Hyperspectral Target Detection via Adaptive Information ... - MDPI

remote sensing Article

Hyperspectral Target Detection via Adaptive Information—Theoretic Metric Learning with Local Constraints Yanni Dong 1 , Bo Du 2, *, Liangpei Zhang 3 and Xiangyun Hu 1 1 2 3

*

Hubei Subsurface Multi–Scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China; [email protected] (Y.D.); [email protected] (X.H.) School of Computer, Wuhan University, Wuhan 430079, China State Key Laboratory of Information Engineering in Surveying, Mapping, and Remote Sensing, Wuhan University, Wuhan 430079, China; [email protected] Correspondence: [email protected]; Tel.: +86-138-7146-1059

Received: 18 July 2018; Accepted: 3 September 2018; Published: 6 September 2018







Abstract: By using the high spectral resolution, hyperspectral images (HSIs) provide significant information for target detection, which is of great interest in HSI processing. However, most classical target detection methods may only perform well based on certain assumptions. Simultaneously, using limited numbers of target samples and preserving the discriminative information is also a challenging problem in hyperspectral target detection. To overcome these shortcomings, this paper proposes a novel adaptive information-theoretic metric learning with local constraints (ITML-ALC) for hyperspectral target detection. The proposed method firstly uses the information-theoretic metric learning (ITML) method as the objective function for learning a Mahalanobis distance to separate similar and dissimilar point-pairs without certain assumptions, needing fewer adjusted parameters. Then, adaptively local constraints are applied to shrink the distances between samples of similar pairs and expand the distances between samples of dissimilar pairs. Finally, target detection decision can be made by considering both the threshold and the changes between the distances before and after metric learning. Experimental results demonstrate that the proposed method can obviously separate target samples from background ones and outperform both the state-of-the-art target detection algorithms and the other classical metric learning methods. Keywords: hyperspectral image; target detection; metric learning; local constraints

1. Introduction A hyperspectral image (HSI) obtained by remote sensing systems can provide significant information. Each pixel of HSI contains a continuous spectrum with hundreds or even thousands of spectral bands, of which the width of each band is about 5–10 nm, to detect and characterize target of interest in the scene [1,2]. Target detection is one of the most wide applications of hyperspectral image processing, and it plays an important role in the real world, such as detecting humanmade objects in reconnaissance applications, searching rare minerals in geology, and researching environmental pollution [3–5]. Based on specific spectral signatures (prior information), the purpose of target detection is to decide whether a target of interest is present or not present (background) in a pixel-under-test, which can be viewed as a binary classifier [6,7]. A number of classical target detection algorithms have been proposed in HSI analysis. Most of them are based on the linear models and statistical hypothesis tests, which can maximize the detection probability for fixed false alarm probability, such as orthogonal subspace projection (OSP) and adaptive cosine/coherence estimator (ACE). The former OSP method proposed by Harsanyi et al. [8] suppresses Remote Sens. 2018, 10, 1415; doi:10.3390/rs10091415

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the background signatures by projecting each pixel’s spectrum onto a subspace, which is orthogonal to the background signatures. The well-known ACE method proposed by Kraut et al. [9] assumes that the additive noise has been included in background, which is an unstructured background detector. However, most classical algorithms depend on the specific statistical hypothesis tests, and may only perform well under certain conditions, e.g., the ACE detector assumes that the background is homogeneous, which is unrealistic in the real world. In recent years, the machine learning techniques have been introduced into HSI target detection, which has been paid great attention [10,11]. Typical examples of these methods are kernel-based detectors, such as the kernel matched subspace detectors (KMSD) [12], kernel spectral matched filter (KSMF) [13], and kernel OSP [14]. The kernel-based methods map the original feature space into a potentially high-dimensional kernel space to solve the linearly inseparable problem in the original space. Apparently, as mentioned in the article [15], kernel-based methods are also based on statistical hypothesis test, and inherit the shortcomings of traditional target detection methods. It can be concluded that kernel-based methods attempt to find a stable and credible feature space (distance metric) for separating potential target pixels and background ones [16–18]. Otherwise, the spectral resolution of HSIs is so high that these spectral bands are often highly correlated. For decreasing spectral redundancy and releasing computational complexity, it is necessary to reduce dimension by discarding redundant features for HSI target detection [19,20]. There are such few target pixels of interest that HSI target detection rarely takes into consideration dimensionality reduction, which may hide the accuracy of detecting targets. That is to say, target detection is usually in a dilemma whether to reduce spectral redundancy or preserve discriminative information [21,22]. Thus, how to develop a proper metric with a low dimensionality for measuring the separability between target pixels and background ones becomes the key for HSI target detection [23]. In fact, metric learning methods have proved to be a more straightforward and effective way to obtain such a distance metric [24–26]. To date, there are a few metric learning methods that have been proposed for HSI target detection. For example, Zhang et al. [15] learned an objective function of the supervised distance maximization by putting a similarity propagation constraint and imposing a manifold smoothness regularization. Dong et al. [27] presented the maximum margin metric learning (MMML) method, which utilizes the maximum margin framework as the objective function to learn distance metric space and can maximally separate target samples from background ones without certain assumptions. Dong et al. [28] presented random forest metric learning (RFML) method, which adopts random forests as the underlying representation of the metric learning, to deal with limited numbers of target samples by merging the standard relative position and the absolute pairwise position. In general, by using metric learning, we can find the distance metric matrix, so as to transform the original space into the metric feature space. Then, we can detect the desired targets, especially when the samples are imbalanced and the number of target samples is very limited. In addition, a number of metric learning methods have been proposed to learn the distance metric, such as neighborhood component analysis (NCA) method [29], large margin nearest neighbor (LMNN) method [30], and so on. For each instance, NCA method expresses the probability of selecting the same class instances as the neighbors, which can maximize the stochastic variance of leave-one-out k-nearest neighbor (KNN) score on the training samples. LMNN method aims to find a distance metric such that the instances from different classes are effectively separated by a large margin within the neighborhood, where the margin is defined as the difference between the between-class and within-class distances. Furthermore, the information-theoretic metric learning (ITML) method, proposed by Davis et al. [31], expresses the weakly supervised metric learning problem as a Bregman optimization problem and can handle a variety of constraints and incorporate a priori information on the distance function. However, the existing metric learning based methods still have some obstacles to be addressed. The major problem is that most methods mentioned above are global metric learning with global constraints, making decisions by comparing their Mahalanobis distance d and judging d is lower or higher than the a fixed threshold b, which is insufficient and suboptimal. Therefore, in this paper, ITML

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method, which works in a weakly supervised manner, is innovatively introduced for hyperspectral target detection with adaptively local constraints (ITML-ALC, for short). The proposed ITML-ALC method explores adaptively local constraints to relax the fixed threshold, which can be used to compute the Mahalanobis distance d and judge if given samples are targets by considering both b and the changes between the distances before and after metric learning. By considering local constraints and avoiding adopting those conflicting constraints, the separability between target samples and background ones can be enhanced. Besides, non-square matrix W can be found for handling high-dimensional data problems by transforming the original space into a metric learning space with a low dimensionality. Compared with existing algorithms, ITML-ALC has several obvious advantages: 1. 2. 3.

The proposed ITML-ALC algorithm can use limited numbers of target samples to detect targets without certain assumptions, compared with traditional target detection methods. ITML-ALC needs only one parameter to be adjusted, and the detection results are relatively stable for different values of parameter. ITML-ALC can remain the locality information and improve the detection performance via considering both the threshold and the changes between the distances before and after metric learning, while existing metric learning based methods uses fixed threshold to make decision.

The rest of this paper is organized as follows. In Section 2, a briefly introduce of the original ITML method is provided, and the proposed ITML-ALC method is then presented. The experimental results of the proposed method using several challenging HSIs are detailed in Section 3, followed by the discussion and conclusions in Sections 4 and 5. 2. Methods 2.1. Related Work The ITML methodology minimizes the LogDet divergence subject to linear constraints. There are two key techniques of ITML. One is the ability to handle a wide variety of constraints and to optionally incorporate a priori information on the distance function. The other key technique is that it is fast and scalable. Suppose that we have a set of L-dimensional training samples {x1 , x2 , · · · , xn } ∈ R L×n , in which n represents the number of training samples and L is the number of feature dimensions. zij ∈ (+1, −1) denotes the relationship between the training samples xi and x j . Considering relationships of the similarity or dissimilarity between pairs of samples, distances between samples in the same class can be constrained as similar, and ones in different classes can be constrained as dissimilar. Then, we have a set of similar constraints S and a set of dissimilar constraints D as Equation (1): S : ∀(xi , x j ) ∈ S D : ∀(xi , x j ) ∈ D

xi , x j ∈ same class, zij = 1, xi , x j ∈ different class, zij = −1.

