  Models related to the Euler's elastica energy have proven to be useful for
many applications including image processing. Extending elastica models to
color images and multi-channel data is a challenging task, as stable and
consistent numerical solvers for these geometric models often involve high
order derivatives. Like the single channel Euler's elastica model and the total
variation (TV) models, geometric measures that involve high order derivatives
could help when considering image formation models that minimize elastic
properties. In the past, the Polyakov action from high energy physics has been
successfully applied to color image processing. Here, we introduce an addition
to the Polyakov action for color images that minimizes the color manifold
curvature. The color image curvature is computed by applying of the
Laplace-Beltrami operator to the color image channels. When reduced to
gray-scale images, while selecting appropriate scaling between space and color,
the proposed model minimizes the Euler's elastica operating on the image level
sets. Finding a minimizer for the proposed nonlinear geometric model is a
challenge we address in this paper. Specifically, we present an
operator-splitting method to minimize the proposed functional. The
non-linearity is decoupled by introducing three vector-valued and matrix-valued
variables. The problem is then converted into solving for the steady state of
an associated initial-value problem. The initial-value problem is time-split
into three fractional steps, such that each sub-problem has a closed form
solution, or can be solved by fast algorithms. The efficiency and robustness of
the proposed method are demonstrated by systematic numerical experiments.
