Mathematics Colloquium: Some Remarks on Total Variation Flow

2010-03-04

4:10 p.m. Neill Hall 5W

Robert Hardt

Abstract: In 1994, R.Hardt and X.Zhou studied the heat equation $u_t = div_x[F_p(Du)]$ corresponding to a convex function F with linear growth at infinity. For an initial, Dirichlet boundary-value problem, they established existence in $L2([0,T],BV)$, a comparison principle, continuity in time, and minimality of the unique time asymptotic limit. Earlier in 1992, E.Fatemi, S.Osher, and L.Rudin had numerical results for the special case of TV flow where $F(p)=|p|$. The last 10 years has seen numerous applications to imaging of the TV flow and its relatives. In work with Rolf Ryham, we examine carefully specific examples and look for further regularity properties and associated constrained problems.