Icarus 319 (2019) 349–362
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Icarus
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Automatic endmember bundle unmixing methodology for lunar regional T area mineral mapping ⁎ Jihao Yina, Chenyu Huanga, Xiaoyan Luo ,a, Qian Dub a Beihang University, 37 Xueyuan Road, Haidian District, Beijing 100191, China b Department of Electrical and Computer Engineering, Mississippi State University, Starkville, MS 39762, USA
ARTICLE INFO ABSTRACT
Keywords: The mineral distribution on lunar surface can contribute to studying lunar evolution, while abundance quan- Linear hyperspectral unmixing tification is still challenging. Unmixing on spectral reflectance data is an effective way for mineralresource Spectral variation explanation, especially in hardly accessible area. In regional area unmixing, some existing unmixing models Endmember bundle extraction mainly rely on spectral libraries, which limits the scene adapability in the absence of some prior information. Lunar mineral mapping Meanwhile, spectral variation is a common phenomenon but often neglected, which may lead to subsequent abundance inversion errors. In this paper, we address a novel automatic image-based endmember bundle un- mixing model, which is called AEBU, to solve these problems. Differently from many unmixing algorithms using a single spectrum to represent a type of mineral, we accommodate spectral variation and construct a set of spectra, i.e., endmember bundle, to represent each material, which will allow for comprehensive endmember expression. The endmember bundles are extracted from the imagery and regarded as a spectra catalog for abundance inversion to avoid the dependence on spectral library. The proposed AEBU model contains two major steps: image-based endmember bundle construction and abundance inversion. To construct endmember bundles effectively, we use pixel-wise sparse representation to extract image pixels as endmember candidates, andthen analyze the shape feature of candidate spectra to separate endmember bundles. In abundance inversion, we consider the extracted endmember bundles as existing spectra library and propose a block sparse representation- based algorithm to automatically select reasonable endmembers for per-pixel unmixing. The performance of AEBU is compared with the state-of-the-art bundle unmixing algorithms on simulated lunar data. The experi- mental results demonstrate excellent performance of the proposed AEBU. Finally, we map the mineral dis- tribution on lunar regional areas by AEBU using interference imaging spectrometer (IIM) data collected by ChangE-1 and moon mineralogy mapper (M3) data collected by Chandrayaan-1, and unmix the Cuprite data to show more application of AEBU.
1. Introduction global mapping and regional area mapping (Bras and Erard, 2003). Global maps of the mineral distribution on the moon are mainly based As the only natural satellite of the Earth, the exploration of the on Hapke radiative transfer analysis (Moussaoui et al., 2008): for ex- moon has a significant meaning to study the origin and evolution ofthe ample, some researchers (Lucey, 2004) mapped the distribution of e.g., Earth (Bharti et al., 2014). There exist many satellites including the clinopyroxene, orthopyroxene, olivine, and plagioclase with five bands SELENE launched by Japan in September 2007, ChangE-1 Lunar Orbiter UVVIS data of Clementine. Regional area mapping pays more attention launched on 24th October 2007 by China, Chandrayaan-1 launched on to the fine research of material distribution on craters and maria. With 22nd October 2008 by India and USAs Lunar Reconnaissance Orbiter/ only five bands of Clementine it is difficult to achieve this mission, Lunar Crater Observation and Sensing Satellite (LRO/LCROSS) laun- while hyperspectral data of IIM onboard ChangE-1 and M3 onboard ched on 18th June 2009, which are important to recent lunar ex- Chandrayaan-1 (Pieters et al., 2009) make it more feasible. ploration missions (Jin et al., 2013). Since several moonshot projects Hyperspectral unmixing is an effective approach to estimate mineral have been achieved till now, a large amount of data is waiting for composition and distribution for large scale areas (Kruse et al., 1985). analysis (Yan et al., 2010). An important task of lunar exploration In general, spectral mixture models can be linear or nonlinear. Linear program is mineral mapping, which can be divided into two categories: spectral mixture model (LSMM) is widely adopted and it is suitable for
