3.4
Through an analysis of the associated Euler equation, a necessary condition is made explicit for MRF or regularization models to be adaptive to discontinuities. On this basis, the DA model is defined by the Euler equation constrained by the class of adaptive interaction functions (AIFs). The definition provides principles for choosing deterministic regularizers and MRF clique potential functions. It also includes many existing models as special instances.
The DA model has its solution in 
and adaptively overrides the smoothness assumption where the assumption is not valid, without the
switching-on/off of discontinuities in the LP model. The DA solution
never contains true discontinuities. The DA model ``preserves''
discontinuities by allowing the solution to have arbitrarily large but
bounded slopes. The LP model ``preserves true discontinuities''
by switching between small and unbounded (or large) slopes.
Owing to its continuous properties, the DA model possesses some theoretical advantages over the LP model. Unlike the LP model, it is stable to changes in parameters and in the data. Therefore it is better than the LP model in solving ill-posed problems. In addition, it is able to deal with problems on a continuous domain. Furthermore, it is better suited for analog VLSI implementation; see Section 9.3.