A Chaotic Dynamical System that Paints

TL;DR

This paper introduces a novel chaotic dynamical system algorithm that accurately reproduces paintings and photographs by capturing their statistical properties, with potential applications in machine learning, sampling, and search strategies.

Contribution

It develops a new chaotic dynamical system combining ergodic and control theory to reproduce images with predetermined statistical features, improving efficiency and accuracy.

Findings

Provides significant acceleration over existing MCMC methods

Achieves higher accuracy in reproducing complex images

Demonstrates potential for diverse applications in data analysis

Abstract

Can a dynamical system paint masterpieces such as Da Vinci's Mona Lisa or Monet's Water Lilies? Moreover, can this dynamical system be chaotic in the sense that although the trajectories are sensitive to initial conditions, the same painting is created every time? Setting aside the creative aspect of painting a picture, in this work, we develop a novel algorithm to reproduce paintings and photographs. Combining ideas from ergodic theory and control theory, we construct a chaotic dynamical system with predetermined statistical properties. If one makes the spatial distribution of colors in the picture the target distribution, akin to a human, the algorithm first captures large scale features and then goes on to refine small scale features. Beyond reproducing paintings, this approach is expected to have a wide variety of applications such as uncertainty quantification, sampling for efficient inference in scalable machine learning for big data, and developing effective strategies for search and rescue. In particular, our preliminary studies demonstrate that this algorithm provides significant acceleration and higher accuracy than competing methods for Markov Chain Monte Carlo (MCMC).