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A variety of analytic and probabilistic models in connection to Markov random fields (MRFs) have been proposed in the last decade for solving low level vision problems involving discontinuities. This paper presents a systematic study of these models and defines a general discontinuity adaptive (DA) MRF model. By analyzing the Euler equation associated with the energy minimization, it shows that the fundamental difference between different models lies in the behavior of interaction between neighboring points, which is determined by the a priori smoothness constraint encoded into the energy function. An important necessary condition is derived for the interaction to be adaptive to discontinuities to avoid oversmoothing. This forms the basis on which a class of adaptive interaction functions (AIFs) is defined. The DA model is defined in terms of the Euler equation constrained by this class of AIFs. Its solution is C1 continuous and allows arbitrarily large but bounded slopes in dealing with discontinuities. Because of the continuous nature, it is stable to changes in parameters and data, a good property for regularizing ill-posed problems. Experimental results are shown.