Vector Nonlocal Euclidean Median: Principal Bundle Captures The Nature of Patch Space

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.