(1)

Metric learning aims to learn metric matrix M, which specifies the Mahalanobis distance dM (xi , x j ) between any pairs of samples xi and x j as: d M ( xi , x j ) =

q

( xi − x j ) T M ( xi − x j ) .

(2)

In order to ensure that dM (xi , x j ) is a meaningful distance, the learned metric matrix M must be symmetric and positive semidefinite (PSD) variance matrix, guaranteeing that dM (xi , x j ) is symmetrical, non-negativite, and has triangle inequality [32,33]. Considering the high dimensional of HSIs and M is PSD matrix, a nonsquare matrix W ∈ R L× D ( D  L), defining a mapping from the high-dimensional space into a low-dimensional embedding, can be established, and M = WWT [34–36]. In the Equation (2), our objective is to find the PSD matrix M (or W) and the corresponding distance threshold b such that for any pairs (xi , x j ) ∈ S the distance between them is smaller than b,

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and for any pairs (xi , x j ) ∈ D the distance between them is greater than b, which can be described as Equation (3): dM (xi , x j ) ≤ b (xi , x j ) ∈ S, (3) dM (xi , x j ) ≥ b (xi , x j ) ∈ D. The ITML method can minimize the differential relative entropy between two multivariate Gaussians and handle a variety of constraints on the distance function via a natural informationtheoretic approach. Thus, given a Mahalanobis distance parameterized by M, its corresponding multivariate Gaussian can be expressed as: p( x; M) =

1 1 exp(− dM (x, µ)), Z 2

(4)

where µ is the mean of Gaussians, Z is a normalizing constant in the Equation (4). By using the bijection, the distance between two Mahalanobis distance functions parameterized by M0 and M can be measured by the differential relative entropy of corresponding multivariate Gaussians: KL( p(x; M0 )|| p(x; M)) =

Z

p(x; M0 ) log

p(x; M0 ) dx, p(x; M)

(5)

In the Equation (5), M0 is a given Mahalanobis distance function, such as identity matrix. In conjunction with given pairs of similar points S and pairs of dissimilar points D, the distance metric learning can be summarized as the following optimization problems: minKL( p(x; M0 )|| p(x; M)) M

subject to dM (xi , x j ) ≤ b1 (xi , x j ) ∈ S, dM (xi , x j ) ≥ b2 (xi , x j ) ∈ D,

(6)

where b1 , b2 are given upper and lower bounds, respectively. Some research has shown that the differential relative entropy of corresponding multivariate Gaussians is equivalent to the LogDet divergence between the covariance matrices [37]: KL( p(x; M0 )|| p(x; M)) =

1 1 d (M−1 , M−1 ) = dlog det (M, M0 ), 2 log det 0 2

(7)

where M0−1 , M−1 are the covariance of the distributions. Taking into account that a feasible solution of Equation (6) may not exist, we incorporate slack variable ξ into Equation (6) to guarantee the existence of the metric matrix M. Thus, the Equation (6) can be represented as the following optimization problem with Equation (7): min dlog det (M, M0 ) + γ·dlog det (diag(ξ), diag(ξ0 ))

M≥0,ξ

s.t. tr(M(xi − x j )(xi − x j )T ) ≤ ξc(i,j) (xi , x j ) ∈ S, tr(M(xi − x j )(xi − x j )T ) ≥ ξc(i,j) (xi , x j ) ∈ D,

(8)

where ξ0 denotes initialized slack variables, and c(i, j) is the index of the (i, j) − th constraint. γ is the tradeoff parameter, which controls the tradeoff between satisfying the constraints and minimizing dlog det . 2.2. Combining ITML and Adaptively Local Constraints The ITML method uses fixed threshold to make decision, which makes it less effective to handle data with complex distributions even if the associated metric is correct. To address this issue, this paper proposes an adaptively local decision rule to design pairwise constraints to relax the fixed threshold for target detection. We design a local decision function f (dij ) to achieve this goal, where dij is the

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distance of a similar or dissimilar pair (xi , x j ). The principle for designing f (dij ) is: that the greater the distance between similar pairs, the more f (dij ) should shrink, while the smaller the distance between dissimilar pairs, the more f (dij ) should expand. Based on this principle, the local constraints can be redefined to make better decision by considering both the threshold b and the changes between the distances before and after metric learning. As a result, we can form the local decision constraints to compute the adaptive upper/lower bounds for (xi , x j ) as: (1/NS )

f S (dij ) = dij − (dij

(xi , x j ) ∈ S, ) (xi , x j ) ∈ D,

/dc ) (1/ND )

f D (dij ) = dij + (dc /dij

(9)

where NS ≥ 1 and ND ≥ 1 are the scale factors that separately control shrinkage and expansion. dc ≥ 1 is the constant. From Equation (9), we can see that the smaller the NS is, the faster f S (dij ) will shrink, while the greater the ND is, the faster f D (dij ) will expand. Then we set dc = dmax (where dmax is the maximal distance between all the pairs). Clearly, we want to maximize the shrinkage and expansion of f (dij ). Considering that NS 1 cannot guarantee that the constraints are positive, we set NS = 1 to ensure that f S (dij ) can shrink as fast as possible, while setting ND = 1/ log2 (dc /(dc − 2)) to guarantee the faster expansion of f D (dij ). Thus, Equation (9) can be transformed as: f S (dij ) = dij − (dij /dmax q)

f D (dij ) = dij + (dmax /

ND

dij )

(xi , x j ) ∈ S, (xi , x j ) ∈ D,

(10)

For relaxing the fixed threshold of ITML method in the Equation (8), we can substitute the original fixed bound of ITML with the above-mentioned local decision constraints of Equation (10), and we finally obtain the adaptive ITML with local constraints (ITML-ALC) detector: min dlog det (M, M0 ) + γ · dlog det (diag(ξ), diag(ξ0 ))

M≥0,ξ

s.t. dM (xi , x j ) ≤ f S (dM (xi , x j )) + ξc(i,j) (xi , x j ) ∈ S, dM (xi , x j ) ≥ f D (dM (xi , x j )) + ξc(i,j) (xi , x j ) ∈ D, ξc(i,j) ≥ 0, ∀(i, j), d ≥ 0,

(11)

In addition, combining Equation (1) with Equation (11), we can set the following Equation (12): ( f (dij , zij ) =

f S (dM (xi , x j )) − f D (dM (xi , x j ))

xi , x j ∈ same class, zij = 1, xi , x j ∈ different class, zij = −1.

(12)

Thus, the final ITML-ALC objective function can be simplified as: min dlog det (M, M0 ) + γ · dlog det (diag(ξ), diag(ξ0 ))

M≥0,ξ

s.t. zij dM (xi , x j ) ≤ f (dij , yij ) + ξc(i,j) ξc(i,j) ≥ 0, ∀(i, j), M ≥ 0,

(13)

Obviously, ITML-ALC has the same complexity and can be solved using the same algorithm as ITML.

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2.3. Final Sketch of ITML-ALC Algorithm In the Equation (13), the metric matrix M can be projected into the PSD cone by adopting spectral decomposition. To compute the linear projections, we solve the following generalized eigenvector problem as: m

M=

∑ λi νTi νi

(14)

i =1

In the Equation (14), we start with setting the negative eigenvalues λ to zero, then we can obtain metric matrix M by using the remaining positive eigenvalues and the corresponding eigenvectors ν. Then, we can further learn a linear projection matrix W ∈ R L× D ( D  L) for the dimensionality reduction. For an arbitrary test pixel vector xi ∈ R L×n , we compute the final metric feature representation in the final ITML-ALC feature space with the equation M = WWT by: x0 = WT x

(15)

By applying Equation (15), the original data x can be transformed into the Mahalanobis metric space. Finally, the target detection result is obtained by using a detection algorithm. In our method, the ACE detector is used because of its simplicity and effectivity. Algorithm 1: Procedure of ITML-ALC n o Input: A set of pairwise training data points (xi , x j , zij ) ∈ (S ∪ D) , the trade–off parameter γ Output: Metric matrix M Step (1): Initialization: input Mahalanobis matrix M0 = I, initialized slack variable ξ0 , dc = dmax , NS = 1, ND = 1/ log2 (dc /(dc − 2)). ( ξc(i,j) ← f S (dij ), (xi , x j ) ∈ S Set: M = M0 , λij = 0. ξc(i,j) ← f D (dij ), (xi , x j ) ∈ D Step (2): Repeat {Main loop} (a) p ← (xi − x j )T M(xi − x j ) (   zij = 1, ( xi , x j ) ∈ S zij 1 γ (b) α ← min(λij , 2 p − ξ ), c(i,j) zij = −1, (xi , x j ) ∈ D (c) β ← zij α/(1 − zij αp)   (d) ξc(i,j) ← γξc(i,j) / γ + zij αξc(i,j) (e) λij ← λij − α (f) M ← M + βM(xi − x j )T (xi − x j )M Until: convergence is attained Return: M