⁎ Corresponding author. E-mail addresses: [email protected] (J. Yin), [email protected] (C. Huang), [email protected] (X. Luo), [email protected] (Q. Du). https://doi.org/10.1016/j.icarus.2018.09.005 Received 22 December 2017; Received in revised form 5 September 2018; Accepted 5 September 2018 Available online 25 September 2018 0019-1035/ © 2018 Elsevier Inc. All rights reserved. J. Yin et al. Icarus 319 (2019) 349–362
Fig. 1. The study areas are shown in yellow boxes, the red line is IIM data and the blue line is M3 data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
medium or low resolution imagery. The major principle of linear spectral variability due to different illumination conditions, variable spectral mixture model is that mineral spectrum features linearly grain size and due to the presence of elemental substitution (Jin et al., combine in proportion to their area fractions and the fractions are 2013; Hiroi and Pieters, 1994; Pilorget et al., 2016). A single spectrum considered as abundance. The mathematical formular of linear spectral may not represent an endmember well in practical applications, espe- mixture model is cially in complex geographical environments (Bateson et al., 2000), and x= aE + n (1) neglecting spectral variation is the major reason of error in abundance inverse model. Therefore, using endmember bundle to represent a where a is the abundance vector of x, E denotes the endmembers material is an alternative and effective approach. spectra of the image and n is the noise vector. Since the lunar surface is Several endmember bundle based algorithms have been proposed composed of minerals and lack of vegetation, we focus on linear spec- recently. For example, Somers et al. (2012) proposes an endmember tral mixture model in this paper. bundle extraction algorithm called EIBE to automatically extract end- Endmembers are spectral signatures of tconstituent materials in an member bundles from imagery, which randomly selects endmember image scene. The mainstream unmixing methods are based on a given candidates from imagery, and then clusters them into bundles. Using catalog of endmember spectra like USGS spectral library to select these extracted endmember bundles as a spectral library, abundances endmember spectra for mapping, in which the spectral adaptability is are estimated via multiple endmember spectral mixture analysis limited by fixed library. In recent years, researchers tend to extract (MESMA) method (Roberts et al., 1998). The process of endmember endmembers from the image, i.e., image-based endmember spectra for bundle extraction in Somerss algorithm ignores spatial homogeneity of mapping (Plaza et al., 2009). These image-based endmember extraction material distribution, assuming similar distribution of minerals in algorithms can not only reduce the cost of extensive spectrometric neighboring pixels. Thus the block-area endmember candidates ex- measurements, but also share the same atmospheric effects with un- traction in EIBE may lead to inaccurate endmember bundle extraction mixed data (Keshava and Mustard, 2002). They mainly contain two and affect subsequent abundance estimation. An improved endmember categories based on whether there exist pure pixel or not per end- bundle extraction method proposed in Xu et al. (2015) considers spatial member in the image. If there exists pure endmember pixels in the and spectral information called SSEBE in this paper, combining the image, the classical endmember extraction algorithms include pixel pixel purity index (PPI) (Boardman et al., 1995) and homogeneity index purity index (PPI) (Boardman et al., 1995), iteration error analysis (HI) for candidate endmembers selection. However, PPI captures the (IEA) (Wang et al., 2014), vertex component analysis (VCA) endmembers lying in the boundary of data simplex and may fail to (Nascimento and Dias, 2005) and N-FINDR (Winter, 1999). With non- extract other variable endmembers within the simplex (Uezato et al., pure-pixel existing hypothesis, the representative algorithms are 2016). These endmember bundle extraction methods classify end- minimum volume-constrained nonnegative matrix factorization (MVC- member bundles by k-means (Hartigan and Wong, 1979) with spectral NMF) (Jia and Qian, 2009) and minimum volume simplex analysis value characteristics. However, spectral value may not be a highly (MVSA) (Li and Bioucas-Dias, 2009). In addition, some unmixing discriminant feature to distinguish spectra into different classes. For methods take advantage of statistic models and consider homogeneous example, some numerically similar spectra belong to different classes, mixture of pixels in statistic formulation, such as the Bayesian model while the same-class spectra vary greatly in value (Zhang et al., 2010). based unmixing algorithm (Chen et al., 2016; Moussaoui et al., 2008). In addition, these endmember bundle-based algorithms estimate Generally, most unmixing methods only estimate a single spectrum abundances with the multiple endmember spectral mixture analysis to represent an endmember for one type of material and ignore the (MESMA) (Roberts et al., 1998), and they randomly select spectra from
