A target detection method for hyperspectral image based on mixture noise model

Author’s Accepted Manuscript A target detection method for hyperspectral image based on mixture noise model Xiangtao Zheng, Yuan Yuan, Xiaoqiang Lu

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S0925-2312(16)30836-0 http://dx.doi.org/10.1016/j.neucom.2016.08.015 NEUCOM17426

To appear in: Neurocomputing Received date: 14 March 2016 Revised date: 28 May 2016 Accepted date: 2 August 2016 Cite this article as: Xiangtao Zheng, Yuan Yuan and Xiaoqiang Lu, A target detection method for hyperspectral image based on mixture noise model, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.08.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A Target Detection Method for Hyperspectral Image Based on Mixture Noise Model Xiangtao Zhenga,b , Yuan Yuana , Xiaoqiang Lua,∗ a Center for OPTical IMagery Analysis and Learning (OPTIMAL), State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, Shaanxi, P. R. China. b University of the Chinese Academy of Sciences, 19A Yuquanlu, Beijing, 100049, P. R. China.

Abstract Subpixel hyperspectral detection is a kind of method which tries to locate targets in a hyperspectral image when the spectrum of the targets is given. Due to its subpixel nature, targets are often smaller than one pixel, which increases the difficulty of detection. Many algorithms have been proposed to tackle this problem, most of which model the noise in all spatial points of hyperspectral image by multivariate normal distribution. However, this model alone may not be an appropriate description of the noise distribution in hyperspectral image. After carefully studying the distribution of hyperspectral image, it is concluded that the gradient of noise also obeys normal distribution. In this paper two detectors are proposed: mixture gradient structured detector (MGSD) and mixture gradient unstructured detector (MGUD). These detectors are based on a new model which takes advantage of the distribution of the gradient of the noise. This makes the detectors more accordant with the practical situation. To evaluate the performance of the proposed detectors, three different datasets, including one synthesized dataset and two real-world ∗

Corresponding author Email address: [email protected] (Xiaoqiang Lu)

Preprint submitted to Elsevier

August 6, 2016

datasets, are used in the experiments. Results show that the proposed detectors have better performance than current subpixel detectors. Keywords: hyperspectral data, subpixel, target detection 1. Introduction With the development of remote sensing and pattern recognition [1, 2, 3, 4], target detection has aroused more and more concerns [5, 6, 7] in hyperspectral image. Hyperspectral target detection [8, 9, 10] is the process of locating certain ground material in a hyperspectral image, when spectral signature of the material is known in advance. Hyperspectral target detection has been widely used for both military and civilian purposes [11, 12]. For instance, it can be used to monitor water quality, forest fire danger, land-utilized condition and enemy military dispositions, which makes it an important research area. In hyperspectral imaging system, the spectrum can be divided into many more narrow and contiguous bands with a wide range of wavelengths. Owning to the high wavelength resolution, a hyperspectral image can be regarded as a set of images. Each image covers a narrow wavelength range. This makes hyperspectral image a three-dimensional data cube with two spatial dimensions and a spectral dimension. Different bands of a pixel can form a continuous spectrum, and each ground material often has its unique spectral signatures, which makes it possible to identify targets. On the other hand, the spatial resolution of hyperspectral images is limited. Sometimes a pixel may consist of more than one material, and target is mixed with background, which is called subpixel target. Subpixel target is difficult to detect, because the spectral spectrum of the mixed pixel is often different from the target spectrum. 2

A number of algorithms have been proposed to solve the subpixel detection problem. One kind of algorithm tried to find a optimal projection vector so that background signatures are suppressed while target signatures are maintained after the projection. In this case, targets and background can be separated. Representatives of this kind of algorithm are orthogonal subspace projection (OSP) [13], kernel orthogonal subspace projection (KOSP) [14], constrained energy minimization (CEM) [15], target-constrained interference-minimized filter (TCIMF) [16], kernel-based TCIMF (KTCIMF) [17], etc. In these algorithms, the noise is assumed to be a zero-mean multivariate normal distribution. Another kind of algorithm is based on hypothesis testing [18, 19, 20, 21]. In these algorithms, firstly a couple of hypotheses including null hypothesis and alternative hypothesis, are formulated. Then a detector is designed to judge whether the pixel in the image belongs to the null hypothesis or the alternative hypothesis. Depending on the model used to describe background signature, hypothesis testing based algorithms can be divided into two classes: structured detector and unstructured detector. One example of structured detector is adaptive matched subspace detector (AMSD) [22]. The AMSD used linear mixing model to represent background. In this case, each pixel can be represented as the product of background endmembers and their corresponding abundance. Although AMSD can correctly find targets from the image in most cases, it has some drawbacks. For AMSD, endmembers are the eigenvectors of the data correlation matrix, and the sum-to-one and nonnegative constraint are not satisfied, which will result in the loss of physical meaning. Adaptive cosine / coherent estimate (ACE) [20] is another example of statistical hypothesis testing based algorithms. Compared with AMSD, ACE assumed that background signature consists of only noise which