The proposed ITML-ALC algorithm is the integration of the ITML method and adaptively local constraints. Thus, ITML-ALC can be solved by using the same algorithm as the procedure of ITML. Refer to the reference [31], and we can obtain the procedure of the proposed ITML-ALC algorithm, summarized as Algorithm 1. 2.4. Workflow of ITML-ALC Algorithm The schematic diagram of the proposed ITML-ALC method for HSI target detection is shown in Figure 1. Given a hyperspectral image, a priori information of the ITML-ALC algorithm including target samples (red points) and background samples (green points and blue points) is needed. The flowchart of the proposed algorithm consists of the following steps: (1) The ITML metric feature framework is constructed, which can transform the original HSI data into the Mahalanobis metric space. (2) The adaptively local decision constraints are applied into ITML framework. Unlike the fixed decision

The schematic diagram of the proposed ITML-ALC method for HSI target detection is shown in Figure 1. Given a hyperspectral image, a priori information of the ITML-ALC algorithm including target samples (red points) and background samples (green points and blue points) is needed. The flowchart of the proposed algorithm consists of the following steps: (1) The ITML metric feature framework is constructed, which can transform the original HSI data into the Mahalanobis metric Remote Sens. 2018, 10, 1415 7 of 16 space. (2) The adaptively local decision constraints are applied into ITML framework. Unlike the fixed decision paradigm, which may be easily misclassified, we can classify the pairs according to the paradigm, which may be constraints, easily misclassified, we the can distance classify the pairs according to the adaptively adaptively local decision with which of the similar pair shrinks while the local decision constraints, with which the distance of the similar pair shrinks while the distance of the distance of the dissimilar pair expands as much as possible, illustrated in the solid box. Thus, it allows dissimilar expands as muchbyasconsidering possible, illustrated the solid box. it allows between us to make us to makepair a correct decision both theinthreshold b andThus, the changes thea correct decision by considering both the threshold b and the changes between the distances before and distances before and after metric learning. (3) For achieving target detection, we transform the after metric achieving target detection,metric we transform the original HSI datasamples into the original HSI learning. data into (3) the For ITML-ALC low-dimensional feature space, in which target ITML-ALC low-dimensional feature space, inones. which samples be maximally can be maximally separated metric from the background (4)target We apply thecan specific detector separated to obtain from the background ones. (4) We apply the specific detector to obtain the target detection results. the target detection results.

Figure 1.1.Schematic illustration of adaptive information-theoretic metric learning local with constraints Figure Schematic illustration of adaptive information-theoretic metric with learning local (ITML-ALC) algorithm for HSI target constraints (ITML-ALC) algorithm fordetection. HSI target detection.

3. Experiments Analysis and Results 3. Experiments Analysis and Results In this section, a synthetic dataset and two real datasets are performed to evaluate the effectiveness In this section, a synthetic dataset and two real datasets are performed to evaluate the of the proposed method. The first one is a synthetic HSI dataset created through implanting alunite effectiveness of the proposed method. The first one is a synthetic HSI dataset created through object spectra into the specific locations. The second and third ones are real HSI datasets with implanting alunite object spectra into the specific locations. The second and third ones are real HSI complex background distributions. For comparison, a series of existing state-of-the-art target detection datasets with complex background distributions. For comparison, a series of existing state-of-the-art algorithms, i.e., ACE and OSP detectors, are used to thoroughly evaluate the performance of the target detection algorithms, i.e., ACE and OSP detectors, are used to thoroughly evaluate the proposed algorithm. We also compare the proposed algorithm with three classical metric learning performance of the proposed algorithm. We also compare the proposed algorithm with three classical methods, i.e., LMNN, NCA, and ITML, which can also be applied to dimensionality reduction. In all metric learning methods, i.e., LMNN, NCA, and ITML, which can also be applied to dimensionality the detectors (except ACE), we apply the same given training samples, including the target signatures reduction. In all the detectors (except ACE), we apply the same given training samples, including the and background signatures, which are randomly selected from the datasets. For ACE algorithm, target signatures and background signatures, which are randomly selected from the datasets. For we only implement the same target signatures as the proposed algorithm. In addition, we apply ACE algorithm, we only implement the same target signatures as the proposed algorithm. In addition, the ACE detector as the basic detector in the LMNN, NCA, and ITML algorithms like the proposed we apply the ACE detector as the basic detector in the LMNN, NCA, and ITML algorithms like the algorithm. As for the adjustment of parameters, trade-off parameters in the LMNN, NCA and ITML proposed algorithm. As for the adjustment of parameters, trade-off parameters in the LMNN, NCA algorithms are tuned via threefold cross validation according to relative references [30], [29], and [31], and ITML algorithms are tuned via threefold cross validation according to relative references [24], respectively. All the experiments are implemented on a computer with an Intel(R) Core(TM) i7-7700 [23], and [25], respectively. All the experiments are implemented on a computer with an Intel(R) Central Processing Unit (CPU) at 3.60 GHz (8 GPUs), 16-GB Random Access Memory (RAM) and Core(TM) i7-7700 Central Processing Unit (CPU) at 3.60 GHz (8 GPUs), 16-GB Random Access 64-bit Windows 10 Operating System (OS). Memory (RAM) and 64-bit Windows 10 Operating System (OS). 3.1. Dataset Description 3.1. Dataset Description Three hyperspectral datasets were used in this study to evaluate the performance of the proposed method introduced in Section 3. (1)

AVIRIS LCVF dataset: This image was acquired by the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) sensor, operated by National Aeronautics and Space Administration (NASA), USA, covering the Lunar Crater Volcanic Field (LCVF) in Northern Nye County, NV,

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Three Three hyperspectral hyperspectral datasets datasets were were used used in in this this study study to to evaluate evaluate the the performance performance of of the the Three hyperspectral datasets were used in this study to evaluate the performance of the proposed proposed method method introduced introduced in in Section Section 3. 3. proposed method introduced in Section 3. (1) (1) AVIRIS AVIRIS LCVF LCVF dataset: dataset: This This image image was was acquired acquired by by the the Airborne Airborne Visible/Infrared Visible/Infrared Imaging Imaging (1) Spectrometer AVIRIS dataset: This image was acquired by the Airborne and Visible/Infrared Imaging Remote Sens. 2018,LCVF 10, 1415 8 of 16 (AVIRIS) sensor, operated by National Aeronautics Space Administration Spectrometer (AVIRIS) sensor, operated by National Aeronautics and Space Administration Spectrometer (AVIRIS) sensor, operated by National Aeronautics and Space Administration (NASA), (NASA), USA, USA, covering covering the the Lunar Lunar Crater Crater Volcanic Volcanic Field Field (LCVF) (LCVF) in in Northern Northern Nye Nye County, County, NV, NV, (NASA), USA, covering the Lunar Crater Volcanic Field (LCVF) in Northern Nye County, NV, USA. This dataset is aa synthetic dataset and is available from the website of NASA website. USA. This dataset is synthetic dataset and is available from the website of NASA website. USA. dataset is aa synthetic dataset and and isis available availablefrom fromthe thewebsite websiteofofNASA NASAwebsite. website. USA. This This dataset is synthetic dataset Many studies used this dataset for HSI processing [30,31]. The spatial resolution of the LCVF Many studies studies used used this this dataset dataset for for HSI HSI processing processing [38,39]. [30,31]. The The spatial spatial resolution resolution of of the the LCVF LCVF Many Many studies used this dataset forspectral HSI processing [30,31]. The spatial resolution of224 the spectral LCVF image is 20 m per pixel, and the resolution of the image is 5 nm, with image is 20 m per pixel, and the spectral resolution of the image is 5 nm, with 224 spectral image the spectral spectral resolution resolution of of the the image image isis 55 nm, nm, with with224 224spectral spectral image is is 20 20 m m per per pixel, pixel, and and the channels in wavelengths ranging from 370 2510 nm. An area of pixels is for 200  channels in in wavelengths wavelengthsranging rangingfrom from370 370toto to2510 2510 nm. An area of pixels is used used for 200  200 200 channels nm. An area of 200 × 200 pixels is used forfor the channels in wavelengths ranging from 370 to 2510 nm. An area of 200  200 pixels is used the experiments, as shown in Figure 2a, including red oxidized basaltic cinders, rhyolite, playa the experiments, as shown in Figure 2a, including red oxidized basaltic cinders, rhyolite, playa experiments, as shown in Figure 2a, including red oxidized basaltic cinders, rhyolite, playa (dry the experiments, as shown in Figure 2a, including red oxidized basaltic cinders, rhyolite, playa (dry lakebed), shade, and vegetation. We implant the alunite spectrum, which is obtained from (dry lakebed), shade, and vegetation. We implant the alunite spectrum, which is obtained from lakebed), shade,shade, and vegetation. We implant the alunite spectrum, which which is obtained from the U.S. (dry lakebed), and vegetation. We implant the alunite spectrum, is obtained from the Survey (USGS) digital spectral library, into image for target the U.S. U.S. Geological Geological Surveydigital (USGS) digitallibrary, spectralinto library, into the the image for simulating simulating target Geological Survey (USGS) spectral the image for simulating target detection the U.S. Geological Survey (USGS) digital spectral library, into the image for simulating target detection in the considered scene. Figure 2b shows corresponding locations of the implanted detection in scene. Figure 2b shows shows corresponding corresponding locations the implanted in the considered scene. Figure 2b shows corresponding locations of locations the implanted target panels. detection in the the considered considered scene. Figure 2b ofofthe implanted target panels. The added target panels have the same size, i.e., two pixels for each target panel, target panels. The added target panels have the same size, i.e., two pixels for each target panel, The added target thepanels same size, and the detailed target panels. Thepanels addedhave target havei.e., the two samepixels size, for i.e.,each two target pixels panel, for each target panel, and the detailed coordinates of all 30 target pixels are given in Table 1. In this table, all the and coordinates of are all 30 30 target target pixels are given Table 1.InInthis thistable, table, the coordinates of all 30 target pixels given in Table 1.are In this table, all the1.implanted targetallall pixels and the the detailed detailed coordinates of all pixels given ininTable the x implanted target pixels are mixed pixels, and each spectrum of the HSI is mixed with the implanted target pixels are mixed pixels, and each spectrum of the HSI is mixed with the x are mixed pixels, each x of the is spectrum mixed with pureHSI prior target spectrum x the implanted target and pixels arespectrum mixed pixels, andHSI each of the is mixed with the b (by prior target spectrum and the original background spectra the following pure prior target spectrum tttspectra and the the original background spectra by the following tpure and the original b byoriginal the following equation: x = pt 1− p)b, where p bb+by pure prior targetbackground spectrum and background spectra the following p x = p t  (1  p ) b equation: ,, where is the implanted fraction, which varies from 10% to 50%, p x = p t  (1  p ) b equation: where is the implanted fraction, which varies from 10% 50%, is the implanted fraction, which varies from 10% to 50%, as indicated in Table 1. The adopted equation: x = pt  (1  p)b , where p is the implanted fraction, which varies from 10% toto50%, as indicated in Table 1. The adopted pure target spectrum and some representative background pure target spectrum representative background spectra (denoted as A to as in 1.and Thesome adopted pure target targetspectrum spectrumand andsamples somerepresentative representative background as indicated indicated in Table Table 1. The adopted pure some background samples spectra (denoted as A to H) are shown in Figure 2c, and the locations of the background H) are shown in Figure 2c, and the locations of the background samples given in Figure 2c are samples spectra (denoted as A to H) are shown in Figure 2c, and the locations of the background samples spectra (denoted as A to H) are shown in Figure 2c, and the locations of the background samples given in Figure 2c are highlighted in Figure 2d. highlighted in Figure 2d.2c samples in Figure 2c are are highlighted highlightedin inFigure Figure2d. 2d. samples given given in Figure 12000 12000 12000 12000