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Fig. 2. Flow chart of the AEBU. each endmember bundle rather than calculating the best matched moon launched by China on 24 October, 2007, and the data can be endmember combination to reduce computational cost. downloaded from Ground application system (http://moon.bao.ac.cn/) This study deals with spectral variation and library lackage pro- for lunar exploration project. The IIM dataset includes 32 bands from blems in unmixing and proposes an innovative automatic image-based 480nm to 960nm with spectral resolution of 15 nm, and the spatial endmember bundle extraction and mineral map inversion unmixing resolution is 200 m/pixel. Here we use 2c level data of IIM, which has model, in the following referred to as AEBU. It offers a complete un- been preprocessed by spectral radiometric correction, geometric cor- mixing framework with three objectives: (1) capturing the most re- rection and photometric correction. The data ID of the research area is presentative endmember candidates; (2) achieving accurate bundle CE1-BMYK-IIM-SCI-N-20081204081022-20081204101804-4458, and classification based on spectral characteristics; (3) selecting proper the data includes 2843 rows and 128 lines. To generate the experi- endmembers and inversing abundance map. mental results universal, we randomly select one area from the IIM dataset, which ranges between 2201 and 2500 row with 128 lines for 2. Study area and dataset unmixing. Chandrayaan-1 is the first Moon mission of India, and theM3 onboard Chandrayaan-1provides enormous data of lunar surface. We Inthis paper, we focus on the unmixing problem for lunar surface use its 2 Level hyperspectral dataset in our research, which includes 85 analysis, and use two datasets collected by Interference Imaging contiguous bands from 460 nm to 2970 nm with 140–280 m spatial Spectrometer (IIM) onboard ChangE-1 and Moon Mineralogy Mapper resolution for global observations. The data ID of the research area is (M3) onboard Chandrayaan-1. ChangE-1 is the first satellite to orbit the M3G20090118T022705-V01-RFL, which can be downloaded from
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to convert radiance to reflectance. Finally, we use the minimum noise fraction (MNF) (Green et al., 1988) transform to estimate the number of endmember classes k.
3.2. Endmember candidates extraction
Endmembers can express all pixels by linear combination. Before obtaining endmember bundles we need to extract endmember candi- dates, and then classify them into different bundles. Sparse representation aims at choosing atoms to express a signal by sparse linear combination with the least-squares error (Iordache et al., 2012). Considering all pixels as the atoms of the dictionary, each pixel can be linearly represented by representative pixels through sparse representation. That is to say, we can use sparse representation to evaluate the interrelationship among pixels, and the sparse coefficients can reflect similarity. If an atom can represent many pixels with large sparse coefficients, it has great contribution to the image and isre- garded as a potential endmember. We control the sparsity as the same as the number of endmember classes. Consequently, sparse coefficients are important evidence for endmember candidate selection. Fig. 3. Spectral shape feature extraction diagram. The solid curve is the original Given a hyperspectral image containing N pixels with m rows, n X={,,,,} x x … x … x spectrum, and the red dotted lines are the fitting lines. (For interpretation of the columns and l bands, it can be denoted as 1 2 i N , where x={,,} x x … x references to colour in this figure legend, the reader is referred to theweb i i1 i 2 il is the ith pixel vector. Using the linear mixture version of this article.) model, xi can be expressed by xi= a i E + n j (2)
where ai is the abundance vector of xi, E denotes the endmembers in the image and n is the noise vector. When we enlarge the endmember matrix with the whole pixels, the abundance matrix will be sparse. Then we can use sparse representation to calculate sparse coefficients. To represent xi, its corresponding overcomplete dictionary Di is con- structed as Tropp et al. (2006)
Di ={,,,,} x1 x 2 … xi 1 x i+ 1 … x N (3)
