The article is devoted to the problem of detecting low contrast small-sized objects in two-color images with a powerful spatially non-stationary background. An increase of the detecting reliability is achieved through a combination of three factors: attenuation of the background based on the construction of its locally stationary model; improving the estimation of model parameters by excluding statistically significant outliers from the initial data; joint processing of two-color images with a weakened background component. A method of constructing a linear boundary for detecting a useful signal in a two-dimensional space is proposed. The performance characteristics of a two-channel detector of small-sized objects are presented.
One of the urgent tasks of the Earth space monitoring is the early detection of potentially dangerous objects in near-earth space by analyzing images generated by onboard spacecraft equipment. A particular version of this problem is the detection of dim "point" objects, the sizes of which are determined mainly by the point spread function (PSF) of the equipment, against a random spatially non-stationary background created by the Earth’s surface and cloud cover. Preliminary suppression of the background, based on knowledge of its statistical characteristics, facilitates to efective selection of objects. In conditions of spatial non-stationarity, this is ensured by constructing a locally stationary background model from a fragment of an image near the supposed presence of an object [1, 2]. If the signal from the object has a wide spectral range, it is advisable to jointly process images of the same area of the observed space obtained in diferent spectral channels with diferent statistical characteristics of the background. The suppression of the main structural features of the background in each of the channels by its local models leads to a decrease in the interchannel correlation of the background component, while maintaining the correlation between the images of objects. Therefore, with joint processing, one can expect a decrease in the probability of a false alarm, or, for a given probability of a false alarm, an increase in the probability of detection.
The most efective way of joint processing would be to build a multichannel local stationary
(, ) = (, ) + · ( − 0, − 0) + (, ).
Here , are the current coordinates, is the background component, is the normalized image of the object with the center located at the point (0, 0) and with the amplitude , is the random uncorrelated registration noise. Then, with the known background model ^(, ), it is advisable to search for objects in the modified image
˜(, ) = (, ) + · ( − 0, − 0) + (, ) − ^(, ) = · ( − 0, − 0) + (, ), (
The minimum of the functional (h) is attained at h^ = (P P)− 1P c,
If we can consider the background in some domain Ω as a realization of a stationary random process, then its model at the point (′, ′) ∈ Ω can be represented as a linear combination of the image samples in its vicinity ^(′, ′) = ∑︁ ( − ′, − ′)ℎ(, ), ,∈ where ∈ Ω is the vicinity of point (′, ′), ℎ(, ) is a set of weight coeficients, which, according to the optimal linear prediction (OLP) [3], can be estimated from the condition ⎧ ⎨ h^ = arg min (h) = ⎩
⎫
∑︁ [︀ (′, ′) − h d′,′ ︀] 2⎬ .
′,′∈Ω
⎭
Here d′,′ is a vector consisting of lexicographically ordered samples of the vicinity , ℎ is a
vector of similarly ordered weight coeficients. This approach allows the model to be adjusted
to the current statistical characteristics of the background.
(
c| ≥ .
h^ = (P KP)− 1P Kc,
=
√︃
(c^ − c) (c^ − c)
− ℎ
,
where is the number of components in the vector c, and ℎ is the number of weight
coeficients in
(
As mentioned in the Introduction, it is assumed here that background suppression is performed
in each spectral channel separately, and at the detection stage, the samples of the modified
images with the same spatial coordinates are considered as vectors in the two-dimensional
space ˜(
× ˜(
.
probability density functions of the background and objects brightness. Unfortunately, in real
conditions, instead of probability density functions, we can only obtain a two-dimensional
histogram of the joint distribution of the background and object brightness. This histogram,
on the assumption that the appearance of objects is a very rare event, can be considered as
an estimate of the discretized probability density function of the background. As for the
twodimensional probability density function characterizing objects, it can be constructed only based
on a priori data on their spectral and brightness characteristics. In this work, we assumed the
presence of minimal information about objects, namely, only knowledge of the ratio of their
average brightness in diferent spectral channels. This is not enough to determine the specific
shape of the boundary, so here we propose to construct a linear boundary. This boundary should
provide, for a given false alarm probability, the maximum detection probability, being guided by
the experimentally obtained two-dimensional histogram of the background and the direction
in two-dimensional space, given by the ratio of the average brightness of objects ¯(
According to expression (
∑︁ (, ) < 1 − ,
=0
+1
∑︁ (, ) ≥ 1 − .
=0
(
Apparently, with such scant information on the distribution of the objects brightness, the only
justified choice of the detection boundary direction is the choice of such an angle at which
the straight line satisfying (
.