Target spectrum Target Target spectrum Target spectrum spectrum Target spectrum A A A A A B B B B B C C C C C D D D D D E E E E E F F F F F G G G G G H H H H H

Spectral Reflectance SpectralReflectance Reflectance Spectral Reflectance Spectral Reflectance Spectral Reflectance Spectral

10000 10000 10000 10000 8000 8000 8000 8000 6000 6000 6000 6000 4000 4000 4000 4000 2000 2000 2000 2000 0 0 500 00 500 500 500

1000 1000 1000 1000

1500 1500 1500 1500

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Wavelength /nm Wavelength /nm Wavelength /nm Wavelength /nm

(a) (a) (a)

(b) (b) (b)

(c) (c) (c)

(d) (d) (d)

Figure 2. AVIRIS LCVF dataset for the experiment. (a) Image scene; (b) Implanted target locations in Figure2. 2.AVIRIS dataset for the Figure 2. AVIRISLCVF LCVFdataset datasetfor forthe theexperiment. experiment.(a) (a)Image Imagescene; scene;(b) (b)Implanted Implantedtarget targetlocations locationsinin in Figure Image scene; (b) Implanted target locations the image; (c) Implanted pure target target spectrum and some representative background samples spectra; theimage; image;(c) pure the image; (c) Implanted Implanted pure target spectrum spectrumand andsome somerepresentative representativebackground backgroundsamples samplesspectra; the and some representative background samples spectra; (d) Locations of the background samples given in (c). (d)Locations Locationsof background samples given in (d) (d) Locations ofthe thebackground backgroundsamples samplesgiven givenin in(c). (c). Table 1. Details of the implanted target panels for the AVIRIS LCVF dataset. Table the implanted target panels for the AVIRIS LCVF dataset. Table 1. 1. Details the implanted implantedtarget targetpanels panelsfor forthe theAVIRIS AVIRISLCVF LCVFdataset. dataset. Details of of the Colorof of Target Target Panel Color Color Panel Color of Target Panel

(2) (2) (2) (2)

Sample Sample Index Sample Index SampleIndex Index

Sample Index 50 50 50 50 75 75 75 75 100 100 100 100 125 125 125 125 150 150 150 150

Line Index Line Index Line Index Line Index

Fraction Fraction Fraction Fraction Line Index (75, 76) (100, 101) (125, 126) 10% (75, 76) (100, 101) (125, 126) 10% (75, 76) (100, 101) (125, 126) (75, 76) (100, 101) (125, 126) 10% 10% (75, 76) (100, 101) (125, 126) 20% (75, 76) (100, 101) (125, 126) 20% (75, 76)76) (100, 101) (125, 126)126) 20% (75, (100, 101) (125, 20% (75, 76) (100, 101) (125, 126) 30% (75, 76) (100, 101) (125, 126) 30% (75, 76) (100, 101) (125, 126) 30% 30% (75, 76) (100, 101) (125, 126) (75, 76) (100, 101) (125, 126) 40% (75, 76) (100, 101) (125, 126) 40% (75, 76) (100, 101) (125, 126) 40% (75, 76) (100, 101) (125, 126) 40% (75, 76) (100, 101) (125, 126) 50% (75, 76) (100, 101) (125, 126) 50% (75, (100, 101) (125, 50% (75, 76)76) (100, 101) (125, 126)126) 50%

AVIRIS San AVIRIS dataset: This image capturing an airport the region San Diego, airport AVIRIS San Diego Diego airport dataset: This This image image capturing capturingan anairport airportinin inthe theregion regionofof ofSan SanDiego, Diego, airport dataset: AVIRIS San Diego airport dataset: This image capturing an airport in the region of San Diego, CA, USA, by the AVIRIS sensor, as shown in Figure 3a [32]. The spatial resolution CA, USA, was recorded the AVIRIS sensor, as shown in Figure 3a [32]. The spatial resolution by CA, USA, was recorded by the AVIRIS sensor, as shown in Figure 3a [32]. The spatial resolution CA, USA, wasis by pixel. the AVIRIS sensor, as224 shown in Figure 3a [40]. The spatial resolution of this this image isrecorded 3.5 m The has channels inin ranging of pixel. The image has 224 spectral channels wavelengths ranging image 3.5 m per of this pixel. The image image has 224 spectral spectral channels inwavelengths wavelengths ranging image is 3.5 m per per from 370 to 2510 nm. A total of 189 bands are used in the experiments after removing the of this image is 3.5 m per pixel. The image has 224 spectral channels in wavelengths ranging from from 370 to 2510 nm. A total of 189 bands are used in the experiments after removing the bands from 370 to 2510 nm. A total of 189 bands are used in the experiments after removing thebands bands thatto correspond tototal the regions, low-signal noise ratio (SNR), 370 2510 nm. A of 189absorption bands are used in the experiments after removing thebad bands that that correspond to the water absorption regions, low-signal noise ratio (SNR), and bad bands that correspond to the water water absorption regions, low-signal noise ratio (SNR),and and badbands bands (1–6, 33–35, 97, 107–113, 153–166, and 221–224). An area of pixels is used for the 100  100 correspond to the water absorption regions, low-signal noise ratio (SNR), and bad bands (1–6, (1–6, (1–6, 33–35, 33–35, 97, 97, 107–113, 107–113, 153–166, 153–166, and and 221–224). 221–224). An An area area of of 100 pixels is is used used for for the the 100   100 100 pixels experiments, including roofs, bare soil, grass, road, airstrip, and shadow, which contains more 33–35, 97, 107–113, 153–166, and 221–224). An area of 100 × 100 pixels is used for the experiments, experiments, including roofs, bare soil, grass, road, airstrip, and shadow, which contains more experiments, including roofs, bare soil, grass, road, airstrip, and shadow, which contains more complicated background classes. There three airplanes ininthe image denoted as including roofs, bare soil,land-cover grass, road, airstrip, andare shadow, contains complicated complicated background land-cover classes. There are three airplanes image denoted complicated background land-cover classes. There are three which airplanes in the themore image denoted as as the targets of interest, which consist of 58 target pixels, and are denoted with the white target background land-cover classes. There are three airplanes in the image denoted as the targets of the the targets targets of of interest, interest, which which consist consist of of 58 58 target target pixels, pixels, and and are are denoted denoted with with the the white white target target interest, which consist of 58 target pixels, and are denoted with the white target mask in Figure 3b. Figure 3c shows the spectra of mean target pixels and some representative background samples. We select the target spectra of the centers of airplanes as a priori target spectra, and we randomly select eight background spectra as the background samples.