Then, xi can be represented by Di with sparse coefficients yi, which is the sparse form of abundance ai, can be estimated from the following objective function 2 min xi yi Di F ,s . t . yi 0 k , i= 1, … , N (4) where k is the maximum number of non-zero coefficients. Many sparse representation algorithms can solve the optimization Fig. 4. Example of sparse coefficients obtained by the MP on a hyperspectral problem in (4). Here, we select the matching pursuit (MP) algorithm, a data. The pixels are arranged column-wise and the row presents the coefficients of each pixel. classical greedy method to estimate sparse coefficients (Chen et al., 2001; Olshausen and Field, 1997; Kreutzdelgado et al., 2014; Tao, 2012). Considering the fact that sparse coefficients indicate the simi- (http://pds-imaging.jpl.nasa.gov/data/m3). We also randomly select larity between signal and corresponding atoms, and a larger coefficient one area from M3 dataset ranges between 6001 and 6200 rows with value means the corresponding atom is more similar to signal (Mallat 200 lines for unmixing. The research areas are shown in Fig. 1. and Zhang, 1993; Aharon et al., 2006), we find the pixels with large
coefficients as endmember candidates. For pixel xi and dictionary Di, 3. Methods the corresponding sparse coefficient is yi. If the largest absolute value
yip = max (| yi1 |,| y i 2 |,… ,| y iN |) is at location p, we vote for the corre- The data processing and analysis were conducted using ENVI 4.5 sponding atom xp as the most correlated pixel to construct endmember and MATLAB R2013b. The entire processing follows four steps: hy- candidates Xcand (Onn and Weissman, 2011). Usually, the number of perspectral image preprocessing; endmember candidates extraction; endmember candidates cn varies according to the complexity of the endmember bundles clustering and map inversion. The flow chart is image. If the number of endmember classes estimated as k, cn is em- shown in Fig. 2. pirically set to 5 k. Thus, the pixels with the top cn of the voting numbers are pooled as endmember candidates. 3.1. Hyperspectral image preprocessing 3.3. Endmember bundles clustering The first step of preprocessing is band selection with bad bandre- moval. The original data of IIM 2c level data and M3 2 level data are To finalize spectral bundles from the endmember candidates Xcand , radiance data which cannot be used directly (Li et al., 2010), so we use we need to distinguish the spectra by their characteristics. Considering internal average relative reflectance (IARR) (Kruse et al., 1985) method the shape of a spectral curve as an important feature (Bateson et al.,
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Fig. 5. Representative reflectance spectra bundle of major components of lunar soil from RELAB Brown University.
2000), we utilize it as the distinguishing characteristic. To describe the Table 1 shape change on each curve more accurately, we divide the curve into
Spectra selected from the Brown University RELAB. segments to extract the inclination. Given each segment contains l0 x =[,,,],x x … x x X MINERAL SPECTRAL ID SPECTRAL NUMBER bands, the spectrum c c1 c 2 cl c cand is divided into S segments, where S = [,]l l0 . If band l is not divisible by l0, the length of clinopyroxene LS-CMP-009 C1LS09 the last segment is set to the reminder. C2LS09 C3LS09 xc=[x c1 , x c 2 ,, … x cp ,, … x cS ], c = 1,2,, … cn olivine LR-CMP-014 C1LR14 xc= [x l×( p 1)+ 1,… ,xl × p] , p = 1, 2, … ,S C2LR14 p 0 0 (5) orthopyroxene LS-CMP-012 C1LR12 S1LS12 As for the determination of l0, we have empirically found that when S2LS12 l is set to 10% of l, the extracted feature can best distinguish the spectra plagioclase LR-CMP-011 C1LS11 (Yin et al., 2017). We fit each segment with a straight line using the C3LS11 S1LS11 least squares method. Generally, the slopes vector is significant to re- th flect shape feature. The slope factor fcp for the p segment of pixel xc, which can be calculated by (6), is remained as the recognizable feature.
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Fig. 6. Left: original lunar imagery; right: crater detecting results.
the atoms with an uncertain number and type. This is more accurate than endmember combination selecting strategy with a fixed number and type. The overcomplete dictionary can be described as
D= [;;;], Db1 Db 2 … Dbi … Db K Db i = Xb i (8)
where Dbi is a block of dictionary and Xbi is the ith endmember bundle. Any sparse representation algorithm with a fixed dictionary can be used. Here we still use matching pursuit (MP) to achieve block sparse
representation. For pixel xi, the unmixing procedure can be summarized as follows.
(1) Define the position of the selected endmembers as indices set χ, the selected endmember combination as E, and initialize the residual r0 = x, the index set 0 = , the endmember combination set E0 = , and the iteration counter t = 1.
