Total-Variation -- Fast Gradient Flow and Relations to Koopman Theory

TL;DR

This paper analyzes the Total Variation flow using mode decomposition techniques, introduces a fast algorithm for 1D flow, and explores connections to Koopman theory, highlighting phase transitions and limitations of direct DMD application.

Contribution

It provides analytic formulations for DMD and KMD of TV-flow, proposes a highly efficient algorithm, and clarifies the relationship between TV spectral modes and Koopman modes.

Findings

Significant speedup in TV-flow computation by three orders of magnitude.

Analytic expressions for DMD and KMD modes of TV-flow in 1D.

Insights into phase transitions affecting mode extraction and flow modeling.

Abstract

The space-discrete Total Variation (TV) flow is analyzed using several mode decomposition techniques. In the one-dimensional case, we provide analytic formulations to Dynamic Mode Decomposition (DMD) and to Koopman Mode Decomposition (KMD) of the TV-flow and compare the obtained modes to TV spectral decomposition. We propose a computationally efficient algorithm to evolve the one-dimensional TV-flow. A significant speedup by three orders of magnitude is obtained, compared to iterative minimizations. A common theme, for both mode analysis and fast algorithm, is the significance of phase transitions during the flow, in which the subgradient changes. We explain why applying DMD directly on TV-flow measurements cannot model the flow or extract modes well. We formulate a more general method for mode decomposition that coincides with the modes of KMD. This method is based on the linear decay profile, typical to TV-flow. These concepts are demonstrated through experiments, where additional extensions to the two-dimensional case are given.