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mask in Figure 3b. Figure 3c shows the spectra of mean target pixels and some representative samples. We select the target spectra of the centers of airplanes as a priori target Remotebackground Sens. 2018, 10, 1415 9 of 16 spectra, and we randomly select eight background spectra as the background samples. 6000 Mean target spectrum Roof Bare soil

Spectral Reflectance

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3000 2000 1000 0

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Figure 3. AVIRIS San Diego Diego airport dataset dataset for the experiment. experiment. (a) Image scene; (b) The true locations of the targets; (c) (c) Spectra Spectra of of mean mean target target pixels pixels and and some some representative representativebackground backgroundsamples. samples.

(3) HYDICE urban urban dataset: dataset: This This image image is a Hyperspectral Digital Imagery Imagery Collection Collection Experiment (3) HYDICE Experiment (HYDICE) airborne airborne sensor sensor dataset, dataset, sponsored sponsored by by the the U.S. U.S. Navy Navy Space and Warfare Systems Systems (HYDICE) Command, covering covering a suburban illustratedin in Figure Figure 4a. 4a. The scene Command, suburban residential residential area area[41,42], [33], asasillustrated mainly covers coversgrass grassfields fieldswith withsome someforest, forest,and andthe the rest scene is mixed with a parking mainly rest ofof thethe scene is mixed with a parking lot lot with some vehicles, a residential area, anda aroadway roadwaywhere wheresome somevehicles vehicles exist. exist. The spatial with some vehicles, a residential area, and resolution of of this this image image is is approximately approximately 3 m, and the spectral resolution of the image is about resolution 10 nm. The image scene contains an area 80 ×of 10080 pixels, 210 spectral bands in wavelengths nm. The image scene contains an ofarea pixels, with 210 spectral bands in  100with ranging fromranging 400 to 2500 There pixelsare of17 desired thetargets, vehicles, are wavelengths from nm. 400 to 2500are nm.17There pixels targets, of desired thewhich vehicles, contained in the parking lot and the road, shown in Figure 4b. Besides, the the spectra of which are contained in the parking lot and theasroad, as shown in Figure 4b. Besides, spectra mean target pixels 4c. of mean target pixelsand andsome somerepresentative representativebackground backgroundsamples samples are are illustrated illustrated in Figure 4c. We randomly select a target spectrum as a priori target spectrum, and seven background spectra as the the background background samples. samples. as 800 Mean target spectrum Grass

Spectral Reflectance

600

Road1 Road2 Bare soil Parking lot

400

200

0

500

1000

1500

2000

2500

Wavelength /nm

(a)

(b)

(c)

Figure 4. 4. HYDICE urban dataset for the experiment. (a) Image scene; (b) The true locations of the targets; (c) Spectra Spectra of of mean mean target target pixels pixels and and some some representative representative background backgroundsamples. samples.

Taken together, together, we we summary summary the the details details of Taken of experimental experimental images images acquired acquired from from different different sensors, sensors, illustrated in Table 2. illustrated in Table 2. Table 2. Details of the experimental images acquired from different sensors. Experimental Dataset

LCVF Dataset

San Diego Airport Dataset

Urban Dataset

Sensor Spectral band Spatial resolution Wavelength Image size

AVIRIS 224 20 m 370–2510 nm 200 × 200

ACIRIS 224 3.5 m 370–2510 nm 100 × 100

HYDICE 210 3m 400–2500 nm 80 × 100

Table 2. Details of the experimental images acquired from different sensors. Experimental Dataset Sensor Spectral band Spatial resolution Remote Sens. 2018, 10, 1415 Wavelength Image size

LCVF Dataset AVIRIS 224 20 m 370–2510 nm

San Diego Airport Dataset ACIRIS 224 3.5 m 370–2510 nm

Urban Dataset HYDICE 210 3m 400–2500 nm

200  200

100  100

80  100

10 of 16

3.2.3.2. Parameter Analysis Parameter Analysis

0.998 0.996 0.994 ITML-ALC

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0

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14

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the area under the ROC curve (AUC)

1.000

the area under the ROC curve (AUC)

the area under the ROC curve (AUC)

In this subsection, we parameterson onthe thedetection detection performance of the In this subsection, weevaluate evaluatethe theeffect effect of of the the parameters performance of the proposed ITML-ALC algorithm. NCAand andITML, ITML,the the proposed algorithm proposed ITML-ALC algorithm.Similar Similarto to LMNN, LMNN, NCA proposed algorithm alsoalso has has a trade-off parameter, i.e.,γ, γwhich , which can tunedvia via threefold threefold cross However, according a trade-off parameter, i.e., can bebetuned crossvalidation. validation. However, according to experimental studies, thatthe theAUCs AUCs different values relatively stable, to experimental studies,ititcan canbe be observed observed that of of different areare relatively stable, γ γvalues which inspired us to set γ = 1 for all the experiments, illustrated in Figure 5. which inspired us to set γ  1 for all the experiments, illustrated in Figure 5. 1.000 0.996 0.992 0.988 ITML-ALC

0.984 0.980

0

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(c)

Figure 5. The AUCs of the proposed ITML-ALC method with respecttotoparameter parameterγ. γ(a). (a) AVIRIS Figure 5. The AUCs of the proposed ITML-ALC method with respect AVIRIS LCVF LCVF(b) dataset; (b)San AVIRIS Sanairport Diego dataset; airport dataset; (c) HYDICE dataset. dataset; AVIRIS Diego (c) HYDICE urbanurban dataset.

3.3.3.3. Detection Results and Validation Detection Results and Validation In this subsection, the of the theproposed proposedITML-ALC ITML-ALC algorithm is evaluated In this subsection, thedetection detectionperformance performance of algorithm is evaluated quantitatively receiveroperating operatingcharacteristic characteristic (ROC) separation quantitatively byby receiver (ROC)curves curvesand andtarget-background target-background separation maps [1,7]. curves are widely as a standard performance evaluation tool for detection target maps [1,8]. ROCROC curves are widely usedused as a standard performance evaluation tool for target detection applications, which can the relationship between the detection probability applications, which can describe thedescribe relationship between the detection probability (the ratio(the of the ratio of the number of corrected-detected target pixels to the total number of target pixels in the image) number of corrected-detected target pixels to the total number of target pixels in the image) and and falserate alarm rate the (FAR, theof ratio the number of background pixels mistaken targets thetotal false alarm (FAR, ratio theof number of background pixels mistaken as as targets toto the total number of pixels in the image) based on the ground truth. Obviously, for the same level of FAR, number of pixels in the image) based on the ground truth. Obviously, for the same level of FAR, the algorithm with the highest detection performs better, which locates further near the top left of the the algorithm with the highest detection performs better, which locates further near the top left of the coordinate plane. Target-background separation maps can intuitively show how the target pixels are coordinate plane. Target-background separation maps can intuitively show how the target pixels are separated from the background ones. Generally speaking, good detector may highlight the targets separated from the backgroundinto ones. Generally good detector may the targets and suppress the background a small range ofspeaking, values while distinguishing the highlight targets. Moreover, andwe suppress background intounder a small values while the targets. Moreover, can alsothe consider the area therange ROC of curve (AUC) to distinguishing assess the accuracy, obtaining the we can also consider the area under the ROC curve (AUC) to assess the accuracy, obtaining the average behavior [34]. Meanwhile, we use the FAR under TDR = 100% to further evaluateaverage the behavior [43]. Meanwhile, detection performance [35].we use the FAR under TDR = 100% to further evaluate the detection performance [44]. For the AVIRIS LCVF dataset, the two-dimensional (2-D) detection maps and target detection test plots, LCVF being obtaining by the output value, of(2-D) all the algorithms in comparison shown test Forstatistic the AVIRIS dataset, the two-dimensional detection maps and target are detection in Figure 6, in which the higher brightness implies that the probability of detecting the desired targets statistic plots, being obtaining by the output value, of all the algorithms in comparison are shown in is higher. As shown in Figure 6, OSP, ITML that and the ITML-ALC obtain a superior in is Figure 6, in which the higher brightness implies probability of detecting theperformance desired targets background suppression, while ITML and ITML-ALC can also obtain a superior performance in higher. As shown in Figure 6, OSP, ITML and ITML-ALC obtain a superior performance in background highlighting target pixels. However, for ITML-ALC, the false alarm pixels are much fewer when the suppression, while ITML and ITML-ALC can also obtain a superior performance in highlighting target target locations of the proposed algorithm are more obvious. Furthermore, the higher test statistic pixels. However, for ITML-ALC, the false alarm pixels are much fewer when the target locations of the indicates a higher level of probability that the desired target presents at a certain pixel, shown in proposed algorithm are more obvious. Furthermore, the higher test statistic indicates a higher level Figure 6a–f. While, from these plots it can be seen that the proposed ITML-ALC can suppress the of probability that the desired target presents at a certain pixel, shown in Figure 6a–f. While, from these plots it can be seen that the proposed ITML-ALC can suppress the background pixels to a low and steady range. In general, for the proposed ITML-ALC algorithm, the background suppression performance is outstanding, while all targets are successfully extracted.