(2) Find the index λt that solves the optimization problem
t =arg max |
fc=[,,,,], f c1 f c 2 … f cp … f c S p culate the new residual rt
=1, 2, … ,S t = argmin x Et 2 (10) ()()()()q2 x q qx f =1000 × q q ,q=l × ( p 1)+ 1, …l × p cp L ()()q2 q 2 0 0
(6) rt = x tE t (11)
The feature matrix Fcand of Xcand can be represented as
Fcand =[,,,, f1 f 2 … fc … f cn ],c = 1,2, … , cn (7) (4) Increment t, and return to step (2) till t = k. (5) Use the obtained endmember candidates to estimate abundance To obtain the endmember bundles with homogenous curve feature, with the abundance sum-to-one (ASC) and nonnegative constrain we divide the extracted endmember candidates into several clusters (ANC). based on Fcand using the k-means algorithm (Kanungo et al., 2002). abund = argmin x abundE 2 Consequently, the feature vectors are clustered into k fitting bundles abund (12) Fbundle =[;;;] Fb1 Fb 2 … Fbi … Fb k . According to the feature bundles, we can gather the corresponding endmember bundles Xbundle = [;;;],[,{}] XbXb1 2 … Xbi … XbXb k i = x h h = fh Fbi . Fig. 4shows an example of constructing an overcomplete dictionary and its sub-dictionary with the obtained sparse coefficients and corre- 3.4. Bundle-aware map inversion sponding selected spectra.
A pixel may not contain all types of endmembers, and the number of 3.5. Accuracy assessment endmembers per pixel is limited and variable. Therefore, when un- mixing with endmember bundles, we need to select the best matching In this study, the average deviation mean (ADM) and root mean endmember combinations for each pixel, and use the selected spectra square error (RMSE) are used to evaluate the unmixing accuracy. The for subsequent abundance estimation. average deviation mean is used to evaluate the capability of end- In this step, we combine sparse representation with block sparsity member bundle extraction, which is defined as method to obtain endmember combination with the least reconstruc- l tion error. We consider each endmember bundle as a block of dictionary 1 ADMq = |meanr ()qd meane ()|,qd q= 1,2, … K and represent each pixel by block sparse representation to select best l d=1 (13) matching atoms. To select an atom from each block dictionary, we take advantage of block sparse representation (Eldar et al., 2009). When the where rqd is the value of reference spectrum of class q at band d, and eqd reconstruction error is less than a threshold, we automatically output is the value of extracted spectrum of class q at band d. The average
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Fig. 8. Estimated endmember spectra of (from left to right) reference spectra, EBE, SSEBE and AEBU. deviation mean evaluates the deviation of the mean of each bundle this study simulates the lunar surface hyperspectral data to test the spectra with the true bundle spectra. algorithms. Meanwhile, we compare the AEBU with two state-of-the-art We use root mean square error to measure the abundance estima- endmember bundle unmixing algorithms, i.e., EIBE (Somers et al., tion error, which is defined as 2012) and SSEBE (Xu et al., 2015). Firstly, to simulate the mineral spectra on lunar surface, we use four ( m n A 2 ) i=1j = 1 qij common types of minerals on lunar surface, which are clinopyroxene, RMSEq = ,q= 1, 2, … K m× n (14) orthopyroxene, olivine, and plagioclase, as endmembers and select corresponding spectra from the RELAB spectra library of Brown where m × n is the size of image, and Aq is the difference between the University (http://www.planetary.brown.edu/relab) as reference estimation abundance and true abundance for class k. spectral bundles (Fig. 5). The information of used spectra are shown in Table. 1. The spectra selected from BU RELAB cover different band 4. Results and discussion ranges, so we retain the common 431 bands from 350 to 2500 nm with 5 nm internal for unified processing. As for abundance maps, to satisfy 4.1. Results on lunar synthetic data the block distribution of minerals, we apply the crater detection algo- rithm in Li et al. (2014) on the image collected by CCD onboard Since the lunar hyperspectral data has no reference abundance maps ChangE-1 to obtain true crater distribution, and consider the detected or reference spectra for unmixing, we cannot evaluate the unmixing craters as the concentrated distribution for a mineral (Zhu et al., 2016). methods quantitatively. To quantitatively evaluate our proposed AEBU,
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Fig. 9. Estimated abundance maps of (from left to right) reference map, EBE, SSEBE and AEBU.
