A Color Elastica Model for Vector-Valued Image Regularization

Models related to the Euler's Elastica energy have proven to be very useful for many applications, including image processing and high energy physics. Extending the Elastica models to color images and multi-channel data is challenging, as numerical solvers for these geometric models are difficult to find. In the past, the Polyakov action from high energy physics has been successfully applied for color image processing. Like the single channel Euler's elastica model and the total variation (TV) models, measures that require high order derivatives could help when considering image formation models that minimize elastic properties, in one way or another. Here, we introduce an addition to the Polyakov action for color images that minimizes the color manifold curvature, that is computed by applying of the Laplace-Beltrami operator to the color image channels. When applied to gray scale images, while selecting appropriate scaling between space and color, the proposed model reduces to minimizing the Euler's Elastica operating on the image level sets. Finding a minimizer for the proposed nonlinear geometric model is the challenge we address in this paper. Specifically, we present an operator-splitting method to minimize the proposed functional. The nonlinearity 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.