## 1. Introduction

Compressive sensing (CS)-based target detection is a hot topic in remote sensing. Two different methods are available to resolve this problem: reconstruction-based detection and handling the image in the compressive domain without reconstruction. Aiming at a one-dimensional measurement model, previous studies [

1,

2,

3] detected and classified targets based on reconstructing the image. Reconstruction algorithms for the two-dimensional (2D) measurement model are priorities in recent research [

4,

5,

6,

7,

8,

9]. Although these reconstruction-based detection algorithms achieved CS-based target detection, they do not only impact the detection results, but also require more memory space and operation time. Detection algorithms in the compressive domain have increasingly attracted attempts to enhance detection efficiency while avoiding the effects of reconstruction algorithms. Although the research outputs are still limited, Ma et al. [

10] proposed a conventional energy detection-based non-reconstruction CS detection algorithm, and Xu et al. [

11] and Li et al. [

12] realized compressive detection by means of directly decoding targets’ spatial positions in the compressive domain. All these algorithms were designed for one-dimensional measurement models, which merely achieve compressive sampling of row vectors while the column vectors keep their original dimensions. Target detection based on 2D measurement models that achieve compressive sampling of both row and column vectors remains a challenge.

To achieve 2D measurement model-based target detection in the compressive domain, we propose a 2D compressive domain adaptive threshold (2D CDAT) algorithm that uses Gram matrix mapping of the compressive subtracted image from the compressive domain to the space domain without reconstructing the image, and detects targets via an advanced adaptive threshold method.

This paper is organized as follows: firstly,

Section 2 illustrates the CS measurement models, especially the two-dimensional measurement model. Our 2D CDAT algorithm is then outlined in

Section 3. Finally,

Section 4 provides the experimental results.

## 2. Compressive Sensing—The Basics

The CS theory was proposed by Donoho in 2006 [

13]. After being vigorously developed, CS theory became a hot research topic in the applied mathematics and signal processing fields. A combined compression and sampling method for sparse or compressible signals was the core concept of CS theory; in other words, the signal was compressed and sampled simultaneously at a rate below the Nyquist rate.

Traditional one-dimensional measurement models only compress and sample the column vectors of the signal, whereas the row vectors of the signal are not compressed. If we want to further compress the signal, we can use a 2D measurement model to compress the signal rows and columns simultaneously [

14,

15].

Supposed that a two-dimensional signal

$X\in {R}^{{N}_{1}\times {N}_{2}}$, called

K (

K < N_{1},

K <

N_{2}) ranks sparse, meaning

**X** only has

K nonzero elements. The compressive sampling signal

$\mathit{Y}\in {R}^{{M}_{1}\times {M}_{2}}$ can be calculated through two measurement matrixes,

${\mathit{\Phi}}_{c}\in {R}^{{M}_{1}\times {N}_{1}}$ and

${\mathit{\Phi}}_{r}\in {R}^{{M}_{2}\times {N}_{2}}$, which were non-adaptive; they can simultaneously compress a signal’s row and column vectors separately. The compressive sampling ratios of rows and columns are

R_{r} =

M_{2}/

N_{2} and

R_{c} =

M_{1}/

N_{1}, respectively, corresponding to the compressive sampling ratio of the 2D measurement model, which is

${R}_{CS2}=\left({M}_{1}\times {M}_{2}\right)/\left({N}_{1}\times {N}_{2}\right)$. Thereby, the 2D measurement equation can be expressed as:

where every element in

**Y** contains the complete information of

**X**, as shown in

Figure 1. Compared with the one-dimensional measurement model, the 2D measurement model can compress the signal’s row and column vectors simultaneously, which more deeply compresses the image, and lowers the compressive sampling ratio. Additionally, the detector’s scale is further decreased, and the memory capacity requirement in the hardware implementation is reduced.

Natural signals are not usually sparse in the time domain, whereas if the original signal is sparse or compressible in the

**Ψ** domain, Equation (1) can be written as:

where

**A**_{c} and

**A**_{r} are the sensing matrix, and

**Ψ**_{c} and

**Ψ**_{r} are sparsity bases. There are several sparsity bases, including the discrete cosine transform (DCT) base, the fast Fourier transform (FFT) base, the discrete wavelet transform (DWT) base, the Curvelets base [

16], the Gabor base [

17], and redundant dictionaries [

18].

