Spectral Variability in Hyperspectral Data Unmixing: A Comprehensive Review
Ricardo Augusto Borsoi, Tales Imbiriba, José Carlos Moreira Bermudez, Cédric Richard, Jocelyn Chanussot, Lucas Drumetz, Jean-Yves Tourneret, Alina Zare, Christian Jutten
11 Spectral Variability in Hyperspectral DataUnmixing: A Comprehensive Review
Ricardo Augusto Borsoi, Tales Imbiriba, José Carlos Moreira Bermudez, Cédric Richard, Jocelyn Chanussot,Lucas Drumetz, Jean-Yves Tourneret, Alina Zare, Christian Jutten
Abstract —The spectral signatures of the materials containedin hyperspectral images, also called endmembers (EM), canbe significantly affected by variations in atmospheric, illumi-nation or environmental conditions typically occurring withinan image. Traditional spectral unmixing (SU) algorithms neglectthe spectral variability of the endmembers, what propagatessignificant mismodeling errors throughout the whole unmixingprocess and compromises the quality of its results. Therefore,large efforts have been recently dedicated to mitigate the effectsof spectral variability in SU. This resulted in the developmentof algorithms that incorporate different strategies to allow theEMs to vary within a hyperspectral image, using, for instance,sets of spectral signatures known a priori , Bayesian, parametric,or local EM models. Each of these approaches has differentcharacteristics and underlying motivations. This paper presentsa comprehensive literature review contextualizing both classicand recent approaches to solve this problem. We give a detailedevaluation of the sources of spectral variability and their effect inimage spectra. Furthermore, we propose a new taxonomy thatorganizes existing work according to a practitioner’s point ofview, based on the necessary amount of supervision and on thecomputational cost they require. We also review methods usedto construct spectral libraries (which are required by many SUtechniques) based on the observed hyperspectral image, as wellas algorithms for library augmentation and reduction. Finally,we conclude the paper with some discussions and an outline ofpossible future directions for the field.
I. I
NTRODUCTION
Hyperspectral cameras are able to sample electromagneticspectra at hundreds of contiguous wavelength intervals. Thehigh spectral resolution of hyperspectral images makes theman important tool for the precise identification and discrimi-nation of different materials in a scene. Hyperspectral imagescontribute significantly to different fields and are now at thecore of a vast number of applications such as space explo-ration [1], land-use analysis, mineral detection, environmentmonitoring, field surveillance [2], [3], disease diagnosis andimage-guided surgery [4].Notwithstanding the advantages brought forth by their highspectral resolution, hyperspectral cameras operate on a delicatetrade-off between spatial resolution and signal-to-noise ratio.This happens since the light observed at the sensor is decom-posed into several spectral bands, which in turn demands thepixel size to be large enough to attain an acceptable signal-to-noise ratio. When combined with a large target-to-sensordistance, which is common in many applications, this leadsto images with low spatial resolution [5]. The limited spatialresolution of hyperspectral images means that each imagepixel is actually a mixture of P different pure materials, whose Figure 1. Spectral variability is ubiquitous in hyperspectral images: the pixelsin regions composed of a single material (e.g., tree, roof and soil in the imageabove) can contain very different spectral signatures. spectra are termed endmembers (EM), present in the scene [6].This mixing process conceals important information about thepure materials and their distribution in an image. Spectralunmixing (SU) aims to solve this problem by decomposinga hyperspectral image into the spectral signatures of theendmembers and their fractional abundance proportions foreach pixel [7].The simplest and most widely used model to represent theinteraction between light and the EMs in the scene is theLinear Mixing Model (LMM) [6], which represents a givenpixel y n indexed by n with L spectral bands as: y n = M a n + e n , subject to (cid:62) a n = 1 and a n ≥ (1)where M = [ m , , . . . , m ,P ] is an L × P matrix whosecolumns are the P endmembers, a n is a vector containing theabundances of every endmember in the pixel y n and e n is anadditive noise vector. Traditionally, the LMM assumes that thesignatures M of the pure materials are the same for all pixels y n , n = 1 , . . . , N in the image. Although this assumptionleads to a well-posed and computationally simpler framework,it limits the applicability of the LMM since it can jeopardizethe accuracy of estimated abundances in many circumstancesdue to the spectral variability of the endmembers. a r X i v : . [ ee ss . I V ] O c t A. Spectral variability in SU
Spectral variability is an effect commonly observed in manyscenes in which the spectral signatures of the pure constituentmaterials vary across the observed hyperspectral image, asillustrated in Fig. 1. It can be caused, for instance, by variableillumination and atmospheric conditions. Variability can alsobe intrinsic to the very definition of a pure material, such assignatures of a single vegetation species varying significantlydue to different growing and environmental conditions [8], [9].In this context, the use of a single matrix M for all pixelsin the LMM (1) leads to problems such as proportion inde-terminacy , where errors in the estimation of the endmemberspectra at each pixel propagate to the estimated abundances.This results in erroneous abundance estimation and in theselection of too many endmembers to represent the spectrumof each pixel y n [8]–[10]. Due to the significant impact ofendmember variability on abundance estimation quality, a lotof effort has recently been dedicated to develop algorithms thatare able to obtain better abundance estimates in this scenario.The most general form of the LMM considering spectralvariability generalizes (1) to allow for a different endmembermatrix for each pixel, resulting in: y n = M n a n + e n , subject to (cid:62) a n = 1 and a n ≥ (2)for n = 1 , . . . , N , where M n ∈ R L × P is the n th pixelendmember matrix.SU considering spectral variability can be generally definedas two complementary problems related, respectively, to therecovery of the abundances and to the recovery of the end-members. These can be defined as: P : To mitigate the adverse effects of spectral variability inthe abundance estimation; P : To estimate the spectral signatures of the endmemberspresent in each pixel of the image.Substantial interest has been recently raised for both of theseproblems. While all SU methods must deal with P whileaccounting for spectral variability, not all of them take P into consideration due to the additional difficulty it entails. B. Contribution, taxonomy and organization
Many SU algorithms have been proposed to address prob-lems P and P . Different algorithms follow different method-ologies to represent the endmembers in the scene. Existingmethods employ Bayesian, parametric, or spatially localizedmodels, as well as libraries containing different instances ofmaterial spectra known a priori . This multiplicity of modelsgives rise to solutions presenting different advantages anddisadvantages in terms of computational complexity, accuracyand amount of user supervision.In this paper we categorize the methods according tocriteria that are most relevant to the practitioner, such as, e.g.,computational complexity, to provide a comprehensive reviewthat complements and updates previous review papers [8], [9],[11], [12]. Since existing SU methods that address spectralvariability have very heterogeneous characteristics, navigatingthe field can be difficult, especially when taking into account both classical algorithms and recent developments. This dif-ficulty motivated the present review, which presents a noveltaxonomy aimed at the practitioner, as well as a comprehensivecategorization of existing approaches. The contributions andhighlights of the present paper are described in the following. A new taxonomy, for the practitioner
We propose a new taxonomy to organize the existingtechniques according to a practitioner’s point of view, basedon the amount of user supervision and on the computationalcomplexity required to solve the SU problem. The resultingtaxonomy is summarized in the form of a decision tree shownin Fig. 2, which can be used to guide the choice of a family ofSU algorithms. The decision tree also dictates the organizationof the rest of the paper. We start from whether a spectrallibrary is known a priori or not, and proceed to differentfamilies of SU methods based on the trade-offs they offerregarding the need for user supervision and computationalcost. Table I summarizes the main characteristics of eachgroup of techniques, and points to illustrations with high-level descriptions of the key ideas on which they are based(Figs. 10–14). Comprehensive overview and recent highlights
We provide a comprehensive review of the methods de-veloped to solve the SU problem with EM variability. Weencompass and contextualize both the classic strategies thathave been reviewed before as well as numerous recent devel-opments in the field. Thus, both classic and recent algorithmsare categorized according to the proposed taxonomy, whichhelps to highlight the recent advances in each area. Spectral libraries, ex situ
A considerable number of SU methods address spectralvariability using libraries of spectra that originally had to beacquired a priori (e.g., through laboratory of in situ measure-ments), which used to limit the applicability of these methods.An important recent development concerns methods that caneither extract spectral libraries directly from the observedimages, or generate them using physics-based mathematicalmodels of material spectra. This allows for the widespread ap-plicability of library-based SU techniques in situations wherespectral libraries are not available or cannot be obtained.Such methods are reviewed in Section V-A, and an illustrativedescription of these techniques can be seen in Fig. 15.Moreover, library pruning techniques, which were originallydevised to reduce the size of libraries so as to improve thecomputational complexity of SU, have evolved to consider alsothe quality of the unmixing results. Recent library pruningmethods aim at removing, before unmixing, either entire EMclasses or individual spectral signatures which are not likely tobe present in observed image. This reduces the ill-posednessof the SU problem and can improve abundance estimation.These techniques are discussed in Section V-B.Table II summarizes the key ideas involved in library extrac-tion and pruning methods, as well as their main characteristics. Experimental aspects and toolbox
The practical aspects related to the evaluation of SU meth-ods when spectral variability is considered are also discussedin Section VI. This includes the generation of realistic syn-thetic data and a list of existing software resources that yes no
BayesianmodelsEM librariesextraction noyes
User-definedspectraltransformation
Is expert knowledgeavailable? supervision less costIs the libraryvery large?