⎯
⎸⎸⎸ 2 ︂[ (︁ ¯(
The minimum of this expression with respect to determines the sought boundary for
detecting objects. The second way to choose the direction of the border is to choose, taking
into account (
Since the angle takes only discrete values, its estimate, provided that relations (
¯(
⃒
We will compare these methods of constructing the boundary in the experimental part of the work.
For the computational experiment, we used multispectral images obtained by the Landsat-7 satellite [6]. The purpose of the experiment was to test the above technique for identifying small-sized objects in two-color images with a complex spatially non-stationary background. Pairs of images obtained synchronously in diferent spectral channels made up two-color images and served as a spatially non-stationary background, on which images of small-sized objects were additively applied.
Since one of the objectives of this study is to assess the probability of detecting objects with
an accuracy of 10− 3, the experiment should contain a representative (∼ 105) sample of their
images. Therefore, firstly, more than 600 objects were simultaneously applied to the background
(
In the experiment, the domain of the model for predicting the background value at the image
point (′, ′) was its square neighborhood of 7 × 7 pixels with a 3 × 3 pixel square cut out in
the center. As the region of stationarity of the background Ω, according to which the parameters
of the model were estimated, the neighborhood of the current point with a size of 13 × 13 pixels
was considered. The sizes and shapes of the domains Ω and were selected experimentally.
Background suppression was performed according to expressions (
Figure 2 illustrates the efectiveness of background suppression without correction and with
correction of the objects occurrence in Ω. Figure 2, a shows a fragment of a satellite image
obtained in the spectral range of 0.45 ÷ 0.515 m with objects applied to it, the average
brightness of objects is ∼ 2.2 RMSD of the background brightness calculated over the entire
image. All data are presented in a single brightness scale. The background standard deviation in
the original image is 100, the average object brightness is 212, and the object brightness standard
deviation is 12.7. In Figure 2, b, this fragment is shown after suppressing the background based
on the model (
Experiments with images of diferent spectral ranges confirm the assumption that the use of a locally stationary background model makes it possible to weaken the background component many times (∼ 10 times) when the quality of object images deteriorates by about half.
The images of “point” objects to be detected, as shown in Figure 1, are small outliers with
a brightness exceeding the brightness of neighboring pixels, therefore, after suppressing the
background, it is advisable to search for them among the local maxima of the modified images.
The first step in processing a pair of images belonging to diferent spectral channels is to split
the images into regions of the background and objects. Since the objects were applied artificially,
b
c
such a partition is easy to implement. Further, for each of the regions, two-dimensional
histograms of the maxima are constructed. Pairs of maxima are selected in each of the images
and have the same (or close) coordinates. Based on the histogram belonging to the background
region, according to expression (
The second stage of the experiment is to estimate the detecting probability of objects that are
applied on the original background images. Similarly to the histogram of the background maxima
for the area of objects, a histogram of the maxima of objects is built and, taking into account
the already found boundaries, the probabilities of detection and are calculated. In
Figure 4 for the above frames are shown in blue — the histogram of the background maxima
distribution, in red — the histogram of the objects maxima distribution, the green straight line
indicates the direction determined by the a priori known ratio of the average brightness of
objects ¯(
The summary result of the experiment with this pair of images is shown in Figure 5, a.
Here, the dependence of the objects detecting probability on the angle , which determines
the orientation of the boundary, is shown for three values of the false alarm probability :
10− 2, 10− 3 and 10− 4. The blue points on the graphs correspond to the boundaries determined
a
b
according to the minimum of the functional (
The paper proposes an approach to solving the problem of detecting small-sized objects in a pair of synchronous images of a scene with a powerful spatially non-stationary background, obtained in diferent spectral ranges. It is shown that the separate application of locally stationary models to describe the background to each of the images makes it possible to weaken their background component many times (up to 10 times). The exclusion of statistically significant outliers from the data used to estimate the parameters of the model makes it possible to significantly reduce (by an average of 20%) the distortion of the objects amplitude. A method for joint processing of a pair of images with a suppressed background is proposed, based on the analysis of twodimensional histograms of the background brightness, which ensures the detection of objects with a given false alarm probability. Assuming that only the ratio of the average values of the brightness of objects in diferent spectral ranges is known, it is proposed to search for the boundary between the background and objects in a two-dimensional color space as a straight line intersecting the line of direction to objects at a minimum distance from the origin in this space. It has been shown by means of modeling that among all linear dividing boundaries that provide a given false alarm probability, this boundary allows one to obtain the maximum of the object detection probability.
The work has performed with the financial support of the Ministry of Education and Science of Russia (project 121022000116-0).