  In this paper, we propose a Sparse Low-rank Quaternion Approximation (SLRQA)
model for color image processing problems with noisy observations. %Different
from the existing color image processing models, The proposed SLRQA is a
quaternion model that combines low-rankness and sparsity priors. Furthermore,
it does not need an initial rank estimate. To solve the SLRQA model, a
nonconvex proximal linearized ADMM (PL-ADMM) algorithm is proposed. %where only
one variable is linearized. Furthermore, the global convergence analysis of the
PL-ADMM under mild assumptions is presented. When the observation is
noise-free, an SLRQA-NF model of the limiting case of the SLRQA is proposed.
Subsequently, a nonconvex proximal linearized ADMM (PL-ADMM-NF) algorithm for
the SLRQA-NF is given. % Extensive numerical experimental results show the
robustness and effectiveness of quaternion representation. In numerical
experiments, we verify the effectiveness of quaternion representation.
Furthermore, for color image denoising and color image inpainting problems,
SLRQA and SLRQA-NF models demonstrate superior performance both quantitatively
and visually when compared with some state-of-the-art methods.
