for image denoising. However, through the literature
study, we ﬁnd that only little work is done on how
to determine regularization parameters, and diffusion
operators for achieving optimal and highﬁdelity im
age restoration results.
In this paper, we extend the variable exponent,
linear growth functional (Chen et al., 2006), (Chen
and Rao, 2003) to double regularized Bayesian es
timation for simultaneously deblurring and denois
ing. The Bayesian framework provides a structured
way to include prior knowledge concerning the quan
tities to be estimated (Freeman and Pasztor, 2000).
Different from traditional “passive” edgepreserving
methods (Geman and Reynolds, 1992), our method
is an “active” datadriven approach which integrates
selfadjusting regularization parameters and dynamic
computed gradient prior for selfadjusting the ﬁdelity
term and multiple image diffusion operators. A new
scheme is designed to select the regularization pa
rameters adaptively on different levels based on the
measurements of local variances. The chosen diffu
sion operators are automatically adjusted following
the strengths of edge gradient. The suggested ap
proach has several important effects: ﬁrstly, it shows
a theoretically and experimentally sound way of how
local diffusion operators are changed automatically
in the BV space. Secondly, the selfadjusting regu
larization parameters also control the diffusion oper
ators simultaneously for image restoration. Finally,
this process is relatively simple and can be easily ex
tended for other regularization or energy optimiza
tion approaches. The experimental results show that
the method yields encouraging results under different
kinds and amounts of noise and degradation.
The paper is organized as follows. In section 2, we
discuss the concepts of BV space, the total variation
(TV) model and its related functionals. In section 3,
we present a Bayesian estimation based adaptive vari
ational regularization with respect to the estimation
of PSFs and images. Numerical approximation and
experimental results are shown in section 4. Conclu
sions are summarized in section 5.
2 RELATED WORK
2.1 The Bv Space and the Tv Method
Following the total variation (TV) functional (Rudin
et al., 1992), (Chambolle and Lions, 1997), (Weickert
and Schn
¨
orr, 2001), (Chan et al., 2002), (Aubert and
Vese, 1997), we study the total variation functional in
the bounded total variation (BV) space.
Deﬁnition 2.1.1 BV(Ω) is the subspace of functions
f ∈ L
1
(Ω) where the quantity is ﬁnite,
TV( f) =
Ω
DfdA = (1)
sup
Ω
f · divϕdA ; ϕ ∈ C
1
c
(Ω,R
N
)
where dA = dxdy, ϕ(A)
L
∞
(Ω)
≤ 1, C
1
c
(Ω,R
N
) is
the space of functions in C
1
(Ω) with compact sup
port Ω. BV(Ω) endowed with the norm k fk
BV(Ω)
=
k fk
L
1
(Ω)
+ TV( f) which is a Banach space.
While one adopts the TV measure for image regular
ization, the posterior energy for Tikhonov Regulariza
tion then takes the form which is also given in (Rudin
et al., 1992),
J ( f) =
λ
2
Ω

g− hf

2
dA+
Ω

Df

dA (2)
where g is the noisy image, f is an ideal image and
λ > 0 is a scaling regularization parameter. When an
image f is discontinuous, the gradient of f has to be
understood as a measure. The TV( f) functional is
often denoted by
Ω
Dfdxdy, with the symbol D re
ferring to the conventional differentiation ∇. One use
f ∈ L
1
(Ω) to simplify the numerical computation (see
(Giusti, 1984), for instance),
Ω
DfdA =
Ω
∇fdA.
In order to study more precisely the inﬂuence of
the smoothing term in the regularization, we need to
make an insight observation of a more general total
variation functional which can help us to understand
the convexity criteria in variational regularization. A
general bounded total variational functional can be
written in the following,
J ( f
(g,h)
) =
λ
2
Ω
(g− hf)
2
dA+
Ω
φ(∇f(x, y))dA
The choice of the function φ is crucial. It determines
the smoothness of the resulting function f in the space
V = { f ∈ L
2
(Ω);∇f ∈ L
1
(Ω)} which is not reﬂexive.
In this variational energy function, the closeness
of the solution to the data is imposed by the penalty
term φ(·) in the energy function. If the energy func
tions are nonconvex, it might become more compli
cated than the convex functionals. Although some
nonconvex φ(·) penalty terms can achieve edge
preserving results, convex penalty terms can help us
to get a global convergence and decrease the complex
ity of computation. In the following, we study φ(·) in
a more general form φ(∇f) → φ(Df) in the BV space.
2.2 Convex LinearGrowth Functional
Let Ω be an open, bounded, and connected subset
of R
N
. We use standard notations for the Sobolev