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Remote Sens. 2018, 10, 1415to a low and steady range. In general, for the proposed ITML-ALC algorithm, the 11 of 16 background pixels

backgroundsuppression pixels to a low and steadyisrange. In general, forall the proposed ITML-ALC algorithm, background performance outstanding, while targets are successfully extracted. the background suppression performance is outstanding, while all targets are successfully extracted. 1.0

1.0

1.0

0.81.0

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0.60.8

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0.60.8

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(d) (e) (f) (d) (e) (f) Figure 6. Detectionmaps mapsand andtarget target detection detection test ofof thethe AVIRIS LCVF dataset. (a) ACE; Figure 6. Detection teststatistic statisticplots plots AVIRIS LCVF dataset. (a) ACE; Figure 6. Detection maps and target detection test statistic plots of the AVIRIS LCVF dataset. (a) ACE; (b) OSP; (c) LMNN; (d) NCA; (e) ITML; (f) ITML–ALC. (b) OSP; (c) LMNN; (d) NCA; (e) ITML; (f) ITML-ALC. 0

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(b) OSP; (c) LMNN; (d) NCA; (e) ITML; (f) ITML–ALC.

Figure furtherprovides provides the ROC curves of allof theallreference algorithms in log- in Figure 7 7further thetarget targetdetection detection ROC curves the reference algorithms Figure 7 further provides the target detection ROC curves of all the reference algorithms logscale. For all datasets, it can be observed that the ITML-ALC algorithm achieves superior detection log-scale. For all datasets, it can be observed that the ITML-ALC algorithm achieves superior in detection scale. For all datasets, it can be observed that the ITML-ALC algorithm achieves superior detection performance, since the100% 100%detection detection rate rate can the FAR is rare (0–0.02), compared performance, since the can be beobtained obtainedwhen when the FAR is rare (0–0.02), compared performance, the 100% detection be obtained when the rare (0–0.02), compared with the other since algorithms. Especially forrate thecan AVIRIS LCVF dataset, theFAR FARis of ITML-ALC is nearly with the other algorithms. Especially for the AVIRIS LCVF dataset, the FAR of ITML-ALC is nearly with to thezero other algorithms. for the LCVFFor dataset, the FAR ITML-ALC is nearly equal when the 100%Especially detection rate canAVIRIS be obtained. the AVIRIS Sanof Diego airport dataset, equal to zero when the 100% detection rate can be obtained. For the AVIRIS San Diego airport dataset, equal to zero the 100% detection canthose be obtained. For the AVIRIS San Diego airport the ROC curvewhen of ITML-ALC is alwaysrate above of the other detectors, besides, the ROC dataset, curves the ROC curve of ITML-ALC is always above those ofofthe detectors, besides, the ROC curves curves of ROC curve of ITML-ALC is always above theother other detectors, besides, the ROC ofthe LMNN and NCA methods are away from thethose top left corner, which indicate they cannot detect all LMNN and NCA methods are away from the top left corner, which indicate they cannot detect of LMNN and NCAthe methods are away from left corner, which indicate they curve cannotofdetect all all target pixels within reasonable FARs. Forthe thetop HYDICE urban dataset, the ROC ITMLtarget pixels within the reasonable FARs. For the HYDICE urban dataset, the ROC curve of ITML-ALC target pixels the reasonable the limited HYDICE urban dataset, the ROC ITMLALC lies underwithin the curve of the NCA FARs. only inFor a very range at the beginning of thecurve FAR.of Besides liesthis, under the curve the NCA a very range at the beginning of of thethe FAR. Besides ALC lies under theofcurve of the NCAinonly in a limited very limited range atobtains the beginning FAR. Besides the ROC curve of NCA is only relatively worse while ITML-ALC best ROC curve when the this, theFAR ROC curve of NCA is relatively worse while ITML-ALC obtains best ROC curve when the FAR this,isthe ROC curve of NCA is relatively worse while ITML-ALC obtains best ROC curve when the is more than 3E–4. FAR is more more than 3E-4. than 3E–4. 1.0 0.8

0.6 0.4 0.4 0.2

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(a) (a)

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ACE OSP ACE LMNN OSP NCA LMNN ITML NCA ITML-ALC ITML 0.1 ITML-ALC1 0.1

0.6 0.4 0.4 0.2

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(b) (b)

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ACE OSP ACE LMNN OSP NCA LMNN ITML NCA ITML-ALC ITML 0.1 ITML-ALC1 0.1

0.6 0.4 0.4 0.2

1

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1E-3

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ACE OSP ACE LMNN OSP NCA LMNN ITML NCA ITML-ALC ITML 0.1 ITML-ALC1 0.1

1

(c) (c)

False alarm rate

Figure 7. ROC curves of the different algorithms. (a) AVIRIS LCVF dataset; (b) AVIRIS San Diego Figure 7. ROC curves the different algorithms. (a) AVIRIS SanSan Diego Figure 7. dataset; ROC curves of of the different algorithms. (a) AVIRIS AVIRISLCVF LCVFdataset; dataset;(b)(b) AVIRIS Diego airport (c) HYDICE urban dataset. airport dataset; HYDICE urbandataset. dataset. airport dataset; (c) (c) HYDICE urban According to ROC curves, we then show the AUC values and the FARs under 100% detection According to curves, then showthe the AUC values andthe the FARs under detection to ROC curves, wewe then and FARs under 100% detection ofAccording all algorithms forROC the three datasets inshow Figures 8AUC and 9,values respectively. The AUC analysis shows that of of all algorithms for the three datasets in Figures 8 and 9, respectively. The AUC analysis shows that the the areas of the comparison algorithms are all less than the ITML-ALC algorithm for all three datasets. all algorithms for the three datasets in Figures 8 and 9, respectively. The AUC analysis shows that the areas of the comparison algorithms are all less than the ITML-ALC algorithm for all three datasets. Theofstatistics of the FARalgorithms under 100%are detection the proposedalgorithm ITML-ALCfor algorithm results areas the comparison all lessshows than that the ITML-ALC all three datasets. of the FAR under 100% detection shows thatdetected. the algorithm results thestatistics lowest FAR when all the target pixels have been TheseITML-ALC two important evaluation TheinThe statistics of the FAR under 100% detection shows that theproposed proposed ITML-ALC algorithm results in the lowest FAR when allthe the target pixels have been detected. These two performance. important evaluation criteria further indicate that ITML-ALC algorithm yields the best detection in the lowest FAR when all the target pixels have been detected. These two important evaluation criteria further indicate that the ITML-ALC algorithm yields the best detection performance.

criteria further indicate that the ITML-ALC algorithm yields the best detection performance. In order to better compare the separability of target and background, the separation diagrams are shown in Figure 10. For the convenience of comparison, all detection results are normalized to [0–1], where the lines at the top and bottom of each column are the extreme values. The red boxes represent the distribution of the target pixels’ values, and the green ones represent the distribution of the background pixels’ values. From Figure 10a, for the AVIRIS LCVF dataset, the gaps between the target box and the background box for ACE, ITML, and ITML-ALC are very obvious, but ITML-ALC can suppress the background information to the smallest range. For the AVIRIS San Diego airport dataset,

Remote Sens. 2018, 6, x FOR PEER REVIEW 0.9993 0.9993

0.9966 0.9978

1.00

Areas of the ROC curves

Areas of the ROC curves

0.9877

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0.9548

Remote Sens. 2018, 10, 1415 0.94

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0.92

0.9932 0.9806

0.98 0.96 0.9395

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0.92

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0.90

as shown in Figure 10b, the ITML-ALC algorithm can separate target and background effectively, 0.90 0.85 0.90 ACE OSP LMNN ITML small ITML-ALC rangeACE ACE OSP LMNN information NCA ITML ITML-ALC OSP LMNN with NCA ITML and the background can be enclosed in aNCAvery compared theITML-ALC other (a) (b) (c) algorithms. For the HYDICE urban dataset, as shown in Figure 10c, it demonstrates that the ITML-ALC algorithm has 8.a more advanced performance when(a) compared withdataset; the other From these Figure The AUCs of the different algorithms. AVIRIS LCVF (b) algorithms. AVIRIS San Diego (c) HYDICE dataset. results, airport we candataset; conclude that theurban proposed ITML-ALC algorithm gives a superior performance relative to distinguishing target from background. Remote Sens. 2018, 2018, 6, 6, xx FOR FOR PEER REVIEW 12 of of 15 15 Remote Sens. PEER REVIEW 12 1.0