EM-model-freemethodsParametric EMmodels no yesPrune signaturesto make the librarysmall? noyessupervision less costno yes
Library-basedspectraltransformation
Apply spectral transformation tothe image andlibrary? noyes
MESMA andvariants SparseUnmixingand EMs for each pixelEstimate abundancesUnmixingFuzzy methodslearningMachine LocalUnmixing supervision orless computationalcost?Less supervision orless computationalcost?LessExtractlibraries from theobserved image?less less Low Medium Highunmixing methods:computational cost of theColor code for thea priori?Are spectrallibraries available
Figure 2. Decision tree for hyperspectral unmixing considering spectral variability. The blue boxes denote families of unmixing algorithms, while the yellowboxes denote additional techniques related to the extraction and processing of spectral libraries.Table IC
HARACTERISTICS OF EACH GROUP OF SU TECHNIQUES AND WHERE THEY ARE REVIEWED IN THE PAPER
MESMA andvariants Fuzzy SU Sparse SU Machine Learning Local SU ParametricEM models EM-model-free BayesianAmount of usersupervision • • •• • • • • • • • • • •• •
Computational cost • • • • • • • •• • •• •• • • •
Requires spectrallibraries? (cid:51) (cid:51) (cid:51) (cid:51) (cid:55) (cid:55) (cid:55) (cid:55)
Estimates pixel-dependend endmembers? (cid:51) (cid:51) (cid:51) (cid:55) (cid:51) (cid:51) (cid:55) (cid:55)
Where to findin the paper: Sec. III-A Sec. III-A Sec. III-B Sec. III-C Sec. IV-A Sec. IV-B Sec. IV-C Sec. IV-DIllustrative descriptionof the key ideas: Fig. 10 Fig. 11 Fig. 12 Fig. 13 Fig. 14 are available to the reader. We also present an illustrativesimulation in order to demonstrate the application of a fewof the SU techniques reviewed in the paper, which werechosen by selecting different paths in the proposed deci-sion tree. This example is made publicly available in theform of a software toolbox at https://github.com/ricardoborsoi/unmixing_spectral_variability and in [13].The paper is organized as follows. In Section II we present adetailed overview of the physical effects that originate spectralvariability and their effects on the endmembers and on thehyperspectral image. In Section III, we review the SU methodsaccounting for spectral variability that use spectral libraries. In Section IV, we describe the blind methods that do not requireEM signatures to be known a priori. We then discuss theconstruction and pruning of spectral libraries in Section V.Section VI discusses the evaluation of SU algorithms whenspectral variability is present, lists existing software resources,and presents an end-to-end illustrative example comparingsome existing techniques selected following the proposed de-cision tree. Finally, we conclude the paper in Section VII withsome discussion and conclusions about the existing methodsand future research directions.
Table IIC
HARACTERISTICS OF SPECTRAL LIBRARY EXTRACTION AND PRUNING TECHNIQUES AND WHERE THEY ARE REVIEWED IN THE PAPER
Library extractiontechniques: Image-basedlibrary extraction Library generationfrom physics models Spatial interpolationof EM signaturesKey idea Extracts multiple EM signaturesfrom the observed image and clus-ter them to construct a library Create synthetic EM signatures us-ing physico-chemical mathematicalmodels describing EM variability Estimate EM signatures for eachpixel by interpolating pure pixelsat known spatial locationsAdapted to the HI? (cid:51) (cid:55) (cid:51)
Amount of usersupervision •• • • • ••
Depends on the existenceof pure pixels? (cid:51) (cid:55) (cid:51)
Where to findin the paper: Sec. V-A1 Sec. V-A2 Sec. V-A3Library pruningtechniques: Library reduction Endmember selection Same-class EM pruningKey idea Remove redundant signatures froman existing library to reduce thecomputational complexity of SU Remove entire EM classes (e.g.,water, tree) not present in the ob-served image from the library Select the signatures from each EMclass most closely related to theobserved image before SUAdapted to the HI? (cid:55) (cid:51) (cid:51)
Amount of usersupervision • •• ••
Improves the computa-tional cost of SU? (cid:51) (cid:51) (cid:51)
Improves SU quality? (cid:55) (cid:51) (cid:51)
Where to findin the paper: Sec. V-B1 Sec. V-B2 Sec. V-B3
II. O
RIGINS OF S PECTRAL V ARIABILITY AND THEIR E FFECTS
The variability in the spectral signatures occurs mainly dueto (a) atmospheric effects, (b) illumination and topographicchanges, and (c) intrinsic variation of the spectral signaturesof the materials (i.e., due to physico-chemical variations). Un-derstanding how these conditions affect the spectral signaturesof the materials and the unmixing results is important in orderto develop informed models and methods to deal with EMvariability.In addition to spectral unmixing, spectral variability alsoaffects other hyperspectral imaging tasks, which promptedextensive investigations into its causes and on how it manifestsin the material spectra. In this context, a recent review articleby James Theiler and his coworkers provides an excellentoverview of spectral variability in hyperspectral target de-tection [14]. In particular, the causes and effects of spectralvariability in target detection are reviewed, with a focus onthe study of environmentally induced variability (caused by,e.g., atmospheric and topographic changes) through an in-depth view of radiative transfer models. A detailed computersimulation is also included to illustrate how the materialspectra are affected by changes in the different parametersof the radiative transfer model.In the following, we review the causes and effects of spectralvariability from a spectral unmixing perspective. Althoughwe also introduce the radiative transfer function interpretation of some atmospheric and topographic effects, we focus ourexposition on a more generic analysis of the consequencesthat spectral variability has on the observed pixel spectra andon the results of spectral unmixing as reported by previousexperimental works (i.e., with a stronger focus on the resultsof, e.g., atmospheric compensation methods as opposed tothe interpretation of the imaging models themselves). Theinterested reader can find a more comprehensive and in-depthanalysis from a radiative transfer function standpoint in [14]. A. Atmospheric effects
One of the main sources of spectral variability is theinterference by the atmosphere when measuring ground re-flectance. Atmospheric gases (such as O , O , CH , CO ,etc.), aerosols and, most prominently water vapor, absorbsignificant amounts of radiation, while other molecules andaerosols scatter incoming light [15]. These effects have animpact on the radiance measured at the sensor, which canbecome significantly different than that corresponding to thedesired ground reflectance. Atmospheric absorption from gasesis also heavily wavelength dependent, whereas aerosol ab-sorption varies smoothly in spectra. These effects must becompensated to achieve an accurate characterization of surfacereflectance.Atmospheric compensation models can be roughly dividedinto statistical (empirical) and physics-based models [15]. Sta-tistical models are based on additional information about the Light source ViewerTerrain(b)(d)(a)(c)
