  In color image processing, image completion aims to restore missing entries
from the incomplete observation image. Recently, great progress has been made
in achieving completion by approximately solving the rank minimization problem.
In this paper, we utilize a novel quaternion matrix logarithmic norm to
approximate rank under the quaternion matrix framework. From one side, unlike
the traditional matrix completion method that handles RGB channels separately,
the quaternion-based method is able to avoid destroying the structure of images
via putting the color image in a pure quaternion matrix. From the other side,
the logarithmic norm induces a more accurate rank surrogate. Based on the
logarithmic norm, we take advantage of not only truncated technique but also
factorization strategy to achieve image restoration. Both strategies are
optimized based on the alternating minimization framework. The experimental
results demonstrate that the use of logarithmic surrogates in the quaternion
domain is more superior in solving the problem of color images completion.