Table 2 Fig. 7. Results of the average deviation between the means of each bundle spectra with In this experiment, we choose the block size in EIBE as 20 × 20, and the true bundle spectra and abundance error RMSE between estimated abun- randomly select 30 pixel blocks for calculation. In SSEBE, the initial dances and reference abundances for each mineral. homogeneity index is set as the maximum homogeneity index value of ADM the selected endmembers. The number of random vectors is set as 10000. In the proposed unmixing method, the sparsity of sparse re- Method clinopyroxene olivine orthopyroxene plagioclase presentation in step A is set to 4 the same as the endmember class
EBE 0.0411 0.0289 0.2616 0.0196 number k = 4. When selecting endmember candidates, considering the SSEBE 0.0257 0.0084 0.0064 0.0112 size of image data, we set the number of spectra in each bundle 5, so the AEBU 0.0205 0.0246 0.0059 0.0202 number of endmember candidates are cn=5 k=20. Since the number RMSE of spectral bands is 431, the length of each segment in step B is set to 40 EBE 0.2766 0.2506 0.4002 0.1055 bands. The accuracy of the estimated endmember spectra showed in SSEBE 0.2124 0.2130 0.1349 0.1755 AEBU 0.0397 0.0268 0.0205 0.0194 Fig. 8 is evaluated using the reference spectra in Fig. 5, and the esti- mated abundances using the reference abundance maps are in Fig. 9, and the results of average deviation mean and RMSE are shown in The crater detection results are shown in Fig. 6. We assume each crater Table. 2. as the concentrated distribution for a mineral, which means the abun- According to the results, EIBE has a good performance in extracting dance of the main mineral of in-crater pixel is set larger than 0.5, and olivine and plagioclase spectra bundles with low average deviation the remaining three abundances are generated randomly. The abun- mean, while it fails to extract accurate spectra bundle of orthopyroxene, dance of remaining pixels outside the craters are generated by four and it also mistakes part of the spectra of orthopyroxene for clinopyr- random numbers with Gaussian distribution. Meanwhile, all pixels sa- oxene. These results lead to an inaccurate estimation of abundance tisfy sum-to-one constraint (ASC) and nonnegative constraint (ANC) maps and cause the high RMSE of clinopyroxene, olivine and ortho- (Onn and Weissman, 2011). Thus, we construct a set of abundance pyroxene. This is because the endmember bundle results are unstable in maps with the size of 200 rows and 400 lines. In the synthetic image, the clustering step by k-means, which heavily depends on the randomly for each pixel the endmember spectra are randomly selected from selected initial clustering centers. In endmember candidate extraction, bundles. Finally, we use the linear mixture model to synthesize hy- the SSEBE shows the lowest average deviation mean for orthopyroxene perspectral data, adding 40-dB signal-to-noise ratio (SNR) White but shows relatelevy inaccurate results for clinopyroxene and olivine. Gaussian noise. An example of true synthetic spectrum is shown in The main reason is that k-means clustering take the reflectance value of
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Fig. 10. (a) the reference spectra obtained from BU RELAB. (b)–(d) The estimated spectra bundles of clinopyroxene, plagioclase and orthopyroxene.
spectra as the only characteristic. The messy endmember bundle results IIM data mainly contains three kinds of minerals and we can detect four lead to the highest RMSE of abundance mapping results. kinds of minerals in the research area in M3 data. Finally, we apply Compared with these methods, AEBU successfully extract end- AEBU on two hyperspectral datasets. After obtaining the spectral bun- member candidates and cluster the endmember bundles to some extent. dles, we measure the similarity between the average spectrum of each Only one spectrum with the similar curve shape as plagioclase is in- bundle with relative spectra in BU RELAB by Euclidean distance to correctly extracted and divided into plagioclase spectral bundle, re- ensure the mineral type of each bundle. The reference spectra from BU sulting in low accuracy of plagioclase average deviation mean. The RELAB are shown in Fig. 10 (a). The spectral bundles results of IIM and estimated abundance maps shown in Fig. 9 demonstrate that AEBU can M3 are shown in Fig. 10 and Fig. 11, and the abundance maps are correctly estimate the abundance maps, which are close to the truth. shown in Figs. 12 and13. The Euclidian distances between bundle and reference spectra of IIM are shown in Table 3 and M3 are shown in 4.2. Automatic unmixing on lunar hyperspectral data Table 4. The spectra bundles are classified to the mineral based onthe largest Euclidean distance, the matching minerals in Tables 3 and4 are Firstly, we remove bad bands of dataset for accurate unmixing, and all accord with this principle. remain 11–30 bands for IIM and 3–85 bands for M3. Then we de- Obviously, the number of minerals in IIM and M3 data are not the termined the number of mineral classes by MNF, the research area in same, and we can automatically extract their corresponding
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Fig. 11. (a)–(d) The estimated spectra bundles of clinopyroxene, plagioclase and orthopyroxene and olivine.