0.5479

1.0

0.5067

0.30

0.96 0.96 0.94 0.94

0.9548 0.9548 0.15

0.9312 0.9312

0.02718 0.01442

0.92 0.92 0.90 0.90

1.00

0.00 ACE

ACE ACE

2.500×10-5

0

False alarm rate

0.98 0.98

1.000.8

Areas Areasofofthe theROC ROCcurves curves

False alarm rate

Areas Areasofofthe theROC ROCcurves curves

1.00 1.00

0.9993 0.9993 0.9993 0.9993

0.9966 0.9978 0.45 0.9966 0.9978

OSP LMNN NCA ITML ITML-ALC

OSP OSP

(a)

LMNN NCA LMNN NCA

0.980.6 0.98 0.960.4 0.96

0.9932 0.9877 0.9932 0.9877

0.9842 0.9806 0.9842 0.9806

0.9395 0.1957 0.9395 0.04031

0.920.0 0.92

ACE

ACE ACE

0.2341

0.01381

OSP LMNN NCA

(b)

OSP LMNN OSP LMNN

ITML ITML-ALC

0.9271

1.00 0.8 1.00

0.9932 0.9932

0.3707

0.940.2 0.94

0.90 0.90

ITML ITML-ALC ITML ITML-ALC

0.8105

Areas ROC curves Areasofofthe the ROC curves False alarm rate

0.60

0.6

0.5756

0.9483 0.9483

0.95 0.95 0.4 0.2

0.2061 0.2346 0.1254

0.8937 0.8937

0.90 0.90

0.01351

0.0 ACE

0.85 0.85

NCA ITML ITML-ALC NCA ITML ITML-ALC

0.9968 0.9872 0.9968 0.9872

0.9821 0.9821 0.9752 0.9752

ACE ACE

OSP

OSP OSP

LMNN

NCA

(c) LMNN LMNN

ITML ITML-ALC

NCA NCA

ITML ITML-ALC ITML ITML-ALC

(a) (b)different algorithms. (a) AVIRIS LCVF (c)dataset; (b) Figure 9.(a) The FARs under 100% detection of the (b) (c) AVIRIS San Diego airport dataset; (c) HYDICE urban dataset. Figure 8. The AUCs of ofthe thedifferent different algorithms. (a) AVIRIS LCVF dataset; (b) AVIRIS AVIRIS San Diego Diego Figure of the different algorithms. AVIRIS LCVF dataset; (b) San Figure8. 8. The The AUCs algorithms. (a)(a) AVIRIS LCVF dataset; (b) AVIRIS San Diego airport airport dataset; (c) HYDICE urban dataset. dataset; (c) HYDICE urban dataset. airport dataset; (c) HYDICE urban dataset. In order to better compare the separability of target and background, the separation diagrams are shown in Figure 10. For the convenience of comparison, all detection results are normalized to 1.0 1.0 0.5479 1.0 0.9271 1.0 0.9271 0.5067 [0–1], where0.5479 the0.5067 lines at the top and bottom of0.8105 each column are the extreme values. The red boxes 0.8105 0.8 0.8 0.8 0.8 0.45 represent the distribution of the target pixels’ values, and the green ones represent the distribution 0.45 0.5756 between 0.6 0.6 of the background pixels’ values. From Figure 10a, for the AVIRIS LCVF dataset, the gaps 0.5756 0.6 0.6 0.30 0.30 the target box and the background 0.4 box0.3707 for ACE, ITML, and ITML-ALC are very obvious, but ITML0.4 0.3707 0.4 0.4 ALC can suppress the background information to0.1957 the 0.2341 smallest range. For the 0.2346 AVIRIS San Diego 0.2061 0.2346 0.15 0.2341 0.15 0.2 0.1957 0.2 0.2061 0.1254 0.2 0.2 airport dataset, as shown in Figure 10b, the ITML-ALC algorithm can separate target and background 0.1254 0.04031 0.02718 0.01442 0.01381 0.01351 0.04031 0.02718 0.01442 2.500×10 0 0.01381 0.0 small range compared 0.01351 0.0 effectively, and the background information can be enclosed in a very with 2.500×10 0 0.00 0.0 0.0 0.00 ACE OSP LMNN NCA ITML ITML-ALC ACE OSP LMNN NCA ITML ITML-ALC ACE OSP LMNN NCA ITML ITML-ALC ACE OSP LMNN NCA ITML ITML-ALC ACE OSPdataset, LMNN NCA ITML ITML-ALC ACE OSP LMNN NCA ITML the other algorithms. ForITML-ALC the HYDICE urban as shown in Figure 10c, it demonstrates that the (a) (b) (c) (a) (b) ITML-ALC algorithm has a more advanced performance when compared with the (c) other algorithms. From these results, we can conclude that the proposed ITML-ALC algorithm gives a superior Figure 9. The FARs under 100% detection of the different algorithms. (a) AVIRIS LCVF dataset; (b) Figure 9. The FARs under 100% detection of the different algorithms. (a) AVIRIS LCVF dataset; (b) Figure 9. The FARs under 100% detection of the different algorithms. (a) AVIRIS LCVF dataset; (b) performance relative to distinguishing target from background. AVIRIS San Diego airport dataset; (c) HYDICE urban dataset. AVIRIS San Diego airport dataset; (c) HYDICE urban dataset. AVIRIS San Diego airport dataset; (c) HYDICE urban dataset. False Falsealarm alarmrate rate

False Falsealarm alarmrate rate

False Falsealarm alarmrate rate

0.60 0.60

-5 -5

1.2

0.2 0.2 0.0 0.0

ACE ACE

OSP OSP

LMNN LMNN

(a) (a)

NCA NCA

ITML ITML-ALC ITML ITML-ALC

Detection Test Statistic Range

0.4 0.4

0.6 0.6

Detection DetectionTest TestStatistic StatisticRange Range

0.6 0.6

Detection Test Statistic Range

Detection DetectionTest TestStatistic StatisticRange Range

Detection DetectionTest TestStatistic StatisticRange Range

Detection Test Statistic Range

Target 1.2 1.2 Targetbackground, Target In order order to to better better compare compare the separability of target and the separation diagrams In Background the separability of target and background, the separation diagrams Background Background 1.0 1.0 results are normalized to are shown in Figure 10. For the convenience of comparison, all detection are shown in Figure 10. For the convenience of comparison, all detection results are normalized to 0.8 0.8 0.8 [0–1], where the the lines lines at at the the top top and and bottom bottom of each each column column are are the the extreme extreme values. The The red red boxes boxes [0–1], where of values. 0.6 0.6 0.6 represent the the distribution distribution of of the the target target pixels’ pixels’ values, values, and and the the green green ones ones represent represent the the distribution distribution represent 0.4 0.4 dataset, the gaps between 0.4 Figure 10a, for the AVIRIS LCVF of the background pixels’ values. From of the background pixels’ values. From Figure 10a, for the AVIRIS LCVF dataset, the gaps between 0.2 0.2 for ACE, ITML, and ITML-ALC are the 0.2 target box box and and the the background background box box very obvious, obvious, but but ITMLITMLthe target for ACE, ITML, and ITML-ALC are very 0.0 0.0 0.0 ALC can can information to theNCAsmallest smallest range. For For the the AVIRIS AVIRIS SanITML Diego ACE suppress OSP LMNN the NCA background ITML ITML-ALC ACE OSP LMNN NCA ITML-ALC ACE OSP to LMNN ITML ITML-ALC ALC suppress the background information the range. San Diego airport dataset, as shown in Figure 10b, the ITML-ALC algorithm can separate target and background (a)shown in Figure 10b, the ITML-ALC (b) algorithm can separate target and (c) background airport dataset, as effectively, and and the the background background information information can can be be enclosed enclosed in in aa very very small small range range compared compared with with effectively, Figure Target-backgroundseparation separationmaps mapsof of the the different different algorithms. dataset; Figure 10.10. Target-background algorithms.(a) (a)AVIRIS AVIRISLCVF LCVF dataset; the other algorithms. For the HYDICE urban dataset, as shown in Figure 10c, it demonstrates that the the other algorithms. For the HYDICE urban dataset, as shown in Figure 10c, it demonstrates that the AVIRIS Diego airport dataset; HYDICE urban dataset. (b) (b) AVIRIS SanSan Diego airport dataset; (c)(c)HYDICE urban dataset. ITML-ALC algorithm has a more advanced performance when compared with the other algorithms. ITML-ALC algorithm has a more advanced performance when compared with the other algorithms. From these results, results, we we can can conclude conclude that that the the proposed proposed ITML-ALC ITML-ALC algorithm algorithm gives gives aa superior superior 4. Discussion 4. Discussion From these performance relative to distinguishing target from background. performance relative to distinguishing target algorithm from background. In this paper, the proposed is to transform transformthe theoriginal originalfeature feature In this paper, the proposedITML-ALC ITML-ALC algorithm is employed employed to space into the metric feature space by introducing adaptively local constraints to the ITML model. space into the metric feature space by introducing adaptively local constraints to the ITML model. 1.2 Target 1.2 1.2 1.2 Target Target Target 1.2 1.2 Target Target the Background According to the experimental results presented in the previous section, we can observe that Background Background Background 1.0 Background Background 1.0 1.0 1.0 1.0 1.0 ITML-ALC algorithm has the better ability of suppressing background pixels and extracting the 0.8 0.8 0.8 0.8 0.8 0.8 target ones. 1.0