CS is an under-sampling measurement, and there are infinite solutions satisfying Equation (2) because the dimensions of

**Y** are much fewer than for

**X**. Therefore, dissolving the original signal

**X** from the measurement data

**Y** is a nondeterministic-polynomial (NP)-hard problem. To reconstruct the sparse signal precisely, Candes and Tao [

19] proved that the sensing matrix

**A** must satisfy the restricted isometry property (RIP). Giving a discretional value of

k = 1,2,…,

K, we defined

**A**’s restricted isometry constants

a_{k} to be the smallest quantity, such that

**A** obeys:

where

x is a

k-order sparse constant,

$0<{a}_{k}<1$, and

**A** satisfies

k-order RIP. We can design the measurement matrix to make

**A = ΦΨ** meet the RIP requirement on the condition that

**Ψ** is constant. There are three kinds of random measurement matrixes [

20] that are frequently used: (1) Measurement matrix elements are independent, and they obey a certain distribution, including the Gauss Random matrix, Bernoulli Random matrix, and Sub-Gauss Random matrix. (2) The measurement matrix is composed of random rows in any orthogonal matrix, including the Part Fourier matrix, Part Hadamard matrix, and Noncorrelation Random matrix. (3) The measurement matrix is composed of one specific signal, including the Toeplitz matrix, Circulant matrix, Binary Sparse matrix, and Structurally Random matrix.

## 4. Simulation and Analysis

The simulation data were derived from a real sky background infrared image collected by a medium-wave infrared thermal imager, and the imager’s parameters are shown in

Table 1. The acquisition time was the morning of 5 May 2016, and the weather was cloudy. After the 2976th frame, there were two targets of about 16 and 11 pixels in the infrared image.

The 2980th frame was selected as the current image, and a 256 × 256 pixel size image, including targets, was captured as experimental data, as shown in

Figure 3.

We subtracted the 2980th from the 2979th frame, shown in

Figure 4a, in which the background points were filtered effectively after background subtraction, and the SNR of the image improved. Then, the effect of the method used to map from the compressive domain to the spatial domain was analyzed, and different types of measurement matrixes were chosen to verify the universality of the algorithm. First, Gauss, Part Hadamard, Bernoulli, Circulant, and Toeplitz were used as measurement matrixes for compressing and sampling the sequential images, with the measurement matrixes

${\mathit{\Phi}}_{\mathit{c}}\in {\mathit{R}}^{{M}_{1}\times {N}_{1}}$,

${\mathit{\Phi}}_{\mathit{r}}\in {\mathit{R}}^{{M}_{2}\times {N}_{2}}$, where

M_{1} =

M_{2} = 128,

N_{1} =

N_{2} = 256, and the compressive sampling ratio

R_{CS}_{2} = 0.25. After compressive sampling, the size of the images was changed to 128 × 128 pixels. Then, the subtraction images in the compressive domain were mapped to the spatial domain, and the sizes of the images were recovered to 256 × 256. As shown in

Figure 4b–f, the Gram matrix could effectively recover the subtraction image. The recovery subtraction image in which target points were prominent and the SNR was high could be used to further detect the targets.

Five indexes, including SNR, the Signal Clutter Ratio (SCR), the Background Suppression Factor (BSF), the Receiver Operating Characteristics (ROC), and the Area Under the Curve (AUC), were used to evaluate the mapping effect of the subtraction image. The definitions of these indexes are as follows:

where

μ_{t} is the mean of the target gray level and

μ_{b} is the mean of the background gray level, and

σ_{b} is the standard deviation of the background gray level. SNR mainly reflects the correlation between the target gray level and the background gray level. The larger the SNR, the smaller the correlation between the target and the background, and the less the target is disturbed by the background:

where

μ_{t} is the mean of the target gray level and

μ_{b} is the mean of the background gray level. SCR mainly reflects the difference between the target gray level and the background gray level. The bigger the SCR, the bigger the gray difference between the target and the background:

where

σ_{in} and

σ_{out} are the standard deviations of background gray level before and after filtering, respectively. The larger the BSF, the stronger the suppression of the background after filtering.

For the filtered residual image, the ROC curve of the detection algorithm can be drawn by changing the detection threshold

T and traversing the probability of false alarm

P_{f} to obtain the corresponding probability of detection

P_{d}, taking

P_{f} as the horizontal axis and

P_{d} as the longitudinal axis.

