In the first part of the talk, we consider two approaches of quaternion geometric weighting Lp averaging working on the exponential and logarithm maps of full quaternions. The first formulation is based on computing the Euclidean weighted Lp center of mass in the tangent space of quaternions. The second method consider gradient descent algorithms for Lp averaging by minimizing the powers of the sum of quaternion geodesic distances, which converges to the Fréchet-Karcher barycenter of quaternions for p = 2 and to the Fermat-Weber point for p = 1.
Besides giving explicit forms of these algorithms, their application for quaternion image processing is shown in the second part of the talk, by introducing the notion of quaternion bilateral filtering. The performance of this approach of locally adaptive (spatially-variant) nonlinear filtering is illustrated using RGB color images, but also using RGB-NIR images and RGB-Depth ones where the image quaternion representation is natural.
The third part of the talk deals with the extension of mathematical morphology operators to quaternion images. The lack of natural ordering in non-Euclidean spaces present an inherent problem when defining morphological operators extended to quaternion-valued images. We analyze how a robust estimate of the center of mass can be used to obtain a notion of quaternion local origin which can be used to compute rank based operators in the quaternion tangent space. Hence, the notions of local supremum and infimum are introduced, which allow to define the quaternion dilation and erosion, and other derived morphological operators. The practical interest of these morphological operators is also illustrated using four components images such as RGB-NIR images and RGB-Depth images.