking for quaternion systems
  • Chaotic dynamics in quaternion signal processing
  • Filter design for multi-dimensional signals

Performance Metrics

The application reports relative residual: 

Residual = ||X ∗ w - y||₂ / ||y||₂

Convergence criteria: - Tolerance: 10⁻⁶ - Maximum iterations: Number of unknowns - Comparison across GMRES, Block GMRES, and Q-GMRES

Getting Started

  1. Run the Q-GMRES application to see quaternion filtering in action
  2. Execute the benchmark to compare different solution methods
  3. Examine output plots showing convergence and filtering results
  4. Modify parameters to explore different Lorenz dynamics

This application provides a comprehensive example of quaternion linear algebra applied to chaotic signal processing, demonstrating the power and efficiency of QuatIca's quaternion-native algorithms.