3

obeys multivariate normal distribution. This makes ACE a very fast algorithm since it does not extract endmembers from background. However, recent study has shown that multivariate normal distribution cannot correctly model the background in hyperspectral imaging [23]. To overcome the shortage of hypothesis testing based algorithms, two hybrid detectors for subpixel targets, including hybrid structured detector (HSD) and hybrid unstructured detector (HUD), have been proposed [24] in recent years. These algorithms use both physics and statistics to model the background, and targets are then detected using statistical hypothesis. By doing so, physical meaning is brought into the original background model, and experiments also show that hybrid detectors perform better than typical hypothesis testing based algorithms. In subpixel detection, one of the key factors that alter the target observations is the estimation of the noise statistics [25, 22, 26]. All the aforementioned algorithms model the noise as a set of independent and identically distributed (i.i.d.) noise random variables for all spatial points, each of which follows a multivariate normal distribution or multivariate Gaussian distribution. However, this model is weak [27] because it does not capture an important property of image noise, which is that image noise exhibits spatial randomness. Recent study [27] has shown an interesting observation that normal distribution alone may not be an appropriate description of the noise in images, the gradient of noise should also obey normal distribution. This observation can be easily explained that the i.i.d. property makes the gradient of noise random variables also follow multivariate Gaussian distributions with different standard deviations 1 . 1

If n1 and n2 are normally distributed and independent, then their difference n1 − n2 is also

distributed normally.

4

0.014 Noise Gradient of Noise

Probability Density

0.012

0.01

0.008

0.006

0.004

0.002

0 −300

−200

−100

0

100

200

300

400

Noise

(a) Hyperspectral image

(b) Distribution of noise

Figure 1: (a) Original hyperspectral image. (b) Distribution of noise and gradient of noise.

To further validate this observation, a simulation experiment is performed on hyperspectral image. In Figure 1(a), 10 dB Gaussian noise is added to the original hyperspectral image, and distributions of noise and gradient of noise are shown in Figure 1(b). From Figure 1 it can be concluded that the gradient of the noise within a pixel among different bands also follows Gaussian distribution. The distribution of noise in the hyperspectral image needs to be exploited in order to better detect targets from backgrounds. However, to our knowledge in current target-detection algorithms, the distribution of the gradient of noise has never been studied. In this paper, a stronger model of noise is proposed to regard the distribution of noise and its gradient. Based on the new model, we then follow the work of hybrid detectors, and propose two new hybrid detectors, mixture gradient structured detector (MGSD) and mixture gradient unstructured detector (MGUD), which make full use of the gradient distribution of the noise. The rest of the paper is organized as follows. Section 2 introduces fully constrained least squares (FCLS), HSD and HUD which are related to the proposed algorithm. Section 3 describes the two proposed detectors MGSD and MGUD. 5

Performance comparisons of different algorithms are given in Section 4, and finally conclusion is given in Section 5. 2. Related Work This section introduces linear mixture model (LMM), fully constrained least squares (FCLS), and reviews the HSD and HUD detection algorithms, which are related to the proposed algorithm. Both LMM and FCLS are important parts of HSD and HUD. HSD is a structured detector which takes advantage of both FCLS and AMSD, while HUD is an unstructured detector which is based on FCLS and ACE. 2.1. Linear Mixture Model (LMM) In order to tackle the subpixel detection problem, a model is needed to describe the inner structure of pixels. The most commonly used model is linear mixing model (LMM) [28, 29]. This model assumes that each pixel is a linear combination of different elements (called endmember), and each element has its own weights (called abundance). The model can be represented as

x = Eα + n,

(1)

where x is a l × 1 vector that represents the spectrum of the current pixel. l is the number of bands. E is a l × p matrix, where the ith column represents the spectrum of the ith endmember, and the jth row represents the spectrum of the jth band. α = [α1 , α2 , . . . , αp ]T is a p × 1 vector representing the abundance of different endmembers. To enhance the physical meaning of LMM, two constraints are added to the model: 6

p P

αi = 1

i=1

(2)