0.4 0.4 0.2 0.2 0.0 0.0

ACE ACE

OSP OSP

LMNN NCA LMNN NCA

(b) (b)

ITML ITML-ALC ITML ITML-ALC

0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0

ACE ACE

OSP OSP

LMNN NCA LMNN NCA

ITML ITML-ALC ITML ITML-ALC

(c) (c)

Figure 10. 10. Target-background Target-background separation separation maps maps of of the the different different algorithms. algorithms. (a) (a) AVIRIS AVIRIS LCVF LCVF dataset; dataset; Figure (b) AVIRIS San Diego airport dataset; (c) HYDICE urban dataset. (b) AVIRIS San Diego airport dataset; (c) HYDICE urban dataset.

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According to the experimental results presented in the previous section, we can observe that the 13 ofthe 16 has the better ability of suppressing background pixels and extracting target ones.

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(1) (1) As metric learning-based methods, LMNN and NCA methods do not not perform perform well well (shown (shown in in Figure 7), and the cause may be that LMNN and NCA methods both have problems with handling 7), and the cause may be that LMNN and NCA methods both have problems with high-dimensional data and are sensitive the selection initial points, leading to inability handling high-dimensional data and aretosensitive to theofselection of initial points, leading to to achieve the optimalthe value if thevalue parameters are not selected appropriately. To date, only few inability to achieve optimal if the parameters are not selected appropriately. Toadate, researchers use distanceuse metric learning forlearning target detection. example, reference [15] adopts only a few researchers distance metric for targetFor detection. For example, reference LMNN and NCA as the comparison to prove the effectiveness of proposed supervised [14] adopts LMNN and NCA as algorithms the comparison algorithms to prove the effectiveness of metric learning (SML) algorithm for (SML) the AVIRIS LCVF Though the implanted target proposed supervised metric learning algorithm fordataset. the AVIRIS LCVF dataset. Though the locations oftarget reference [15] are little different compared to this paper, LMNNtoand have poor implanted locations ofareference [14] are a little different compared thisNCA paper, LMNN 4 , the detection similarity in both papers. When the valuepapers. of FARWhen is equal tovalue 10 × 10 and NCA performance have poor similarity performance in both the of−FAR is equal to probabilities LMNN in reference [15] and thisin paper are both 80%.paper Similarly, in theabout two 10 × 10−4, theof detection probabilities of LMNN reference [14]about and this are both articles, the detection probability reaches 100% when the FAR is about respectively. 80%. Similarly, in the two articles, of theNCA detection probability of NCA reaches 100%1, when the FAR Moreover, although SML algorithmalthough can recognize target pixels the target background is about 1, respectively. Moreover, SML algorithm can easily, recognize pixels pixels easily, cannot be suppressed to an even lower value, compared with the ITML-ALC algorithm. This the background pixels cannot be suppressed to an even lower value, compared with the ITMLcouldalgorithm. be becauseThis thatcould SML be only use a similarity constraint to simultaneously link ALC because that SML propagation only use a similarity propagation constraint to target pixels and background ones, ITML-ALC adopts constraints to simultaneously link target pixels andwhile background ones, whileadaptively ITML-ALClocal adopts adaptively separate similar and dissimilar point-pairs. local constraints to separate similar and dissimilar point-pairs. –2 (2) For the HYDICE urban dataset, the FAR is reduced to a 10−to2 level the detection (2) the HYDICE urban dataset, the FAR is reduced a 10when level when theprobability detection − 2 –2 of ACE is at of 80% in this FAR is reduced a 10 is level when detection probability ACE is paper, at 80%while in this paper, whiletoFAR reduced tothe a 10 levelprobability when the of ACE is probability at 70% in the they use [36]. different priorthey information, the ACE detection of reference ACE is at[45]. 70%Though in the reference Though use different prior results of the different articles achieve a slightly different performance. Niu et al. [45] propose information, the ACE results of the different articles achieve a slightly different performance. an adaptive weighted method (AWLM) usingmethod a self-completed dictionary Niu et al. [36] proposelearning an adaptive weighted learning (AWLM) background using a self-completed (SCBD) to extract the accurate target spectrum for hyperspectral target detection. When thetarget FAR background dictionary (SCBD) to extract the accurate target spectrum for hyperspectral − 3 is reduced When to 10 ×the 10 FAR , the probability thedetection proposedprobability algorithm in [45] detection. is detection reduced to 10 × 10−3,ofthe of reference the proposed (named AWLM_SCBD+ACE) is at nearly 90%, while the detection probability of ITML-ALC algorithm in reference [36] (named AWLM_SCBD+ACE) is at nearly 90%, while the detection in this paperofisITML-ALC also at 90%. onlyHowever, has trade-off parameter to has be adjusted, probability inHowever, this paperITML-ALC is also at 90%. ITML-ALC only trade-off but AWLM_SCBD+ACE additional parameters, i.e., κadditional and τ of adaptive weights,  be parameter to be adjusted, but AWLM_SCBD+ACE parameters, i.e., which of  andcan used to suppress the influence of irrelevant pixels. adaptive weights, which can be used to suppress the influence of irrelevant pixels. (3) addition, we we compare compareITML-ALC ITML-ALCwith withMMML MMMLand andRFML, RFML,which which are previously proposed (3) In addition, are previously proposed in in [27] [28], respectively. Figure 11 shows corresponding ROC curves for AVIRIS the AVIRIS [21] andand [22], respectively. Figure 11 shows the the corresponding ROC curves for the San San Diego airport dataset. be found that ITML-ALC has similar performance when the Diego airport dataset. It canItbecan found that ITML-ALC has similar performance when the FAR is − 3 FARthan is less × 10 and then ITML-ALC outperforms otherMaybe methods. Maybe because less 10 than × 10−310 , and then, ITML-ALC outperforms other methods. because ITML-ALC ITML-ALC can distances shrink thebetween distancessamples betweenofsamples similar andthe expand the distances can shrink the similar of pairs and pairs expand distances between between of samples of dissimilar pairs compared with and MMML and RFML. samples dissimilar pairs compared with MMML RFML. 1.0

Probability of detection

0.8 0.6 0.4 RFML MMML ITML-ALC

0.2 0.0 1E-4

1E-3

0.01

0.1

1

False alarm rate

Figure 11. ROC curves of the different different algorithms algorithms for for AVIRIS AVIRISSan SanDiego Diegoairport airportdataset. dataset.

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5. Conclusions In this paper, the adaptive information-theoretic metric learning with local constraints (ITML-ALC) algorithm has been proposed. Based on limited numbers of prior samples, the ITML-ALC algorithm constructs an efficient metric learning-based method without certain assumptions. Adaptively local constraints are then introduced to indicate the discriminative information for separating similar and dissimilar point-pairs with fewer parameters to be adjusted. By combining the ITML framework and adaptively local constraints, decision can be made by considering both the threshold and the changes between the distances before and after metric learning. Extensive experiments, which are carried on three hyperspectral datasets for target detection, confirm the superior performance of the proposed ITML-ALC algorithm, which can obviously separate target samples from background ones. In general, the ITML-ALC algorithm presents a better detection performance and separability than the other classical target detectors. Author Contributions: Y.D. and B.D. conceived, designed and performed the experiments; L.Z. and X.H. provided advice for the preparation and revision of the paper. Funding: This work was supported in part by the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) CUG170687; in part by the National Natural Science Foundation of China under Grant 61801444 and Grant 61471274. Acknowledgments: The authors would like to thank the handling editor and anonymous reviewers for their careful reading and helpful remarks. Conflicts of Interest: The authors declare no conflicts of interest.

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