![](https://csdnimg.cn/release/download_crawler_static/15810903/bg4.jpg)
DAI AND WU: REWEIGHTED INFRARED PATCH-TENSOR MODEL WITH BOTH NONLOCAL AND LOCAL PRIORS 3755
Noise patch-tensor N : In this paper, we just assume that the
noise is additive white Gaussian noise and N
F
≤ δ for some
δ>0.Thus,wehave
F−B−T
F
≤ δ. (5)
It should be noted that although the values of parameters k, r, δ
are different depending on the images, we would not directly
use them in the following sections.
Ideally, we would like to obtain a low-rank and sparse de-
composition and solve the following problem:
min
B,T
rank(B)+λT
0
, s.t. B + T = F . (6)
Unfortunately, the rank computation of a given tensor is a
NP-hard problem in general [33]. In [34], Goldfarb and Qin
proposed the higher order RPCA (robust tensor recovery)
through replacing the rank by a convex surrogate Tucker-rank
CTrank(B), and T
0
by T
1
to make the above problem
tractable. In the singleton model, the tensor rank regularization
term is defined as the sum of all the nuclear norms of the mode-
i unfoldings, i.e., CTrank(B)=
i
B
(i)
∗
,i=1, 2, 3. With
this relaxation, our proposed IPT model with random noise as-
sumption can be solved by minimizing the following convex
problem:
min
B,T
3
i=1
B
(i)
∗
+ λT
1
, s.t. F − B − T
F
≤ δ. (7)
λ is a weighting parameter that controls the global tradeoff be-
tween the background patch-tensor and the target patch-tensor.
Larger λ can shrink those nontarget but sparse components
to zeros in the target patch-tensor. Nevertheless, it will also
overshrink the dim target which should be preserved. On the
contrary, a smaller λ does retain the dim target, but it keeps
the strong cloud edges as well. Therefore, adopting a global
constant weighting parameter λ is not a reasonable scheme
for detecting the infrared small target in a complex scene.
Naturally, it motivates us to design an entrywise weighting
scheme.
III. I
NCORPORATING LOCAL STRUCTURE PRIOR
In this section, we focus our emphasis on combining the local
structure prior and nonlocal correlation prior together into our
model. We construct a local structure weight and interpret it as
an edge salience measure. For the sake of simplicity, the local
structure weight is designed on the basis of the image structure
tensor. Structure tensor is widely used in many partial differ-
ential equation (PDE)-based methods [35], [36] to estimate the
local structure information in the image, including edge orien-
tation. To integrate the local information, the structure tensor is
constructed based on a local regularization of a tensorial prod-
uct, which is defined as
J
α
(∇u
σ
)=G
α
∗ (∇u
σ
⊗∇u
σ
)=
J
11
J
12
J
21
J
22
(8)
where u
σ
is a Gaussian-smoothed version of a given im-
age u. σ>0 is the standard deviation of the Gaussian
kernel which denotes the noise scale, making the edge
Fig. 3. Illustrations of the structure tensor-based local structure weight map.
(a)–(d) Illustrations of Fig. 5(a)–(d).
detector ignorant of small details. J
α
is a symmetric
and positive semidefinite matrix, which has two orthonor-
mal eigenvectors denoted as w and v = w
⊥
, where w =
(2J
12
,J
22
− J
11
+
(J
22
− J
11
)
2
+4J
2
12
)
, w = w/|w|. w
points to the maximum contrast direction of the geom-
etry structure while v points to the minimum direction
[37]. Their corresponding eigenvalues λ
1
and λ
2
can be
calculated via
λ
1
, λ
2
=(J
11
+ J
22
) ±
(J
22
− J
11
)
2
+4J
2
12
. (9)
These two values can be used as two feature descriptors of
the local geometry structure, where at the flat region, λ
1
≈
λ
2
≈ 0; at the edge region, λ
1
λ
2
≈ 0; at the corner region,
λ
1
≥ λ
2
0. For the sake of low computational cost, we take
λ
1
− λ
2
as the edge awareness feature since its value of the
edge pixel is much larger than that belongs to the flat region and
corner. By applying (8) and (9) to every pixel in the input image
f
F
, two matrices L
1
and L
2
can be obtained, which consists
of the large and small structure tensor eigenvalues of all the
pixels, respectively. Then, we can transform L
1
and L
2
to their
corresponding patch-tensors L
1
and L
2
. Finally, we define the
local structure weight patch-tensor as follows:
W
LS
= exp
h ·
(L
1
− L
2
) − d
min
d
max
− d
min
(10)
where h is a weight stretching parameter, d
max
and d
min
are
the maximum and minimum of L
1
− L
2
, respectively. Fig. 3
displays the edge awareness maps of Fig. 5, which demonstrates
that the structure tensor-based local structure weight has a good
performance in identifying the edges. It should be noticed that
for the sake of displaying effect, Fig. 3 is created via a nor-
malized version of −exp
−h ·
(L
1
−L
2
)−d
min
d
max
−d
min
. In the proposed
algorithm, we still calculate the local structure weight via (10).
With the help of W
LS
, we can rewrite (7) into a weighted
IPT model:
min
B,T
3
i=1
B
(i)
∗
+ λW
LS
T
1
, s.t. F − B − T
F
≤ δ
(11)
where is the notation of Hadamard product. Thanks to the
weighted IPT model (11), the strong edges could be well sup-
pressed in the target image.