Fig. 12. (a)–(c) The estimated abundance map of IIM data of clinopyroxene,orthopyroxene and plagioclase by AEBU.
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Fig. 13. (a)–(d) The estimated abundance map of M3 data by AEBU of clinopyroxene,orthopyroxene and plagioclase and olivine by AEBU.
Table 3 Euclidian distance between spectral bundle obtained from IIM and reference data.
Reference spectra matching
clinopyroxene orthopyroxene plagioclase mineral
bundle 1 0.189 0.194 0.256 clinopyroxene bundle 2 0.156 0.151 0.222 orthopyroxene bundle 3 0.264 0.272 0.197 plagioclase endmember spectral bundles. Due to the limited spectral bands of hy- perspectral data and the different research areas, there are certain differences between the extracted spectral bundles and the reference spectra. However, the extracted spectra from IIM data and M3 data, Fig. 14. AVIRIS Cuprite false color map. which belong to the same mineral are consistent in Euclidean distance and spectral signature. This phenomenon indicates the correctness of our results. Limited to the scale of research area, though the spectral number of each bundle is set to 5, the extracted bundle of orthopyr- oxene still contains only one spectrum. This result demonstrates that Table 5 the AEBU is robust and keeps consistence with endmember extraction The average deviation between the means of the true and the extracted end- member spectra for each class. algorithms when there is rare spectral variation. In IIM endmember bundle spectral results, the plagioclase varies greatly on spectral value Average Deviation Mean (ADM) for Cuprite while the spectral curves are in consistency, as well as the clinopyr- 1 Minerals EBU-iiSA N-FINDR L NMF oxene and olivine spectral results of M3. These spectral bundle results 2 show that spectra variation in imagery is common and the bundle ALUNITE 0.1266 0.1661 0.1625 construction effectiveness of AEBU. Comparing the spectral bundle re- ANDRADITE 0.2526 0.0652 0.2135 sults with reference spectra, the spectral variation in the study area is BUDDINGTONITE 0.0775 0.1410 0.1506 obvious, so that the methods rely on the spectra library used here may DUMORTIERTE 0.0869 0.0599 0.0808 lead to abundance mapping errors. KAOLINITE1 0.1660 0.0694 0.1635 In the abundance maps, the pixel with red color represents a higher KAOLINITE2 0.1239 0.0307 0.0515 MUSCOVITE 0.0919 0.2245 0.0863 fraction, while blue represents a lower fraction. According to the MONTMORILLON 0.0884 0.0407 0.0514 abundance maps of IIM, the researching area is mainly composed of NONTRONITE 0.0976 0.0147 0.0577 clinopyroxene and orthopyroxene. The distribution of plagioclase is PYROPE 0.0165 0.0235 0.0748 dispersed and discontinuous. Most lunar craters and the lunar mares in SPHNE 0.0217 0.0526 0.2730 CHALCEDONY 0.0922 0.1025 0.1030 this area are composed by orthopyroxene, while the highland is com- posed of clinopyroxene. As for the abundance results of M3, we can see most craters are mainly composed of clinopyroxene. The lunar
Table 4 Euclidian distance between spectral bundle obtained from M3 and reference data.
Reference spectra matching
clinopyroxene orthopyroxene plagioclase olivine mineral
bundle 1 0.161 0.262 0.275 0.556 clinopyroxene bundle 2 0.154 0.148 0.215 0.491 orthopyroxene bundle 3 0.296 0.262 0.136 0.148 plagioclase bundle 4 0.311 0.274 0.197 0.095 olivine
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highlands contain widespread olivine, plagioclase and orthopyroxene are less and mainly concentrate in the vicinity of the craters.
