A Fast Minimization Algorithm for the Euler Elastica Model Based on a
  Bilinear Decomposition
Zhifang Liu Baochen Sun Xue-Cheng Tai Qi Wang Huibin Chang
The Euler Elastica (EE) model with surface curvature can generate
artifact-free results compared with the traditional total variation
regularization model in image processing. However, strong nonlinearity and
singularity due to the curvature term in the EE model pose a great challenge
for one to design fast and stable algorithms for the EE model. In this paper,
we propose a new, fast, hybrid alternating minimization (HALM) algorithm for
the EE model based on a bilinear decomposition of the gradient of the
underlying image and prove the global convergence of the minimizing sequence
generated by the algorithm under mild conditions. The HALM algorithm comprises
three sub-minimization problems and each is either solved in the closed form or
approximated by fast solvers making the new algorithm highly accurate and
efficient. We also discuss the extension of the HALM strategy to deal with
general curvature-based variational models, especially with a Lipschitz smooth
functional of the curvature. A host of numerical experiments are conducted to
show that the new algorithm produces good results with much-improved efficiency
compared to other state-of-the-art algorithms for the EE model. As one of the
benchmarks, we show that the average running time of the HALM algorithm is at
most one-quarter of that of the fast operator-splitting-based
Deng-Glowinski-Tai algorithm.