Abstract: The hierarchical ($BV, L^2$) multiscale image representation of Tadmor, Nezzar and Vese, [2004], gives rise to a novelintegro-differential equation (IDE) for a multiscale image representation.To this end, one integrates in inverse scale space a succession ofrefined, recursive `slices' of the image, which are balanced by a typicalcurvature term at the finer scale. The importance of the IDE lies in thefact that even though its motivation comes from a variational problem, wecan manipulate the IDE suitable to our image processing needs. We proposedifferent forms of the IDE with filtering, tangential smoothing, anddeblurring. I will also present (time permitting) an IDE based on Chan, Esedoglu's (BV, L^1) image decomposition with some new results.