which the existing methods for infrared small target detection in a
single image are based within these four categories.
1.2. Motivation
In Ref. [26], the authors attributed the unsatisfying performance
of the state-of-the-art IPI model facing extremely complex back-
ground with strong cloud edges to the sparsity measure of the tar-
gets based on ‘
1
norm. Based on this observation, WIPI model
introduced a target unlikelihood coefficient, serving as an adaptive
column-wise weight and threshold for the target patch-image. In
this subsection, we point out that the explanation of IPI model’s
deficiency in Ref. [26] is merely an indirect phenomenon and the
adaptive soft-thresholding for each column in the target patch-
image is just a remediation approach to remedy the undesired
results. The direct reason is the inaccurate estimation of back-
ground patch-image, which does not recover the strong edges of
heavy clouds and then they are all taken as target components.
Since these strong edges are also sparse under ‘
1
norm and might
be brighter than the targets, they are impossible to be eliminated
by the global threshold in IPI model. Although as a remediation
approach, WIPI model does can suppress the strong edges while
preserving the dim targets, the column-wise weight calculation
requires additional computing time. And this additional computing
time is much more than the WIPI model saves from the subsequent
separation process compared with the IPI model (see Ref. [26] for
details).
In fact, the intrinsic reason behind the inaccurate estimation of
background patch-image lies in the mismatching of IPI model’s
implicit assumption of large amount of observations with the real-
ity of deficient observations of strong cloud edges. The successful
target-background separation of the IPI model via nuclear norm
minimization (NNM) based robust principal component analysis
(RPCA) roots in the blessing of dimensionality of the input patch-
image, implying large amount of observations. However, in practi-
cal infrared images, the patch number containing strong edges is
very limited, the separated results of these patches via RPCA would
degenerate, e.g., the strong edges would be considered as outliers,
namely targets. It is because nuclear norm minimization can easily
cause bias since the number of true samples is insufficient to sup-
port a large variance of the background patch-image containing
patches of strong edges. On the other hand, WIPI model over-
emphasizes on the sparsity of the strong edges. Although the
strong edge is indeed sparse in a patch just as it points out, it
neglects their limited repetitions among the overlapped patches.
For this reason, WIPI model is designed to remedy the inaccurate
estimation of background patch-image rather than improve the
estimation accuracy. Actually, this inaccurate estimation problem
caused by insufficient samples could be overcome by an addendum
to the standard nuclear norm minimization approach called partial
sum minimization of singular values (PSMSV) [29,30]. Different
from NNM, PSMSV preserves the large singular values and only
minimizes variances in residual rank related to the noise observed
data, not the variance of the whole input matrix. In this paper, we
propose to reformulate the infrared patch-image model as a prob-
lem of partial sum minimization of singular values rather than
nuclear norm minimization problem, in order to prevent the strong
edges from occurring in the separated target image and reduce the
false alarms.
Another key observation that motivates us is the overlook of
the characteristics of the infrared small targets by the IPI model
[24] and LRR based method [27]. Even the WIPI model [26] which
tries to incorporate the structure prior only exploits the spot-like
structure of the small target to distinguish it from the strong
edges. These three methods all neglect the fact that the infrared
small target is brighter than the adjacent background neighbor-
hood, which is basic and fundamental for the first three cate-
gories of methods in the above subsection. The direct
embodiment of this neglect is simply using ‘
1
norm to depict
the target, as same as RPCA depicting the sparse noise. The ‘
1
norm constraint implicates the small target could be either
brighter or darker than the neighborhood, which does not corre-
spond to the physical truth. Actually, if we append an additional
non-negative constraint to the target sparsity measure, then the
constraint coheres with both the hot and spot-like characteristics
of the small target. Therefore, a Non-negative Infrared Patch-
image model based on Partial Sum minimization of singular val-
ues (NIPPS) is proposed in this paper for infrared small target
detection in a single image. Finally, in order to solve the partial
sum minimization problem with non-negative constraint, a solv-
ing algorithm based on the Inexact Augmented Lagrange Multi-
plier (IALM) Method [31,32] is provided. In summary, the
contributions of this work include:
We proposed a new infrared small target detection model ter-
med NIPPS, which takes both the deficient observation of cloud
edges and the hot-spot characteristics of the targets into
account, achieving a better background suppression perfor-
mance than the state-of-the-art IPI model [24] and WIPI model
[26].
An algorithm based on IALM is presented to solve the proposed
partial sum minimization problem with non-negative
constraint.
Since the new constraints for both background and target
patch-images lead to a more accurate of the background
patch-image, the convergence rate of the proposed method is
accelerated.
In Section 2, we illustrate the problem of deficient observations
firstly, and the proposed method is also described as well as its
solution. Detailed experiments are carried out comparing the pro-
posed NIPPS model with several state-of-the-art methods on the
practical infrared images containing small and dim targets in Sec-
tion 3. Finally, Section 4 gives the conclusion.
Table 1
Summary of existing single-frame detection algorithms.
Assumption Advantage Disadvantage
Background consistency Easy implementation Sensitive to complex edges
Need a pre-set target size
Target saliency Easy implementation Poor performance when the target is not salient enough
Sparse target Good ability to distinguish target from noise Need a pre-set target size
Great computational complexity since a large dictionary is involved
Low-rank background Fit reality well
State-of-the-art background suppressing performance
Computationally expensive
Equal thresholding to the target and background residual
184 Y. Dai et al. / Infrared Physics & Technology 81 (2017) 182–194
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