  Nonlinear variational methods have become very powerful tools for many image
processing tasks. Recently a new line of research has emerged, dealing with
nonlinear eigenfunctions induced by convex functionals. This has provided new
insights and better theoretical understanding of convex regularization and
introduced new processing methods. However, the theory of nonlinear eigenvalue
problems is still at its infancy. We present a new flow that can generate
nonlinear eigenfunctions of the form $T(u)=\lambda u$, where $T(u)$ is a
nonlinear operator and $\lambda \in \mathbb{R} $ is the eigenvalue. We develop
the theory where $T(u)$ is a subgradient element of a regularizing
one-homogeneous functional, such as total-variation (TV) or
total-generalized-variation (TGV). We introduce two flows: a forward flow and
an inverse flow; for which the steady state solution is a nonlinear
eigenfunction. The forward flow monotonically smooths the solution (with
respect to the regularizer) and simultaneously increases the $L^2$ norm. The
inverse flow has the opposite characteristics. For both flows, the steady state
depends on the initial condition, thus different initial conditions yield
different eigenfunctions. This enables a deeper investigation into the space of
nonlinear eigenfunctions, allowing to produce numerically diverse examples,
which may be unknown yet. In addition we suggest an indicator to measure the
affinity of a function to an eigenfunction and relate it to
pseudo-eigenfunctions in the linear case.
