A Study of Non-smooth Convex Flow Decomposition

We present a mathematical and computational feasibility study of the variational convex decomposition of 2D vector fields into coherent structures and additively superposed flow textures. Such decompositions are of interest for the analysis of image sequences in experimental fluid dynamics and for highly non-rigid image flows in computer vision. Our work extends current research on image decomposition into structural and textural parts in a twofold way. Firstly, based on Gauss’ integral theorem, we decompose flows into three components related to the flow’s divergence, curl, and the boundary flow. To this end, we use proper operator discretizations that yield exact analogs of the basic continuous relations of vector analysis. Secondly, we decompose simultaneously both the divergence and the curl component into respective structural and textural parts. We show that the variational problem to achieve this decomposition together with necessary compatibility constraints can be reliably solved using a single convex second-order conic program.