Appl Math Optim 43:245–258 (2001)
2001 Springer-Verlag New York Inc.
Optimal Stopping Time Formulation of Adaptive Image Filtering
I. Capuzzo Dolcetta
and R. Ferretti
Dipartimento di Matematica, Universit`a di Roma “La Sapienza”,
Piazzale A.Moro 2, I 00185 Roma, Italy
Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”,
via della Ricerca Scientiﬁca, I 00133 Roma, Italy
Communicated by A. Bensoussan
Abstract. This paper presents an approach to image ﬁltering based on an optimal
stopping time problem for the evolution equation describing the ﬁltering kernel.
This approach allows us to obtain easily an adaptivity of the ﬁlter with respect to
the noise level. Well-posedness of the problem and convergence of fully discrete
approximations are proved and numerical examples are presented and discussed.
Key Words. Image ﬁltering, Scale choice, Optimal stopping time.
AMS Classiﬁcation. Primary 68U10, Secondary 49N20, 93C20.
It is well known (see [M]) that ﬁltering a noisy image modelled by a function x → y
(x ∈ R, a rectangle in R
) can be performed by the convolution
y(t, x) = y
(x) ∗ h(t, x), (1)
h(t, x) =
Here, t > 0 plays the role of the scale factor of the ﬁlter and 2t is the second moment of
h, which is inversely related to the bandwidth.