P_{d} and

P_{f} are defined as:

where

N_{t} represents the number of pixels that the algorithm detects as the correct target,

S_{t} represents the number of real pixels of the target,

N_{b} represents the number of pixels where the algorithm detects the wrong target, and

S_{b} represents the number of real pixels of the background. Under the same

P_{f}, if the

P_{d} of the algorithm is higher than that of the others, it means the algorithm has better performance.

The area AUC under the ROC curve can be divided into several trapezoids. Let the point on the ROC curve be set as (

x_{i}, y_{i})(

i = 1,…,

n), where

n is the total number of points on the ROC curve. AUC can be expressed as:

The larger the AUC value, the better the performance of the algorithm and the better the detection performance. The average ROC curve of the 100 experiments was compared due to the randomness of the measurement matrix.

Figure 5 depicts the average ROC curve of the 100 tests.

The recovery subtraction image was compared with the original 2980th frame image, and the evaluation indexes are shown in

Table 2. The table shows that the images mapped to the spatial domain through the measurement matrix were far higher in SNR and SCR than in the original 2980th frame image. The SNR values in the Part Hadamard measurement and the Circulant measurement were higher, which indicates that the correlation between the target and the background is smaller, and that the targets are less disturbed by the background. The values of SCR and BSF in the Part Hadamard measurement and Toeplitz measurement were higher, which indicates that the difference between the gray level of the targets and the background is larger, and that the background is strongly suppressed after filtering. The AUC value of the Part Hadamard measurement and Toeplitz measurement was higher, which shows that the two measurements are better in general, and that they produce better detection performance.

The detection effect of the adaptive threshold partition was further analyzed.

Figure 6a–e are images of the initial threshold partition, in which many false alarm points exist. The accuracy of detection decreased because the accumulation of false alarm points was mistaken for targets in small target detection.

Figure 6f–j are the adaptive threshold partition images when

$k=0$, in which the false alarm points were filtered effectively and did not accumulate. Targets in the adaptive threshold partition images were still outstanding. This shows that the adaptive threshold partition method could effectively reduce the probability of false alarms by guaranteeing the detection rate.

Compared the proposed 2D CDAT algorithm to the traditional detection algorithm using reconstruction, the 2D field was established using the Part Hadamard Random matrix model and the compressive subtracted image was reconstructed using the 2D iterative adaptive approach (2D IAA) algorithm [

4] and the 2D Smoothed L0 (2D SL0) algorithm [

5]. The sparse base was the DCT base. The parameters were set to LL = 10 and Ite = 500 in the 2D IAA algorithm, and to L = 3 in the 2D SL0 algorithm after repeated experiments to achieve better results.

$k=0$ was set in the 2D CDAT algorithm.

Figure 7a,b shows the reconstruction of the compressive subtracted image, and

Figure 7d,e show the target detection results produced by the 2D SL0 algorithm and the 2D IAA algorithm, respectively.

Figure 7c,f compare the 2D CDAT algorithm, the 2D IAA algorithm, and the 2D SL0 algorithm. Each algorithm was tested 100 times to reduce the random error, and the average ROC curve was drawn as shown in

Figure 8.

Table 3 shows the operation time, AUC, SNR, SCR, and BSF of each algorithm. The operation time, AUC, SNR, and SCR of the 2D CDAT algorithm were superior to the 2D IAA and 2D SL0 algorithms because the 2D SL0 algorithm and the 2D IAA algorithm needed many iterative optimizations during the reconstruction process. As the 2D CDAT algorithm needed no reconstruction to directly map the compressive subtracted image to the spatial domain, it was more efficient. AUC showed that the 2D CDAT algorithm was better overall. SNR and SCR indicated that targets in the subtraction image were more prominent and obvious after being mapped to the spatial domain by the 2D CDAT algorithm and that the difference between the targets and the background increased.

However, the BSF value for the 2D CDAT algorithm was lower than that of the 2D SL0 and the 2D IAA algorithms, which shows that the background was restrained weakly. This was due to the high sparsity of the compressive subtracted image. The original subtraction image can be recovered efficiently through the reconstruction of the compressive subtracted image using the 2D SL0 and 2D IAA algorithms, whereas 2D CDAT algorithm was mapped to the compressive subtracted image using the property of the Gram matrix, and could not completely recover the original subtraction image.

The focus was to separate the target from the background for the target detection problem. We did not need to excessively pursue the recovery degree of the image. From the detection results, the adaptive threshold detection method used by the 2D CDAT algorithm could effectively filter the false alarm points by guaranteeing the detection rate compared with the 2D SL0 and 2D IAA algorithms.