One of the classical approaches to the energy equation is the calculus of variations (Aubert and Kornprobst 2006; Courant and Hilbert 1953; Scherzer *et al.* 2008), especially functional minimization. In this paradigm, the solution is the function that minimizes the given energy function and, usually, the solution is in the form of an integro-differential equation. The resulting architecture, relaxation, belongs to the four major architectures: relaxation, DP, BP, and GC, which are used in general energy problems. Starting from an initial set of values, the concept underlying the relaxation architecture is the reuse of previous values to update new values recursively, so that the operation is a contraction mapping. The relaxation architecture is the poorest of the four but is often the starting point in designing a vision circuit because it is fast, simple, and general. It is particularly pertinent here because we are considering a machine for stereo matching that can be used in other vision problems after some modifications.

This chapter is a continuation of Chapter 8. In it, we learned how to interpret the energy equation in terms of the calculus of variation, how to derive the Euler–Lagrange equation, and how to derive relaxation equations from it. We will design the derived relaxation equation with the Verilog HDL (IEEE 2005), using the simulator FVSIM, introduced in Chapter 4: the relaxation equation is inherently frame-based computation. ...

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