Generalizing the over operator for parallelization and order-independency

The over operator is commonly used for α-blending in various visualization techniques. In the current form, it is a binary operator and must strictly follow a specific composition order of all participating operands, hence posing a significant performance limit. In this paper, we derive a set of generic formulas for the over operator that work with any number of operands and completely remove the restriction on the composition order. We prove the correctness of these formulas and provide a step-by-step illustration in a blending context. We implement both a sequential and a parallel version of the improved over operator and apply them to the image composition process where operands are received out of order with different arrival time intervals. The performance superiority of the improved over operator over the original one is established by rigorous theoretical analyses and further validated by extensive experimental results.