αi > 0. The former is called abundance sum-to-one constraint (ASC) and the latter is called abundance nonnegative constraint (ANC). By using those two constraints, the true abundance can be extracted from mixed pixels. The estimated abundance is helpful for analyzing the distribution of different materials in the scene. 2.2. Fully Constrained Least Squares (FCLS) FCLS is an algorithm which is used to estimate the abundance of endmembers in the image. The algorithm begins with ANC, and tries to minimize the least squares error while guaranteeing that the abundance is nonnegative. This can be expressed as:

min (x − Eα)T (x − Eα), αi ≥ 0 ∀i. α

(3)

Eq. (3) is a constrained optimization problem, which can be solved by using Lagrange multipliers: α ˆ = (E T E)−1 E T x − (E T E)−1 λ,

(4)

λ = E T (x − E α ˆ ).

(5)

where

To solve Eq. (4) and Eq. (5), an active set based algorithm is adopted to ensure that the solution meets the Karush-Kuhn-Tucker conditions. The iterating step begins with an unconstrained least squares estimation of α . Then two index sets, 7

including passive set P and active set R, are built. Indices (Lagrange multipliers) corresponding to the positive abundance values are put in P , while the remaining indices corresponding to the negative and zero abundance values are put in R. Then Eq. (4) and Eq. (5) is iterated until all elements in set P are zero and all elements in set R are zero or negative. In this case, the Karush-Kuhn-Tucker conditions can be satisfied, and the optimal result is obtained. The aforementioned step only copes with the ANC. To ensure that ASC is satisfied, a simple solution has been proposed to the above equation. Specifically, endmember matrix E and pixel spectrum x are concatenated with 1, which can be written as  E˜ =  and

 x˜ = 

δE T

1

δx 1

 ,

(6)

 ,

(7)

where δ is a weighting factor which controls how strictly the solution will follow the sum to one constraint. A smaller δ will result in a more strict solution, but will decelerate the convergence at the same time. Then by iterating (4) and (5), the final solution will satisfy both the ANC and ASC. 2.3. Hybrid Structured Detector Hybrid structured detector (HSD) is a hypothesis testing based algorithm. It is based on AMSD and FCLS. The noise is assumed to obey the zero-mean multivariate normal distribution, namely n ∼ N (0, σ 2 Γ) where σ 2 Γ is the covariance matrix of noise. 8

In AMSD, sensor noise is further assumed as a Gaussian random vector with uncorrelated components of equal variance: n ∼ N (0, σ 2 Γ), Γ = I, and I is the identity matrix. The procedure starts with a pair of hypotheses written as follows:

H0 : x ∼ N (Bαb,0 , σ02 I),

(8)

H1 : x ∼ N (Sαs + Bαb,1 , σ12 I),

(9)

where H0 is a null hypothesis when no target exists the pixel, H1 is the alternative hypothesis when a target exists in the pixel, B is a L × U matrix with each column representing one background signature, S is a L×V matrix with each column representing one target signature, αb,0 and αb,1 represent the abundance of different background endmembers in H0 and H1 , respectively, αs represents the abundance of targets, and n is the noise. Thus the likelihood equations can be written as L

L(x|H0 ) = (2πσ02 )− 2 × 

 1 T exp − 2 (x − Bαb,0 ) (x − Bαb,0 ) , 2σ0

(10)

L

L(x|H1 ) = (2πσ02 )− 2 × 

 1 T exp − 2 (x − Eαe ) (x − Eαe ) , 2σ0

(11)

where E is the concatenation of B and S: E = [S, B]. αe is the concatenation   αs . In (8) and (9), the abundance αb,0 , αs , αb,1 and of αs and αb,1 : αe =  αb,1 covariance σ02 , σ12 can be estimated by setting the derivatives of the logarithms of (10) and (11) with respect to the corresponding parameter to zeros. After finishing the estimation step, the estimation of the aforementioned parameters are then 9

substituted back into the likelihood function. The result is  L(x|H0 ) =  L(x|H0 ) =


2π T x I − B(B T B)−1 B T x L
2π T x I − E(E T E)−1 E T x L

− L2

− L2

L exp − 2









L exp − 2

,

(12)

.

