  We extensively study the rotational group structure inside the patch space by
introducing the fiber bundle structure. The rotational group structure leads to
a new image denoising algorithm called the \textit{vector non-local Euclidean
median} (VNLEM). The theoretical aspect of VNLEM is studied, which explains why
the VNLEM and traditional non-local mean/non-local Euclidean median (NLEM)
algorithm work. The numerical issue of the VNLEM is improved by taking the
orientation feature in the commonly applied scale-invariant feature transform
(SIFT), and a theoretical analysis of the robustness of the orientation feature
in the SIFT is provided. The VNLEM is applied to an image database of 1,361
images and compared with the NLEM. Different image quality assessments based on
the error-sensitivity or the human visual system are applied to evaluate the
performance. The results confirmed the potential of the VNLEM algorithm.
