  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.