(13)

Specifically, a detector is designed by using the Neyman-Pearson Lemma [30, 31, 32], which can ensure a most powerful test for a given false alarm rate. This is done by taking the ratio of the likelihood functions, namely
xT I − B(B T B)−1 B T x xT PB⊥ x . = xT (I − E(E T E)−1 E T ) x xT PZ⊥ x

(14)

By subtracting one from (14), the AMSD detector is obtained as
xT I − B(B T B)−1 B T x xT (PB⊥ − PZ⊥ )x = . xT (I − E(E T E)−1 E T ) x xT PZ⊥ x

(15)

The hybrid structured detector (HSD) is a derivation of AMSD. It uses a similar background structure, while the abundance α is estimated by using FCLS. The hypotheses of HSD are

H0 : x ∼ N (Bαb,0 , σ02 Γ),

(16)

H1 : x ∼ N (Sαs + Bαb,1 , σ12 Γ),

(17)

which suppose that the covariance of noise in these two hypotheses are σ02 Γ and σ12 Γ, respectively. In (16) and (17), Γ is estimated using a i.i.d. training set which obeys Y = {yi |yi ∼ N (0, Γ), i = 1, 2, . . . , N }, and then the estimation of Γ is ˆ = 1 PN yi y T . σ 2 and σ 2 are estimated using maximum likelihood obtained as Γ i 0 1 i=1 N 10

estimation (MLE). αb,o and αe are estimated using a variance of FCLS, which can be written as min(x − Eα)T Γ−1 (x − Eα), αi ≥ 0∀i. α

(18)

Finally the HSD detector is obtained based on AMSD:

DHSD (x) =

(x − B α ˆ b,0 )T Γ−1 (x − B α ˆ b,0 ) . T −1 (x − E α ˆ e ) Γ (x − E α ˆe)

(19)

Then a single threshold η is applied to (19) to distinguish targets from background. If DHSD (x) > η then x is regarded as target, and vice versa. 2.4. Hybrid Unstructured Detector Hybrid unstructured detector (HUD) is another hypothesis testing based algorithm. HUD is derived from ACE. Unlike AMSD, ACE assumes unstructured background. It assumes that background signature obeys zero-mean multivariate normal distribution, and the hypotheses are

H0 : x ∼ N (0, σ02 Γ),

(20)

H1 : x ∼ N (Sαs , σ12 Γ).

(21)

In the above hypotheses, all the structured information from background are removed, and ASC and ANC constraints cannot be guaranteed. Besides the test data, it is also assumed that an independent data set Y is available which obeys

Y = {yi |yi ∼ N (0, Γ), i = 1, . . . , N } . If N is large enough, then the covariance matrix Γ can be estimated as 11

(22)

N X ˆ= 1 Γ yi yiT . N i=1

(23)

Thus the joint likelihood equations under different hypotheses are L

L

L

L(x|H0 ) = (2π)− 2 (N +1) |Γ|− 2 (N +1) (σ02 )− 2 ×   1 T −1 exp − 2 x Γ x , 2σ0 L

L

(24)

L

L(x|H1 ) = (2π)− 2 (N +1) |Γ|− 2 (N +1) (σ02 )− 2 ×   1 T −1 exp − 2 (x − Sαs ) Γ (x − Sαs ) 2σ0

(25)

The unknown parameters σ02 , σ12 and αs are first estimated using MLE, and then substituted back into the likelihood function. Finally the ratio of the likelihood functions are taken to get Generalized Likelihood Ratio Test (GLRT). The final result of ACE detector is as follow

DACE (x) =

ˆ −1 Sˆ ˆ −1 S(S T Γ−1 S)−1 S T Γ−1 x xT Γ a xT Γ = . ˆ −1 x ˆ −1 x xT Γ xT Γ

(26)

HUD is similar to ACE, except that the abundance estimation a ˆ is replaced by its FCLS counterpart. The HUD can be written as ˆ −1 S aF ˆCLS xT Γ DHU D (x) = ˆ −1 x xT Γ in which α ˆ F CLS is calculated using (18). It is worth noticing that the abundance estimation step still requires the unmixing of endmembers, which indirectly introduces endmember information to the algorithm.