Figure 3. Illustration of the effects of the atmosphere on the acquiredhyperspectral image. The sources of radiation are represented by (a) lightdirectly reflected by the atmosphere to the sensor, (b) light scattered by theatmosphere and reflected by the ground, (c) light directly reflected by theground and (d) light reflected by surrounding regions on the ground and thenscattered to the sensor. atmospheric influence, usually obtained by means of referenceobjects or calibration panels in the scene. This information isused to find a relationship (e.g., linear) between the radiancesobserved at the sensor and at the surface of the scene [15].This results in a gain and an offset factor for each spectralband, which are then uniformly applied to every image pixelto compensate for the atmospheric effects [15]. Sometimes,when a reference object is not present in the scene, naturallyoccurring objects can be employed as reference spectra, mostcommonly consisting of smooth bodies of water, which exhibitlow reflectance and can be considered as dark objects [5]. Thedownsides of this approach are that the true reflectance of areference object must be accurately known, and that it does notaccount for the spatial variability of the distribution of gasesand aerosols. This variability can be very significant, and thuscan introduce spatially-dependent residual atmospheric effects.A classical example of statistical methods is the empirical linemethod (ELM) [5].Physics-based models, on the other hand, are robust alter-natives to empirical methods which do not assume additionalinformation about the scene to be known. These methods arecurrently mature and widely used, addressing the limitationsof empirical methods by employing a rigorous model thatexplicitly describes the absorption and scattering effects dueto atmospheric gases and aerosols [16]. Popular examplesinclude the Atmospheric Removal (ATREM) and the FastLine-of-Sight Atmospheric Analysis of Spectral Hypercubes(FLAASH) algorithms [15].Assuming a ground terrain illuminated by the sun, the lightincident on a pixel in the sensor can be roughly characterizedby four sources: solar radiation directly reflected off theground, light directly reflected off the atmosphere into thesensor, light scattered by the atmosphere and reflected offthe ground, and light that is reflected off surrounding regionson the ground and then scattered before reaching the sensor(constituting the adjacency effect) [17], [18]. These effects areillustrated in Fig. 3. A model for the reflectance at the sensor y sensor is given by [15]: y sensor = y atm T g + y s T g T ↓ T ↑ + ( y avg − y s ) T g T ↓ T ↑ r − y avg s (3)where y s is the surface reflectance, T g is the gaseous trans-mittance, y atm the reflectance of the atmosphere, T ↓ and T ↑ are the upward and downward scattering transmittances, r is the ratio between diffuse and total transmittance forthe ground-to-sensor path, s is the spherical albedo of theatmosphere, and y avg is the average surface reflectance in aregion around a pixel, which is used to account for scattering(adjacency) effects [15].Physics-based atmospheric correction algorithms then try toobtain the ground reflectance y s from the at-sensor reflectance y sensor by solving (3). In the overall working of these algo-rithms the first step for atmospheric compensation consists inretrieving the atmospheric parameters necessary to representthe quantities in (3), mainly consisting of aerosol description(visibility and type of aerosol) and amount of water vaporfor each pixel [19]. They are typically based on variationsof the so-called three-band ratio technique, which is an im-portant step used to quantify the amount of water vapor foreach pixel. The three-band ratio technique basically comparesratios of radiances measured near the edges of a numberof spectral wavelengths which are known to present heavywater-vapor absorption (e.g., at around 0.91 µm , 0.94 µm and 1.14 µm ), using this information to derive the columnwater vapor information for each pixel [5], [20]. After thenecessary parameters have been estimated, (3) can be solvedfor the ground reflectance and an optional post-processing stepcan be employed (called spectral polishing) to remove artifactsfrom the correction process [19].Physics-based models can represent and account for theinteraction between solar radiation and the atmosphere veryaccurately. However, for this accuracy to translate into mean-ingful surface reflectance estimates, these models requireprecise information about atmospheric properties, which arevery difficult to obtain in practice. This is specially truefor scattering and absorption by aerosols, which are hardto characterize accurately due to their spatial and temporalvariability [21]. Inaccuracies in the estimation of these pa-rameters (which include the atmospheric visibility, aerosolmodel type and an atmospheric model) introduce errors inthe retrieved surface reflectance spectra that can be significantand spectrally non-uniform [22].Furthermore, unlike water vapor compensation, which isperformed on a pixel-by-pixel basis, most methods assumethat individual aerosol and gas concentrations are uniformacross the scene (resulting in a single transmittance spectrumbeing computed for each gas) [19], [22]. While this is truefor some gases (such as NH , O , CH , CO , etc.) thatare fairly constant in the atmosphere [20], it is far fromtrue for aerosols, which may show significant variation inspace [23], [24]. Aerosol concentration can vary depending onthe environment (e.g., in large cities and rural areas), and thusmust be informed by the user to the existing algorithms [20].Moreover, standard aerosol types often do not adequatelyrepresent the scene being processed, leading to inaccuracies in the retrieved spectra [25]. Furthermore, experimental studieshave found that aerosol optical thickness has a significantspatial variability within a single scene [23], [26] and is oftencorrelated with cloud concentrations [26].Some works attempted to estimate aerosol optical thicknessfor smaller patches of the image individually using shadowdetection results [27], which depends on the presence of alarge number of shadowed pixels. However, acquiring precisedata for an accurate and possibly spatially variable atmosphericcorrection is generally difficult, which means that the resultsof common atmospheric compensation methods can be subjectto significant errors [23]. For instance, a number of studieshave investigated the residual errors in surface reflectance dataafter the application of atmospheric compensation methodsby comparing the processed results with in situ data orusing simulations. These studies found that generally thereis still an appreciable error in the retrieved reflectances. Asan example, errors in the retrieved reflectance by atmosphericcorrections due to the spatial variability of aerosol opticalthickness over southern England were found to be of up to1.7%, with 5% errors in the normalized difference vegetationindex (NDVI) [23]. This can be significant for practicalapplications, as it corresponds to errors of up to 30% inbiomass production estimates [23], [28]. Furthermore, standardmethods for column water vapor retrieval loose accuracy whenthe aerosol optical thickness is high, leading to errors of upto 10% if aerosol effects are not properly compensated [29].Note that experimental measurements in a water quality man-agement application found significant differences between thetrue and retrieved spectral responses. Errors of up to 15%in reflectance spectra were found, more prominently concen-trated in short ( <
450 nm) and long ( >
750 nm) wavelengthintervals [30]. Another study evaluated a number of physics-based atmospheric correction methods in an experiment for a playa and canola target and found that although the averagerelative differences were moderate, ranging between 0.023and 0.042, larger deviations of up to 0.12 occurred in the near-infrared region [31]. A study with simulated data found thatincorrectly supplying input parameters to the model used inthe FLAASH algorithm can lead to considerable errors in theretrieved reflectance, with an absolute difference of up to 0.11,and a strong sensitivity to moisture/optical depth (visibility)errors [22]. Also, very large errors can be introduced bya bad specification of the aerosol model type, with highererrors generally present in short wavelengths where scatteringprocesses are most significant [22].The influence of uncertainties in column water vapor andaerosol optical depth specification on SU was investigatedin [24] (given their influence in the retrieved reflectances).The performance degradation was found to be more severe inabundance than in reflectance estimation, with degradation ofup to 30% in high scattering conditions. The results were moreseverely affected due to uncertainties in water vapor amountthan in aerosol optical thickness, although the latter showed astrong influence on the quality of the reconstructed abundancemaps when the endmembers were spectrally similar.Finally, it is interesting to highlight that two characteristicswere noticed from these studies. First, the errors in the retrieved reflectances are fairly non-uniform in spectral bands,with large spikes often concentrated near bands where there issignificant gas/water absorption [22], [24]. Second, errors dueto bad aerosol specification are quite significant in short wave-lengths (450 nm-750 nm), where they are concentrated [22],[30]. All these effects are illustrated in Fig. 4.
