In this talk we shall discuss two types of nonlinear diffusions which have been proposed as mild regularizers for the well-known Perona-Malik equation (PME). One is nonlocal in nature and has the advantage to deliver a locally well-posed model (in the sense of classical solutions) without altering the dynamical behavior of PME solutions. The other is even milder in that it preserves the forward-backward feature of PME but admits global weak Young measure solutions the behavior of which is in complete agreement with numerical experiments. It also provides a suggestive explanation for the well-known staircasing phenomenon widely observed in numerical simulations of PME.
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