12

3. Methodology In Section 2 we introduced FCLS, which directly uses abundance to detect target, and HSD and HUD, two hybrid detectors which enjoy the benefit of both FCLS and hypothesis testing. All the algorithms introduced in Section 2 are based on the simple model that each noise n is an i.i.d. random variable following zeromean multivariate normal distribution, namely n ∼ N (0, σ12 Γ). Although this model is valid and accordant with the practical situation to some degree, it does not capture all the spatial randomness [27] that could be expected in noise. This simple noise model only describes the noise distribution with a simple multivariate Gaussian distribution, which ignores the distribution of the gradient of noise. As shown in [27, 33], in nature image gradients of noise can be approximated by normal distribution, which can be written as ∇n ∼ N (0, σ22 Γ). Similar to the aforementioned work, when performing hyperspectral image simulation experiments, it is found that not only the noise among different bands within a pixel obeys multivariate normal distribution, the gradients of noise should also satisfy normal distribution. If this prior can be exploited completely, the performance of detectors will be better. Under such condition, a new model regarding the distribution of the gradient of noise is proposed, and new detectors based on this model are developed. To be more specific, two new detectors: mixture gradient structured detector (MGSD) and mixture gradient unstructured detector (MGUD) are proposed, which are summarized in Figure 2. In MGSD, we follow the work of HSD and the hypotheses are

2 2 H0 : x ∼ N (Bαb , σ0,0 Γ) ∩ ∇x ∼ N (∇(Bαb ), σ1,0 Γ),

13

(27)

2 2 H 0 : x ~ N ( Bα b , σ 0,0 Γ) ∩ ∇x ~ N (∇( Bα b ), σ 1,0 Γ)

Background Hypothesis Parameters Estimation

Two Structured Hypotheses

DS =

2 2 H1 : x ~ N ( Eα E , σ 0,1 Γ) ∩ ∇x ~ N (∇( Eα E ), σ 1,1 Γ)

Hyperspectral Image

2 PS ,0 + ω PS ,1 2QS ,0 + ωQS ,1

MGSD

Target Hypothesis

(a) MGSD 2 2 H 0 : x ~ N (0, σ 0,0 Γ) ∩ ∇x ~ N (0, σ 1,0 Γ)

Background Hypothesis Parameters Estimation

Two Unstructured Hypotheses 2 2 H1 : x ~ N ( Sα s , σ 0,1 Γ) ∩ ∇x ~ N (∇( Sα s ), σ 1,1 Γ)

Hyperspectral Image

DU =

2 PU ,0 + ω PU ,1 2QU ,0 + ωQU ,1

MGUD

Target Hypothesis

(b) MGUD Figure 2: The algorithm chart of two proposed detectors. (a) MGSD. (b) MGUD.

2 2 H1 : x ∼ N (EαE , σ0,1 Γ) ∩ ∇x ∼ N (∇(EαE ), σ1,1 Γ),

(28)

where the symbol ∩ means that x must meet all the constraints simultaneously. The likelihood functions under above two hypotheses are then built as L

1

L

2 −2 L(x|H0 ) = (2π)− 2 Γ− 2 (σ0,0 ) ×

exp{−

1 (x − Bαb )T Γ−1 (x − Bαb )}× 2 2σ0,0 L

1

L

2 −2 ) × (2π)− 2 Γ− 2 (σ1,0

exp{−ω

1 (∇x − ∇(Bαb ))T Γ−1 (∇x − ∇(Bαb ))}, 2 2σ1,0

and

14

(29)

L

1

L

2 −2 ) × L(x|H1 ) = (2π)− 2 Γ− 2 (σ0,1

exp{−

1 (x − Eα)T Γ−1 (x − Eα)}× 2 2σ0,1 −L 2

(2π) exp{−ω

Γ

− 12

2 −L (σ1,1 ) 2×

(30)

1 (∇x − ∇(Eα))T Γ−1 (∇x − ∇(Eα))}, 2 2σ1,1

where ω is the weighting factor that controls the balance between different con2 2 2 2 straints, and the variance are set to be σ1,0 = 2σ0,0 , ,σ1,1 = 2σ0,1 , empirically.