4.3. Additional application
In order to demonstrate the value of our method for in real appli- cations, we apply our method on a common hyperspectral data for unmixing. The hyperspectral data covering the Cuprite scene is cap- tured by the Airborne Visible Infrared Imaging Spectrometer (AVIRIS) sensor. The data has endmember spectral ground truth, while there is only one true spectrum for each endmember. Therefore, it is not sui- table to run endmember bundle algorithms on this dataset. Here, we compare the algorithm with classical unmixing algorithms, i.e., N- L1 NMF FINDR (Winter, 1999), 2 (Qian et al., 2014), and SUnSPI (Tang et al., 2016), which do not consider the spectral variability, to evaluate the endmember extraction capability of EBU-iiSA on real da- taset. The size of the cuprite data is 150 × 100, with 224 bands cov- ering 370 to 2480 nm. After removing the noisy (1 to 2 and 221 to 224) and water absorption channels (104 to 113 and 148 to 167), 188 channels remain. The false color map is shown in Fig. 14. In this experiment, the spectral number of each endmember bundles should be small, so we set the number of endmember candidates as 60, which is 5 times of endmember class number (12). The length of each segment in step B is still set to 20 bands. The initial endmember value of is obtained by N-FINDR and the spectral library of SUnSPI is obtained from USGS. As to endmember extraction algorithms like N- FINDR, after obtaining endmembers, abundances are estimated through the least squares method with the two abundance constraints. We compare each bundle spectra obtained by the EBU-iiSA and endmember extraction results by three compared algorithms with spectral truth. The Average Deviation Mean (ADM) results for each endmember are shown in Table 5. The extracted endmembers spectra are shown in Fig. 15, and the estimated abundance maps are in Fig. 16. SUnSPI needs a spectral library as priori information, so Fig. 15 does not show the extracted endmembers of SUnSPI. According to the endmember ex- traction results in Fig. 15., the three algorithms can generally extract the endmember spectra correctly. For quantitative analysis (see L1 NMF, Table 5), the ADMs of N-FINDR are less than 2 which illustrates N-FINDR has better performance in endmember extraction than . Although the ADM results of EBU-iiSA limited to the mean of spectral bundle, some ADM values are still nearly or even smaller than that of N-FINDR, such as alunite buddingtonite, pyrope, sphne and chalcedony. As for the estimated abundance maps in Fig. 16 , we can only qualitatively analyze according to the false color maps due to lack of ground truth. Intuitively, the distribution results of EBU-iiSA is closer to the false color map, and the resulting abundance maps for different minerals can be easily distinguished.
5. Conclusions
This paper proposes an image-based automatically endmember bundle unmixing method (AEBU) for regional mineral mapping. Compared with existing unmixing methods, it has three unique ad- vantages: (1) achieving automatical unmixing without prior informa- tion; (2) introducing endmember bundles to solve spectral variation; (3) selecting best matching endmembers with given bundles for accurate abundance inversion. The experiment on synthetic lunar surface data quantitatively shows that the AEBU can achieve high accuracy mea- surement and has a better performance than state-of-the-art methods. Fig. 15. Endmember bundle results for Cuprite data, from left to right are The reasonable mineral distributions on ChangE-1 IIM data and L1 NMF Chandrayaan-1 M3 data further demonstrate that our AEBU has high groundtruth, EBU-iiSA, 2 and N-FINDR. From top to bottom are Alunite, Andradite, Buddingtonite, Dumortierte, Kaolinite1, Kaolinite2, Muscovite, robustness and wide adaptiveness for fine mineral distribution detec- Montmorillon, Nontronite, Pyrope, Sphne, Chalcedony, the x-axis represents tion. The unmixing results of Cuprite dataset demonstrate the value of the reflectance and the y-axis represents the wavelength. our method in real applications.
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L1 NMF Fig. 16. Abundance maps for AVIRIS Cuprite data, from left to right are groundtruth, EBU-iiSA, 2 and N-FINDR. From top to bottom are Alunite and Andradite, Buddingtonite and Dumortierte, Kaolinite1 and Kaolinite2, Muscovite and Montmorillon, Nontronite and Pyrope, Sphne and Chalcedony.
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