Figure 4. Illustration of variability caused by atmospheric effects. B. Illumination and topographic effects
Varying illumination conditions are one of the main sourcesof spectral variability in spectral mixture analysis [32]. Illumi-nation changes are mainly due to two effects: varying terraintopography, which affects the angles of the incident radiation,and occlusion of the light source by other objects (leading toshaded areas).A number of work handled the presence of heavily shadedareas by considering the presence of an additional endmemberrepresenting shadow [33]–[39]. Although this approach is verysimple, its effectiveness is certainly limited since a singlespectral signature can be insufficient to adequately representall pixels affected by shadow [40]. For instance, there might bemany shadow endmembers since shadows in different regionsof the image are influenced by both the material that is beingshaded and by the absorption properties of the material that isblocking the light, what might lead to significantly differentspectral signatures [41]. Furthermore, besides presenting alower reflectance amplitude, the shadow EM is also usuallysignificantly affected by nonlinear atmospheric scattering andmultipath effects, since these areas are illuminated by a largeproportion of diffuse irradiation scattered by the atmosphere(i.e., skylight) and by other nearby objects. This impliesthat the shadow endmember is sensitive to the state of theatmosphere and can vary significantly in space depending onthe amount of scattered light being reflected from the sky ateach position [42], [43].When illumination predominantly comes from scatteredradiation, the spectrum not only presents a lower amplitudebut is also skewed to short (e.g., blue) wavelengths [44],[45]. This means that the signal amplitudes in the shorter(blue) wavelengths are considerably larger than in the rest ofthe spectra [45].Furthermore, since the shadow spectral signature is a func-tion of diffuse illumination, it depends on the neighboringimage area (where the skylight is scattered) [45] and on thecloud cover. Moreover, variations of ground reflectance maynot be easily discernible from atmospheric effects since botheffects are observed jointly and are not easily separable [45].
Figure 5. Examples of 30 pixel instances classified as red roof in thePavia image (in gray), which are primarily affected by illumination, and theirspectral average (in red). The average Pearson correlation coefficient betweeneach signature and the scaled version of the mean spectra that is closest toit is about . , indicating a good agreement between illumination-basedspectral variability and the constant scaling model. These facts introduce a strong dependence of the shadowsignature to the spatial position, and go against the commonnotion that shadow endmembers can be adequately representedby scaled versions of true endmembers [5] (that is only truefor small illumination variations).This makes the detection, correction or quantification ofshadow a challenging task, since physical-based inversion ofthese atmospheric effects turns out to be a hard problem. How-ever, this task is still necessary since linear SU with a singledark endmember usually does not successfully quantifies thepresence of shadow in the scene [45].Although the presence of shadows is common in hyper-spectral images, a more prominent source of variability comesfrom the varying topography of the scene, which introducescomplex fluctuations of the relative angles between the in-coming light source and the sensor for each pixel of the scene.Topographic variations have been shown to significantly affectspectral reflectance values of soil and green vegetation [46] aswell as rocks in lithologic mapping [47], expanding endmem-ber clusters and causing overlap between classes, hinderingthe endmember identification and unmixing processes.Considering that only the amplitude of the incident radia-tion changes along the scene, the reflectance spectra of theobserved pixels in the LMM becomes scaled by a constantpositive factor. This model agrees with the observation thatmost of the variability in a hyperspectral image can berepresented by a constant scaling of reference endmembers [5].As a simple empirical verification, we plot a random subsetof 30 pixels of red roofs from the Pavia image, which arepure pixels mostly affected by illumination effects. The results,which are depicted in Fig. 5, indicate that these pixels differmostly by a scaling factor.Although a constant scaling model is intuitive and simple,a more rigorous conclusion can be achieved by analyzing thedependence of radiative transfer models with the topographyof the scene. To this end, one could resort to the model devel-oped by Hapke [48], [49], which describes the bidirectionalreflectance (i.e., the reflectance as a function of the incidenceangles of the light source and observer/viewer depicted inFig. 6) as a function of the single scattering albedo and ofphotometric parameters of the material [50]. (cid:73)(cid:110)(cid:99)(cid:111)(cid:109)(cid:105)(cid:110)(cid:103)(cid:97)(cid:110)(cid:103)(cid:108)(cid:101) (cid:79)(cid:117)(cid:116)(cid:103)(cid:111)(cid:105)(cid:110)(cid:103)(cid:97)(cid:110)(cid:103)(cid:108)(cid:101)(cid:76)(cid:105)(cid:103)(cid:104)(cid:116)(cid:32)(cid:115)(cid:111)(cid:117)(cid:114)(cid:99)(cid:101) (cid:86)(cid:105)(cid:101)(cid:119)(cid:101)(cid:114)(cid:84)(cid:101)(cid:114)(cid:114)(cid:97)(cid:105)(cid:110)
Figure 6. Hapke’s model relates the reflectance to the incidence angles ofthe light source and observer/viewer shown in this figure, given the material’ssingle scattering albedo and photometric parameters [50].