Next, the unknown parameters are estimated. Covariance is estimated by using MLE, and abundance is estimated by using a variant of FCLS, namely α ˆ = arg min(x − Eα)T Γ−1 (x − Eα)+ α

(31) T

−1

ε(∇x − ∇(Eα)) Γ (∇x − ∇(Eα)), where ε is the weighting factor. (31) can be solved using active-set algorithm. Estimated parameters are then substituted back into (29) and (30). Finally, GLRT is obtained by taking the ratio of the likelihood functions. The final detector is (32).

DS =

2PS,0 + ωPS,1 , 2QS,0 + ωQS,1

(32)

where PS,0 = (x−Bα)T Γ−1 (x−Bα), PS,1 = (∇x−∇(Bα))T Γ−1 (∇x−∇(Bα)), QS,0 = (x − Eαe )T Γ−1 (x − Eαe ), and QS,1 = (∇x − ∇(Eαe ))T Γ−1 (∇x − ∇(Eαe )). The unstructured version of the proposed algorithm, namely MGUD, is similar to HUD. The hypotheses are

15

2 2 H0 : x ∼ N (0, σ0,0 Γ) ∩ ∇x ∼ N (0, σ1,0 Γ),

(33)

2 2 H1 : x ∼ N (Sαs , σ0,1 Γ) ∩ ∇x ∼ N (∇(Sαs ), σ1,1 Γ),

(34)

then we get the likelihood equation L

L

1

2 −2 L(x|H0 ) = (2π)− 2 Γ− 2 (σ0,0 ) ×

exp{−

1 T −1 x Γ x}× 2 2σ0,0

−L 2

(2π)

exp{−ω

Γ

− 12

(35)

2 −L ) 2× (σ1,0

1 ∇xT Γ∇x}, 2 2σ1,0

and L

1

L

2 −2 L(x|H1 ) = (2π)− 2 Γ− 2 (σ0,1 ) ×

exp{−

1 (x − Sαs )T Γ−1 (x − Sαs )}× 2 2σ0,1 L

1

L

(36)

2 −2 (2π)− 2 Γ− 2 (σ1,1 ) ×

exp{−ω

1 (∇x − ∇(Sαs ))T Γ−1 (∇x − ∇(Sαs ))}. 2 2σ1,1

The MLE is used to estimate covariance σ, and abundance α is replaced with (31). Finally the unstructured detector is obtained as (37).

DU =

2PU,0 + ωPU,1 , 2QU,0 + ωQU,1

(37)

where PU,0 = xT Γ−1 x, PU,1 = ∇xT Γ−1 ∇x, QU,0 = (x − Sαs )T Γ−1 (x − Sαs ), and QU,1 = (∇x − ∇(Sαs ))T Γ−1 (∇x − ∇(Sαs )).

16

4. Experiments In this section, three subsections are presented to verify the effectiveness of the proposed algorithm. 1) We first describe three experimental data sets and evaluation indexes in Subsection 4.1. 2) Then we conduct a thorough study on the main parameters of the proposed algorithm in Subsection 4.2. 3) In Subsection 4.3, we conduct the experiments on three data sets to demonstrate the benefits of the proposed detectors for target detection in details and compare them with the current subpixel detectors. 4.1. Data Description and Evaluation Indexes Data Description: To demonstrate the capability of the proposed method for hyperspectral image target detection, three data sets are employed in experiments. • Synthetic data: In the first data set, 400 pixels are synthesized using different spectra from USGS Digital Spectral Library [34]. The spectra have 224 bands, and the spectra ranges between 0.35 and 2.5 micrometers. Three ground materials: Carnallite NMNH98011, Ammonioalunite NMNH145596, Actinolite NMNHR16485, are chosen as the background endmember, and Andradite WS487 is used as the target. The spectra of background and target are shown in Figure 3(a). The 400 pixels are firstly generated using background endmembers with random abundance, then 9 of them are selected where target spectrum is added. The abundance of target spectrum in the 9 pixels decreases from 50% to 10% in steps of 5%, while the abundance of background endmembers increases from 50% to 90% in steps of 5%. Finally, Gaussian white noise is added to each pixel to achieve a sig-

17

1 0.9 0.8

Reflectance

0.7 0.6 0.5 0.4 0.3 0.2

Carnallite Ammonioalunite Actinolite Andradite

0.1 0

400

600

800

1000 1200 1400 1600 1800 2000 2200 2400 2600 Wavelength (µm)

