In this paper we generalize the iterated refinement method,
introduced by the authors in a recent work, to a time-continuous
inverse scale-space formulation. The iterated refinement procedure
yields a sequence of convex variational problems, evolving toward
the noisy image.
¶ The inverse scale space method arises as a limit for a
penalization parameter tending to zero, while the number of
iteration steps tends to infinity. For the limiting flow, similar
properties as for the iterated refinement procedure hold.
Specifically, when a discrepancy principle is used as the stopping
criterion, the error between the reconstruction and the noise-free
image decreases until termination, even if only the noisy image is
available and a bound on the variance of the noise is known.
¶ The inverse flow is computed directly for one-dimensional
signals, yielding high quality restorations. In higher spatial
dimensions, we introduce a relaxation technique using two
evolution equations. These equations allow fast, accurate,
efficient and straightforward implementation. We investigate the
properties of these new types of flows and show their excellent
denoising capabilities, wherein noise can be well removed with
minimal loss of contrast of larger objects.