Hapke’s model suggests a more complex relationship be-tween the endmember signatures and the topography. In thiscontext, the mixture of materials is assumed to happen atthe macroscopic level, allowing for the consideration of theLMM in the albedo domain, where Hapke’s model actsseparately on each endmember. Besides the dependency onthe spectral signature with photometric parameters, whichshall be discussed in the next section, the dependence on thesingle scattering albedo indicates that changes in incidentangles can affect each material in a pixel differently fromthe others, since the behavior of the reflectance as a functionof the angle is different for each material. This indicatesthat each endmember/material in a pixel can be differentlyaffected by topographic effects. Furthermore, the nontrivialrelationship between geometry and the spectral signaturesleads to a more complex variation than single scaling for eachendmember for high albedo materials [51], [52]. Besides, evensmall topographic variations can significantly affect the groundreflectance. For instance, in [53] experimental studies foundthat even small slopes (of less than 10 degrees) originatingfrom irregularities in tree canopy can lead to appreciable(enough to influence the results of subsequent tasks) changesin the measured reflectance of vegetation spectra. C. Intrinsic spectral variability
Another important source of spectral variability is the in-trinsic variation pertaining the definition of a material, whichis also called intrinsic variability . The characterization of thistype of variability has been prominently studied in the areaof vegetation monitoring, where it poses a huge challengeto the ability to identify tree species from spectral measure-ments [54], [55], and also to the characterization of soil andmineral spectra. Vegetation spectral signature can change dueto many factors, including micro-climates, soil characteristics,precipitation, presence of heavy metals and drought, foliageage and colonization by leaf pathogens [54]. The spectralsignature of soil is also heavily affected by variations in itscomposition and moisture content [56]. Furthermore, intrinsicspectral variability is also common in mineral spectra due to i.e. the ratio between reflected and received radiation, as a function of theviewing angle. (a) (b) (d) Figure 7. Samples of variation of spectra from the USGS library. (a) Alunite.(b) Muscovite. (c) Pyrite. variations in the grain size distribution and the presence ofvariable amounts of impurities [57], [58]. Moreover, it alsodepends on what level of detail is adopted to represent a givenmaterial (e.g., a tree endmember may possibly be split into trunk and leaf endmembers), which is generally applicationdependent [59]. Although imposing a large impact on theendmember spectral signatures, the dependence of intrinsicspectral variability on physico-chemical parameters, which areusually unknown, makes it very hard to tackle.One characteristic consistently observed in experimentalstudies is the smoothness of the observed spectra (i.e., thereflectance varies slowly between spectral bands). This behav-ior can be taken into account when designing SU algorithms.Moreover, unlike spectral changes caused by illuminationand topography effects, intrinsic spectral variability frequentlypresents a considerable dependence of the variability ampli-tude with the spectral wavelength. For instance, the signaturesof different instances of minerals in the USGS library depictedin Fig. 7 show complex dependence between the reflectancevariation and wavelength. The samples from alunite and mus-covite show a variability that is far from uniform across thespectrum. Moreover, different instances from pyrite displaycomplex variation, which is not consistent across all samples,occurring independently in different regions of the spectra.This behavior has been verified in similar experimental studiesin other works, and poses a significant challenge for differen-tiating mineral classes based on their spectral signatures [60].These characteristics are even more prominent and wellknown in the spectral variation of vegetation reflectance,which shows significant dependency on the wavelength andbehaves very differently in visible, near-IR, and short-wave-IR ranges [61]. This means that a simple scaling of a referencespectral signature is usually not sufficient to account forvariations within tree species [54]. Extensive experimentalstudies support this claim. In [54] the author found that thevariation of spectral reflectance in the visible and near-infraredregions can occur independently when measuring tropicalforest canopy in Brazil. Similar inhomogeneity in spectralvariation was also observed in other studies with tropicaltree species [62] and also in many distinctive environments,including conifer [63] and boreal tree species [64]. Similarnon-uniform variation trends are also consistently observed inseasonal changes as indicated by many experiments, includingin salt marshes [65], semi-arid environments [66] and borealtree species [67]. Furthermore, nonuniform spectral variationshave also been observed in samples from mineral, soil androck spectra [60]. (a) (b) (c)
Figure 8. Reflectance spectra for vegetation generated with the PROSTECT-Dmodel [76] for varying degrees of (a) chlorophyll content, (b) equivalent waterthickness, and (c) dry matter content.
Numerous works model the spectral signature of materialsas a function of photometric or chemical properties of themedium, being based on either radiative transfer modelling orin empirical approaches. A well known example is Hapke’smodel, which describes the spectra of a surface composed ofparticles as a function of parameters such as surface roughnessand density and size of the particles [48], [49].Another prominent line of work models the spectral char-acteristics of vegetation and soil samples as a function ofbiophysical parameters [68]. Models of this kind have beenapplied for the estimation of leaf biochemistry from theobserved spectra. An important example consists of the char-acterization of leaf reflectance spectra as a function of leafbiophysical parameters [68], for which a wide variety ofmodels have been used, ranging from a simple description ofleaf scattering and absorption properties to complex modelswhich perform a detailed description of the plant cells’ shape,size, position, and biochemical content [68]. Some instancesof those models include the characterization of the spectra ofbroadleaf vegetation as a function of leaf mesophyll structure,pigment and water concentration [69] or as a function of leafangular profiles [70], and of pine needles as a function of cel-lulose, lignin and water content [71]. Other works model soilreflectance spectra as functions of moisture conditions [72]–[74], and snow albedo as a function of snow grain sizes andliquid equivalent depth [75].As an illustrative example, we generated spectral signaturesof vegetation spectra using the PROSPECT-D model [76] as afunction of varying degrees of chlorophyll content, equivalentwater thickness and dry matter content. The resulting signa-tures, depicted in Fig. 8, show that intrinsic spectral variabilitycan present complex patterns and non-uniformity, as it is oftenconcentrated in specific regions of the spectrum.Through their analytical characterization of EM spectra,these kinds of models confine spectral variability to lie ona low-dimensional manifold. This constitutes important infor-mation that can be leveraged to alleviate/reduce the severeill-posedness of unsupervised SU problems accounting forspectral variability.Another important characteristic is that endmembers af-fected by intrinsic spectral variability usually display signifi-cant spatial correlation [77]. For instance, many experimentalgeostatistical works evaluating the spatial distribution and vari-ability of the physico-chemical properties of the soil (e.g., sandand clay concentration, electrical conductivity, pH, compactionand available elements such as nitrogen, phosphorus and potas-sium) have reported significant spatial correlation/smoothness (a) (b) (c)
Figure 9. Spatial behavior of endmember variability. (a) Soil subregion ofthe Samson image (highlighted by a red square). (b) Euclidean distance and(c) spectral angle between each pixel and the average spectra of the region. in these properties. Reports include measurements performedin Rhodes grass crop terrain [78], calcareous soils [79], ricefields [80] and tobacco plantations [81]. Besides directlyimpacting the spectral signature of the soil, these character-istics have been widely acknowledged to directly influencevegetation growth (e.g., they show strong correlation with cropproductivity [78]), and hence their spectral signature [61], [78].Therefore, spatial correlation in the variability is expected bothin soil/terrain and in vegetation signatures. A similar behaviorhas also been observed in mineral spectra in the presenceof spatially correlated grain size distributions and impurityconcentrations [57], [58]. This implies that the variability tendsto be small in small spatial neighborhoods, even though it maybe large across a large scene. This fact can be leveraged todesign SU algorithms since it supplies information that can beused to reduce the severe ill-posedness of the problem.To illustrate this effect, we performed an experiment bymeasuring the spectral variability in a homogeneous region(composed by mostly pure pixels) of soil in the Samson image,depicted in Fig. 9-(a). We then computed the Euclidean distance and the spectral angle between each soil pixel andthe average spectra of all pixels in the subregion, whichwas used as a reference material signature. The results aredepicted in Figs. 9-(b) and 9-(c), where it can be seen that thevariability shows strong spatial correlation, as observed bothin the Euclidean distance and spectral angle.III. U NMIXING M ETHODS T HAT U SE S PECTRAL L IBRARIES