(a)

(b)

Figure 3: (a) The spectra of the endmembers used in the synthesized image. (b) Synthetic data used in Experiment 1.

nal to noise ratio (SNR) of 30 dB. Image of the Synthetic data is shown in Figure 3(b). • Nuance Cri: The second data set is collected by a Nuance Cri hyperspectral sensor. This device acquires data in 24 bands of 20 nm width with wavelengths from 650-1100 nm. Due to a relatively small spatial range of the device, image data are acquired in a small scale scene. Two images are used in this experiment, in which the background consists of bare soil, grass and dry grass, and the targets are stone and paper. To evaluated the performance of algorithms under different difficulties, in each image five targets with different sizes are placed. These images are shown in Figure 4. • HyMap: The dataset used in this section is from RIT self-test dataset, a standard hyperspectral dataset which is free and available online [35]. The data was collected by a HyMap visible near infrared and shortwave infrared 18

(a)

(b)

Figure 4: Cri images used in Experiment 3. (a) Image1. (b) Image2.

(VNIR/SWIR) airborne sensor around the Cooke City, Montanta, and it was collected on July 4, 2006. The spectral range covers from 0.45 to 2.5 micrometers and there are 126 bands in all. The ground sampling distance (GSD) is about 3m. Both radiance and reflectance images are provided, and the latter was obtained by using the HyCorr software package [36]. In the experiments the reflectance image is chosen to test the performance of different algorithms. Various kinds of materials are used as targets in this dataset. In this experiment different algorithms are tested on four kinds of fabric panel samples, which are referred to as F1, F2, F3 and F4. Among them, F1 and F2 are 3×3 targets, F3 and F4 each includes two targets whose size are 2 × 2 and 1 × 1, and lied in different places, respectively. The samples are taken back to the laboratory where a Cary 500 spectrophotometer is used to measure the reflectance spectra. Then by using high resolution color images as well as global positioning system (GPS) information, the location of the targets are also determined. Figure 5 is the image of the scene. It can be seen that many objects in the images looks very like the target, which

19

(a)

(b)

Figure 5: RIT self-test dataset used in Experiment 2. (a) Image scene. (b) Enlarged image of the red rectangle area in (a).

makes it more challenging to detect the targets. These data sets were imaged in different real-world scenes where we had positioned enough subpixel targets and these data can be considered as real-world data. Evaluation indexes: To evaluate the ability of different algorithms to detect targets, the receiver-operating curve (ROC) for each detector on each target are drawn. ROC curve is a commonly used criterion for target detection algorithm, which shows the detector’s probability of detection under certain false alarm rate. To further investigate the overall performance of different algorithms, the areaunder-the-curve (AUC) is also computed. 4.2. Determination and Sensitivity of the Weighting Factors In the two proposed detectors, the weighting factors ε and ω are very important parameters. They adjust the gradient constraint in the proposed model. Too low a

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value would not affect the gradient of noise, while too high a value would cause the omission of other noise constraint. Therefore, the suitable values of weighting factors ε and ω are necessary. In our experiments, we use the Synthetic data to determine the suitable values of the weighting factors so that they could be used in the experiments employing real-world hyperspectral data. The produced detection results are analyzed by area-under-the-curve (AUC). Additionally, we evaluate the sensitivity of the proposed algorithm to noise with two different data: the Synthetic data and the noisy Synthetic data. The SNR of the noisy Synthetic data is 30 dB. Experimental results of two proposed detectors for various weighting factors are shown in Figure 6. Considering the MGSD, we change the value of the weighting factors {ε, ω} from {0, 0} to {1000, 1000} and obtain a series of detection results. The AUC values under different {ε, ω} are shown in Figure 6(a)-(b). We find in Figure 6(a) that when the values of weighting factors {ε, ω} are in {0 : 0.1, 0 : 10}, the AUC almost remains the best values. However, when MGSD is conducted on the noisy Synthetic data (SNR=30 dB), the suitable values of weighting factors {ε, ω} are in {0 : 0.1, 0.1 : 1000}. In practice, the proposed MGSD should be not so sensitive to noise. Therefore, the robust and suitable values {ε, ω} are in {0 : 0.1, 0.1 : 10}. In the case of the MGUD, the analysis procedure is similar and the resulting determinations of {ε, ω} are {0 : 0.1, 0.1 : 1000}. To simultaneously consider the suitable values {ε, ω} for MGSD and MGUD, their intersection are selected as the final suitable values {ε, ω}. In the following experiments, both ε and ω are set to 0.1.

