This chapter introduces the relaxation equation and architecture, one of the basic computing methods in energy minimization: relaxation, dynamic programming, message passing, graph cuts, and linear programming relaxation (LPR).
Early to intermediate vision has a common computation structure. One structure has the attributes to be computed defined on the pixels or a group of pixels. Another structure has the attribute in a pixel correlated with its neighbor values, as often modeled by MRF. Yet another structure has the attributes being obtainable by iteration, relaxing intermediate values, and reusing them for a better solution in a recursive way. We examine a computation structure that combines all these together in terms of representation and architecture.
First, we represent the relaxation equation in time and space in terms of its common structures – iteration, neighborhood computation, and concurrency – and thereby observe how the general computational architectures, Gauss–Seidel and Jacobi methods, can be combined and the numerous architectural variations that can be positioned between the two ends of the spectrum.
Next, we represent the relaxation in a graph, which is the product space of the image plane and iteration. The computation can be viewed in different ways, such as edge connections and spanning orders, in the graph. Deforming the graph in an affine manner, we obtain a new graph in which the connections enable different spanning ...