4 J. Dong, J. Pan, D. Sun, Z. Su, and M. Yang
estimate the latent image from the blurred image using the MAP criterion
l = argmax
l
p(l|k, b) = argmax
l
p(b|k, l)p(l), (2)
where the first (data) term models the distribution of the residue image log p(b|k, l)=
−
1
λ
R(b − l ∗ k), and R(·) measures the data fitting error. In addition, p(l) is the
latent image prior log p(l)=−P(l). The objective function in (2) can be equiva-
lently solved by minimizing the following energy function
min
l
E(l|k, b) = min
l
R(b − l ∗ k) + λP(l). (3)
As our focus is on the data term, we use a standard hyper-Laplacian prior
and set P(l) = k∇lk
p
=
P
i
|(∇
h
l)
x
|
p
+ |(∇
v
l)
x
|
p
, where ∇
h
and ∇
v
denote the
horizontal and vertical differential operators respectively and x is the pixel index.
In this paper, we fix p = 1 for the image prior in (3) and show that even with a
simple total variation prior, the proposed algorithm is able to deblur images with
significant amounts of noise and saturation. We show that this algorithm requires
low computational load and can be easily integrated with more expressive image
priors for better performance. Different from most existing methods that use `
1
or `
2
norm for R, we assume a flexible form for R that can be learned along
with the tradeoff parameter λ from images.
3.1 Data Term
To enforce the modeling capacity, the data term R is parameterized to charac-
terize the spatial information and the complex distribution of the residue image
b − l ∗ k,
R(b − l ∗ k) =
N
f
X
i=0
R
i
(f
i
∗ (b − l ∗ k)), (4)
where f
i
is the i-th linear filter (particularly, f
0
is set as the delta filter to
exploit the information of the residue image in the raw data space), R
i
is the i-
th corresponding non-linear penalty function that models the i-th filter response,
and N
f
is the number of non-trivial linear filters for the data term.
3.2 Inference
Formulation. We first describe the scheme to minimize the energy function (3)
and then explain how to learn the data term. We use the half-quadratic opti-
mization method and introduce auxiliary variables z, v
i
, and u, corresponding
to b − Kl, F
i
(b − Kl), and ∇l = [∇
h
l, ∇
v
l]. Here, K, F
i
, l, and b denote the
matrix/vector forms of k, f
i
, l, and b. Thus, the energy function (3) becomes
min
z,v,u,l
τ
2
kb−Kl−zk
2
2
+R
0
(z)+
N
f
X
i=1
(
β
2
kF
i
(b−Kl)−v
i
k
2
2
+R
i
(v
i
))+λ(
γ
2
k∇l−uk
2
2
+kuk
1
),
(5)