Total variation time flow with quantile regression for image restoration

We analyze the time flow of a modification of the standard Rudin-Osher-Fatemi (ROF) model for image restoration using the bounded variation semi-norm. The solution of the minimization problem is a restored image in the space of bounded variation (BV) recovered from a corrupted one, and the functional may include a more general term S(x,u) rather than the L2 penalty term as used in the original ROF model. These new models were introduced in Osher and Scherzer (2004) [31]. In particular, we will examine the non-differentiable quantile regression penalty. We will define an appropriate weak solution of the time flow and in particular prove the existence in L∞([0,∞);BV) and L∞ stability.