Methods that use Spectral Libraries: + The methods are usually conceptually simple andeasy to interpret − The quality of the SU results depends strongly onthe spectral libraryOne of the main approaches to address spectral variabilityin SU is to consider large libraries of spectra acquired a priori .These libraries contain different instances of each materialin a scene, and the unmixing problem becomes generallyequivalent to finding which of these signatures can bestrepresent each pixel in the scene. Different algorithms havebeen proposed for this task, which we review in the sequel. The Euclidean distance between x and y is computed as (cid:113) N (cid:80) Ni =1 ( x i − y i ) . The spectral libraries used by these methods are sometimescalled bundles , and should in principle account for all possiblevariations of each material. Mathematically, they are repre-sented as M p = (cid:8)(cid:102) m p, , . . . , (cid:102) m p,M p (cid:9) , p = 1 , . . . , P (4)where M p is a library/bundle containing M p reference spectralsignatures (cid:102) m p,i ∈ R L of the p th material, and P is thenumber of materials in the scene. The spectral signature ofeach material in the n th pixel y n of a hyperspectral imageis then represented as an unknown element m n,p ∈ M p belonging to this bundle.Those sets can be readily used to constrain the endmembermatrices of the LMM for the N pixels to belong to a new set M n ∈ M , with n = 1 , . . . , N , where M = (cid:110) [ m , . . . , m P ] , m p ∈ M p , p = 1 , . . . , P (cid:111) , (5)is the set of all possible endmember matrices, with (cid:81) Pp =1 M p elements. This definition assumes that only one signaturefrom each library M p , p = 1 , . . . , P is present in eachpixel. However, other representations of the EM signaturesas, e.g., sparse or convex combinations of the elements in M p can also be considered in order to obtain more flexibility(see, e.g., [82]–[84]). Such strategies will be discussed inSections III-A and III-B.Different methods have been proposed to solve the SUproblem using spectral libraries. These can be roughly di-vided into four groups of formulations: MESMA, Sparse SU,machine learning, and spectral transformations. The MESMAalgorithm and its variants formulate SU as a computationallydemanding optimization problem, and achieve good quality.Sparse SU formulations use mathematical relaxations to theMESMA problem that are computationally easier to solve.Machine learning algorithms provide more flexible ways todo SU but also at a large computational complexity. Spectraltransformations are empirically-oriented techniques that canbe used to improve methods from the first three categories.We review each of these approaches in the following. A. MESMA and variants for small spectral libraries
MESMA and Variants: + Generally provide good SU results + Are easy to setup (few or no parameters) − Have a very high computational complexity − The results depend strongly on the quality of thespectral library availableThe Multiple Endmember Spectral Mixture Analysis(MESMA) algorithm [37] and its variants (sometimes alsoreferred to as iterative mixture analysis cycles ) are amongthe most widely used algorithms for this task. These methodsallow the endmember signatures to vary on a per-pixel basiswhile following the model in (4). The unmixing problemis solved by searching for the endmember and abundance MESMA and
Sparse SU are the mainmethods based on spectral libraries. Thebasic principle behind MESMA is to iter-atively search for the combination of EMsignatures in the library which, amongall possibilities, allows for the closestreconstruction of each observed pixelunder the LMM. Sparse SU, on the otherhand, performs EM selection and abun-dance estimation in a single optimizationproblem using sparsity and structuringconstraints and penalties, which allowsfor faster processing times.
Figure 10. Illustrative description of MESMA, fuzzy and Sparse SU techniques. combinations that result in the smallest reconstruction error(RE) for each observed pixel, i.e., arg min a n , M n (cid:107) y n − M n a n (cid:107) subject to M n ∈ M , a n ≥ , (cid:62) a n = 1 . (6)The endmember matrices M n constructed by taking spectrafrom the bundles are sometimes called endmember models .The MESMA algorithm has been employed in a widevariety of situations, including natural, urban and extra-terrestrial environments [9, p.1607], and in single and multi-date scenarios [85]. However, even though MESMA is veryamenable to parallelization [86], it consists of a combinatorialoptimization problem whose associated computational cost canbecome very high. More specifically, its computational costscales as the product of the sizes of the individual libraries, asit consists of solving an FCLS problem (cid:81) Pp =1 M p times [87].This can make the complexity of unmixing unrealistic forlarge library sizes. Furthermore, the problem (6) can becomeill-posed when there are many endmembers in the bundles,since different material combinations can lead to very similarreconstruction errors. In order to circumvent these limitations,several modifications to the original MESMA algorithm havebeen proposed.Many variants of MESMA aim to provide computationallyefficient approximate solutions to (6). Early simplificationsconsist of an early stop of the exhaustive search optimizationprocedure (6) by selecting the first EM model that presents areconstruction error that is both below a threshold and well dis-tributed across spectral bands [37]. Another approach proposedis to solve (6) approximately by performing unconstrainedleast squares with every possible endmember model, and thenselect the solution that yields positive abundances and thesmallest reconstruction error [88].Although these simple modifications successfully reducethe computational complexity of MESMA, the approximationsinvolved can also negatively impact the abundance reconstruc-tion results [89], what imposes practical limitations on theselection of the thresholds and tolerances. This motivated theconsideration of more elaborated strategies to provide moresignificant reductions of its complexity without impacting theunmixing performance. An alternative approach to MESMA attempted to lessen thecomputational complexity by solving an angle minimizationproblem with respect to each library separately [87], [90].Although not guaranteed to converge to the optimal solutionof (6), this strategy performed similarly to MESMA on prac-tical experiments, and scales linearly with the library sizes,leading to computational improvements for large numbers ofsignatures M p , p = 1 , . . . , P , in the EM bundles. Anotherwork considered a mixed integer linear program (MILP)reformulation of the MESMA problem. This approach allowsfor a more efficient computation of an exact solution to (6)for small to medium scale problems [91].A simple approach which is largely employed to reducethe computational complexity of MESMA is to perform acareful pruning of the spectral libraries M p , p = 1 , . . . , P .This process attempts to remove redundant or irrelevant spectrafrom the libraries before unmixing. These approaches will bedescribed in detail in Section V-B1.Besides reducing its complexity, other approaches modifyMESMA with the purpose of improving its accuracy. For in-stance, an early practice attempts to alleviate the ill-posednessof MESMA by prioritizing models with a smaller numberof endmembers, for otherwise comparable reconstruction er-rors [92], [93]. This avoids increasing the complexity of themodel for marginal gains. When consideration of materialnuances is important, it may be important to allow multiplesignatures of the same broader endmember class in the model.This was the case in [94], where effects such as differentvegetation species in a single pixel were of interest. Spatialinformation has also been considered with MESMA by usingsegmentation algorithms to divide the image into different ho-mogeneous objects, which are then unmixed individually usinga library also constructed from object-based spectra [95], [96].A different formulation attempted to increase the flexibilityof MESMA by allowing the endmembers of each pixel tobe represented as a sparse, non-negative combination of thesignatures contained in the library for their respective materialclass [82], [83]. Under this model, SU was then formulatedas a non-convex optimization problem with different sparsityconstraints, including both L / [82] and L -norm-basedpenalties [83]. This problem was solved using a multiplica-tive update rule in [82], and using the proximal alternating linearized minimization method in [83].Another set of approaches related to MESMA is referredto as fuzzy unmixing . These methods consider a measure ofuncertainty or indeterminacy in the estimated abundances bycomputing quantities such as average, maximum and minimumcover fractions. One of the first approaches of this kindused linear programming methods to determine maximumand minimum fractional abundances for each material usingspectral libraries extracted from the observed image [97].Another approach attempted to determine the abundance inde-terminacy (i.e., its fuzzy membership amount for each valueof abundance fractions) by evaluating how close syntheticallymixed spectra with all possible endmember combinationswere to the observed pixel spectra y n [98]. This procedure,however, required the discretization of the abundance valuesand its computational complexity does not scale well with thenumber P of endmembers classes.Other approaches performed linear SU with a large numberof endmember models selected at random from the library.Afterwards, measures of uncertainty in the estimated fractionalabundances such as maximum, minimum and average coverfractions were computed from these results, providing a moredetailed characterization of the abundances [99]–[101]. A sim-ilar work proposed to compute the final abundance fractionsas a weighted sum of the abundances obtained from SU witheach possible combination of signatures drawn from the library M [92]. The weights corresponded to the probability of eachEM model being actually present in the scene, which wassupposed to be known a priori . B. Sparse unmixing