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Figure 6: AUC performance versus weighting factor ε and ω. (a) MGSD for Synthetic data. (b) MGSD for noisy Synthetic data with SNR=30 dB. (c) MGUD for Synthetic data. (d) MGUD for noisy Synthetic data with SNR=30 dB.

4.3. Results To validate the effectiveness of our proposed detectors, extensive comparative experiments are conducted on three different data sets. In the proposed two new detectors, the prior knowledges of background endmembers are required. Depending on whether the background endmembers are known, three different cases 22

are considered: Experiment 1 (Synthetic data), Experiment 2 (Nuance Cri) and Experiment 3 (HyMap). In Experiment 1, the prior knowledge of background endmembers is known. In Experiment 2, only the number of background endmembers is known and the estimation of background endmembers is required. In Experiment 3, both background endmembers and the number of background endmembers are unknown. Thus, the both prior knowledges need to be estimated.

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Figure 7: Detection results for the Synthetic data. (a) OSP. (b) HSD. (c) HUD. (d) TCIMF. (e) MGSD. (f) MGUD.

Experiment 1: The proposed algorithm is compared with OSP, TCIMF, HSD and HUD. Results are shown in Figure 7. Since this is a simple synthetic data

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and background endmembers are known in advance, all the algorithms perform well. OSP, TCIMF, HUD and MGUD successfully detect all the nine targets; HSD detects five of them, while MGSD, which is based on HSD, can find eight out of nine. The fact that MGSD performs better than its hybrid detector counterpart can validate our model of the distribution of the gradient of noise in hyperspectral image. Experiment 2: In real data, the background endmembers cannot be known in advance, so an estimation algorithm is necessary. In fact the estimation of background endmembers (or hyperspectral unmixing) is a research area. Many algorithms have been proposed, including vertex component analysis (VCA) [37], N-Finder [38], pixel purity index (PPI) [39], etc. In this paper VCA is chosen because VCA performs much better than PPI and better than or comparable to N-FINDR [37]. For the sake of fairness, the unmixing results of VCA are applied to all the detection algorithms. 1

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It can be seen from the ROC curves of HSD, MGSD, HUD, and MGUD on Nuance Cri images in Figure 8 that MGSD is better than HSD. On the other hand, 24

when probability of detection is not very high (below 0.9), HUD performs a little better than MGUD. However, MGUD has a lower false alarm rate when higher probability of detection is demanded. In general the proposed algorithm performs better. To further investigate the overall performance of different algorithms, the area-under-the-curve (AUC) is also computed. Table 1 shows the AUC of Cri images of different algorithms. It can be seen that the proposed algorithm performs better than OSP, TCIMF, HSD and HUD, which once again proves the validity of the proposed noise model. Table 1: The AUC of different algorithms on Cri images.

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Experiment 3: In Experiment 3, the number of background endmembers is also unknown. Also noticing that in unmixing algorithms, the number of the background endmembers must be given, which is not provided with the dataset, so it should also be estimated before the unmixing step. In the experiment, the Hysime [40] algorithm is adopted to estimate the number of endmembers. In Figure 9, results are shown for OSP, TCIMF, HSD, HUD, MGSD, MGUD in terms of ROCs. Generally all the algorithms perform very low false alarm rate when the sensitivities of targets are very high. In general, MGSD works better than HSD, and MGUD works better than HUD. It is worth noticing that in F3, HSD failed to detect any target, while MGSD, which is also a structured detector, outperforms all the other algorithms in this experiment. This shows that the proposed noise model in hyperspectral is appropriate. 25

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5. Conclusions In this paper, the distribution of noise in hyperspectral image is further investigated. It is argued that a better characterization of the noise can improve the detection results. Based on the new noise model, two detectors: MGSD and MGUD are proposed, which assume that the gradient of noise within a pixel should obey Gaussian distribution. In the experiment, three different datasets are used to evaluate the proposed detectors, and experiment results show that the proposed detectors outperform previously developed algorithms. The good results prove that the proposed detectors as well as the noise model can better exploit the information

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