  We present directional operator splitting schemes for the numerical solution
of a fourth-order, nonlinear partial differential evolution equation which
arises in image processing. This equation constitutes the $H^{-1}$-gradient
flow of the total variation and represents a prototype of higher-order
equations of similar type which are popular in imaging for denoising,
deblurring and inpainting problems. The efficient numerical solution of this
equation is very challenging due to the stiffness of most numerical schemes. We
show that the combination of directional splitting schemes with implicit
time-stepping provides a stable and computationally cheap numerical realisation
of the equation.
