  We propose an adaptive regularization scheme in a variational framework where
a convex composite energy functional is optimized. We consider a number of
imaging problems including denoising, segmentation and motion estimation, which
are considered as optimal solutions of the energy functionals that mainly
consist of data fidelity, regularization and a control parameter for their
trade-off. We presents an algorithm to determine the relative weight between
data fidelity and regularization based on the residual that measures how well
the observation fits the model. Our adaptive regularization scheme is designed
to locally control the regularization at each pixel based on the assumption
that the diversity of the residual of a given imaging model spatially varies.
The energy optimization is presented in the alternating direction method of
multipliers (ADMM) framework where the adaptive regularization is iteratively
applied along with mathematical analysis of the proposed algorithm. We
demonstrate the robustness and effectiveness of our adaptive regularization
through experimental results presenting that the qualitative and quantitative
evaluation results of each imaging task are superior to the results with a
constant regularization scheme. The desired properties, robustness and
effectiveness, of the regularization parameter selection in a variational
framework for imaging problems are achieved by merely replacing the static
regularization parameter with our adaptive one.