Sparse Unmixing: + Generally is very computationally efficient (espe-cially compared to MESMA) − SU results might not be as accurate as MESMA − Can be harder to interpret (e.g., it might selectmultiple signatures of the same material to repre-sent a given pixel) − SU results are sensitive to the selection of theregularization coefficientsAn alternative approach to perform spectral unmixing withspectral libraries is to formulate the SU as a sparse regressionproblem, where we want to select a small number of spectralsignatures from the library which can best represent eachobserved pixel according to the LMM.Most sparse unmixing methods are based on an unstructured library, which can be derived from (4) by concatenating all thesignatures in a single matrix M Lib , defined as: M Lib = (cid:2)(cid:102) m , , . . . , (cid:102) m p,k , (cid:102) m p,k +1 , . . . , (cid:102) m P,M P (cid:3) . (7)Using the spectral library defined in (7), the sparse un-mixing problem can be formulated as the optimization prob-lem [102], [103]: arg min a n ≥ (cid:107) y n − M Lib a n (cid:107) subject to (cid:107) a n (cid:107) ≤ P, (cid:62) a n = 1 , (8) where (cid:107) · (cid:107) is the L pseudo-norm, which counts the numberof non-zero elements in a vector. Different strategies have beenproposed to solve the sparse SU problem using the L pseudo-norm using, for instance, greedy (e.g., matching pursuit orforward-backward) algorithms [104], [105], Lagrangian func-tion (regularized) formulations [106], or multi-objective opti-mization procedures that consider the reconstruction error andthe sparsity of the solution jointly [107]–[109]. Note that (8)would be equivalent to MESMA if we added an additionallinear structuring constraint to enforce the occurrence of onlya single nonzero abundance per material class [91].The optimization problem (8) is, however, non-convex andgenerally NP-hard to solve. It is therefore common to relax the L pseudo-norm constraint into its convex surrogate, leadingto the following optimization problem [102]: arg min a n ≥ (cid:107) y n − M Lib a n (cid:107) + λ (cid:107) a n (cid:107) (9)where (cid:107) · (cid:107) is the L norm and the parameter λ controls thelevel of sparsity of the estimated abundances. The sum-to-oneconstraint is not used in (9) due to its incompatibility withthe L norm [102]. Although problem (9) is non-smooth, it isconvex and can be solved very efficiently. Besides, it producedgood experimental performance. This motivated a great dealof interest in sparse unmixing methods, resulting in a numberof works proposing improvements such as the use of alter-native sparsity promoting penalties [110], [111] or differentmeans of spatial regularization [112], [113]. Sparse unmixingmethods would merit a more comprehensive review, whichis beyond the scope of this paper. Thus, in the following werestrict ourselves to modifications of the sparse SU frameworkspecifically aimed at dealing with spectral variability or withstructured libraries.In [114], L , -norm based group sparsity constraints havebeen used to favor the selection of abundance vectors con-taining many entire material classes with zero proportions.A later formulation considered a fractional group ( p, q ) -normsparsity constraint as a generalization of the approaches basedon the L , -norm [84]. The ( p, q ) -norm penalty permits abetter control of the sparsity within each group of variables,as well as the addition of the sum-to-one constraint. However,this comes at the expense of making the optimization problemnon-convex.Another sparse SU formulation [115] proposed to explicitlyrepresent mismatches between the library spectra and thehyperspectral image caused by different acquisition conditions.In this case, the spectral signatures of the library are alsoestimated in the SU process. However, they are constrainedto be within a given Euclidean distance of a corresponding el-ement of the library known a priori . This allows the estimatedsignatures to vary arbitrarily within Euclidean balls centeredat the library elements to compensate for spectral mismatches.A different approach [116] proposed to modify the LMMfor unmixing mineral spectra in mining applications by in-cluding an additional term representing the mixture of the“background” spectrum of the endmembers. This backgroundspectrum was defined as the low-frequency part of the spectralsignatures, and was estimated a priori from the library as a parametric function of smooth splines. The performance of an L -norm based sparse SU framework under this model wasreported to be similar to MESMA, albeit at a much smallercomputational cost. C. Machine Learning Algorithms
Machine Learning Algorithms: + Very flexible approaches, in principle can dealwith any effect that is represented in the trainingdata − Most methods either have a large computationalcomplexity or do not have a clear physical moti-vation − The SU quality depends on the representativenesof the training data (which is usually generatedusing a spectral library) − Generally do not return an EM spectra for eachpixelSome works propose to address spectral variability usingmachine learning methods by formulating SU as a supervisedregression problem. The objective is to learn transforma-tions mapping the observed (mixed) pixel to the abundancefractions [117]–[120] using a supervised training procedure.Mixed pixels with known proportions are employed as trainingdata for algorithms such as neural networks, random forestsor Support Vector Machines (SVMs). These techniques canbe straightforwardly adapted to address spectral variability byconsidering multiple spectral signatures for each endmemberwhen generating the synthetic training dataset. This has beendone either by directly applying regression methods [121] orby converting SU into a classification problem by quantizingthe solution space of abundance values and using a one-against-all strategy [122]. Another work modified the SVMcost function to directly minimize the unmixing reconstructionerror during the training process [123].Usually, these approaches result in extremely large trainingsets for large spectral libraries. Thus, even though somestrategies such as bootstrap aggregation have been employed tospeed-up the training process [124], [125], the computationalcost is still very high. Although methodologies to discardirrelevant (regarding the impact on performance) subsets ofthe training data [126], [127] could in principle be appliedto accelerate training, recent works have instead focused onmodifying the algorithms to reduce their complexity.One of the main reasons for this large complexity is thatthe training data must jointly describe spectral variations dueto changes in both the abundances and in the endmembers.Recent works have tried to address this issue by using onlypure pixels from a spectral library as training data. One suchapproach, which received considerable attention, consists ofextended SVMs. Extended SVMs employ hybrid soft-hardclassification or regression to address spectral variability. Itis assumed that the spectral space is separable by hyperplanesdelimiting two complementary regions containing only pureand only mixed pixels, respectively [128]. The extended SVM is then trained to find a soft-hard classifier containing both1) a hard classification rule consisting of the hyperplanesdelimiting the regions in which the pixels are consideredpure, and 2) a soft classification rule which determines theabundances of the pixels considered to be mixed.Different forms of the extended SVM have been considered,using either a single [128] or multiple kernels [129], consider-ing the abundance indeterminacy by computing the maximumand minimum proportion values similarly to the fuzzy SUprocedures [130], or using Fisher discriminant analysis toreduce the within-class spectral signature variability in thespectral library before training [131]. Although hybrid soft-hard classification methods can be fast to train, they lack aclear physical interpretation of the results since they have nodirect relation to the physical mixing model. Moreover, theinfluence of spectral variability on the regions of the spectralspace containing mixed pixels is limited since it only comesfrom the marginal hyperplanes that separate the pure from themixed pixels regions [128].A related strategy that also uses only pure pixels in thetraining process consists of modeling the latent function fromthe mixed pixel spectra to the abundance maps in a proba-bilistic framework as a multi-task Gaussian Process [132]. Inthis case, the abundance means and covariance matrices areobtained through the posterior distribution of the abundancesconditioned on the training set (i.e., the spectral library) andon the mixed pixels. This strategy was also extended toconsider spatial correlation in a two step process by usingthe Gaussian Process results from [132] as input to the abun-dances prior information in a maximum a posteriori estimationproblem [133]. Although this strategy has a strong statisticalmotivation, the introduction of additional constraints (e.g.,abundance non-negativity and sum-to-unity) is not straight-forward and results in high computational complexity. D. Spectral transformations
Spectral Transformations: + Can be seen as a “pre-processing” strategy thatcan be used jointly with other library-based SUmethods + Are conceptually simple and of low-complexity − Many of the methods are empirical and require asignificant degree of expert knowledge about theunderlying application − The performance of the less-supervised methodsdepends strongly on the representativeness/qualityof the libraryAn approach frequently used to mitigate the effect ofspectral variability in library-based SU consists of selectinga subspace of the spectral space that is minimally influencedby the variability of the endmembers to be prioritized in theunmixing process. This idea was first introduced to improvethe classification of materials under varying atmospheric illu-mination conditions [17], [20]. The flexibility and representation powerof
Machine Learning algorithms canbe exploited to address spectral variabil-ity by formulating SU as a supervisedlearning problem. One simple approachis to learn a mapping between the mixedpixels and the abundance and EMs basedon training data generated syntheticallyusing a spectral library. However, theincorporation of expert knowledge aboutthe SU problem in the design of themachine learning algorithm is importantto obtain a better performance and toaddress spectral variability effectively.
HiddenLayers + +
Figure 11. Illustrative description of machine learning-based SU techniques.
The majority of these methods are based on affine transfor-mations of the observed pixels defined as W λ y n + b n = W λ M n a n + W λ e n + b n (10)where the matrix W λ and the affine term b n are deter-mined to minimize the effects of endmember variability inthe subsequent SU process. Besides modifying the observedpixel spectrum y n , this transformation is also applied to theelements of the spectral library, yielding: W λ M p + b n (cid:44) (cid:8) W λ m + b n : m ∈ M p (cid:9) , (11)for p = 1 , . . . , P . Different particular cases of this modelhave been considered in the literature, most notably with W λ being a diagonal matrix with positive real (band weighting)or binary (band selection) elements. Note that although tradi-tional dimensionality reduction (e.g., PCA) or band selectionmethods used to compress the hyperspectral image could beimplemented using this transformation with b n = , the directapplication of compression techniques does not necessarilyimprove the robustness to spectral variability [134].Spectral transformation approaches can be generally dividedinto two major groups: those defined a priori based on expertknowledge by the user, and those constructed automaticallyusing information in a spectral library. We will review eachcase in the following. User-defined spectral transformations
The first user-defined spectral transformations were pro-posed to normalize the effects of illumination and brightnessvariations, or to emphasize useful spectral features. Theseapproaches include subtracting the reflectance value of aselected (specific) spectral band from all remaining bands [99],subtracting from each endmember its mean value in the spec-tral dimension to reduce the variability due to differences inbrightness [135], or normalizing/dividing the reflectance valueat each wavelength by the corresponding value of the convexhull of the spectral signatures [136]. Other examples alsoinclude using the first or second derivatives [137], [138], orthe wavelet transform of the spectral signatures [134] for SU.A later spectrum-based approach that has become very pop-ular for solving this problem consists of using band selection methods. These methods basically work by performing SUusing only selected wavelength intervals in which there is littlespectral variability between different spectral signatures of thesame material [9], [99]. Although many of these approachesrely on expert knowledge about the specific underlying ap-plication, they are simple and easily interpretable and alsohelp in reducing the computational cost of the SU problem.Examples of band selection methods defined a priori by theuser include the selection of the the SWIR2 spectral region(2100–2400 nm) for unmixing of soil and vegetation in aridand semi-arid environments [99], and the combination ofvarious spectral regions such as visible, NIR and SWIR forother applications [100]. Library-based spectral transformations
Spectral transformations proposed more recently leverageinformation contained in the spectral library to compute theterms W λ and b n of the affine transformation. This circum-vents one of the main downsides of the previous approachesby making the process automated instead of delegating thechoice to the user. These techniques can be further dividedin three groups, namely band selection, band weighting, andmore general spectral transformations. a) Band selection
Band selection methods proposed more recently seek toidentify the robust spectral regions based on the samples inthe spectral library. Different strategies have been proposed.One of the first approaches is based on the analysis ofthe spectral residuals obtained by performing a preliminaryunmixing of the image using the LMM with an averageEM matrix [139]. Only the spectral bands with minimalresidual variance are then used for SU, based on the empiricalobservation that they correspond to more robust spectral zones.Another method, called stable zone unmixing, proposes toselect spectral bands that are robust to spectral variability byminimizing an instability index defined as the ratio betweenthe intra-class and the inter-class endmember variances (com-puted based on a spectral library) [140]. This method waslater extended in order to minimize both the instability indexand the correlation between signatures of different endmemberclasses at the same time, aiming to improve the numerical conditioning of the SU problem [141]–[143].The work [144] proposed to improve the separability be-tween classes by employing the stable zone unmixing frame-work to select an individual set of spectral bands for eachpossible subset of endmember/material classes that could betested with MESMA when considering endmember modelswith fewer then P signatures in the SU process. b) Band weighting
Band weighting methods are more flexible techniques whichallow one to prioritize the spectral bands in the unmixingprocess according to their reliability or significance using acontinuous weight term. This is usually done by weightingthe reconstruction error of each band in the SU cost function.Different approaches have been proposed to compute theweight to be applied to each band. For instance, a weightingstrategy based on two terms was proposed in [145]. One termnormalizes the energy of the reflectance spectra to equalize thecontributions to low- and high-reflectance bands, and anotherterm accounts for the robustness of each band to spectralvariability using its instability index (i.e., ratio between intra-class and inter-class endmember variance). This approachwas later applied to monitor both the level of defoliation inEucalyptus plantations [146] and invasive plant species usingmulti-temporal data [147]. It was also later extended in [148]to consider SU integrating both reflectance and derivativespectra. Band weights based on the instability index werealso used to prioritize the more stable spectral bands whendesigning spectral filters robust to spectral variability, whichare low-complexity alternatives that approximate the solutionof the SU problem as a direct application of a single lineartransformation [149]. c) General spectral transformations
Another group of approaches proposed to use more flexiblelinear transformations to better mitigate the effect of spectralvariability. These techniques consist of variations of the FisherDiscriminant Analysis (FDA), which is widely used for patternclassification. FDA aims to find a transformation of the datato obtain a feature space with the best separability betweendifferent classes [150]. In the context of SU, this amountsto minimizing the variance of the signatures of each materialwhile also maximizing the distance between the mean valuesof the different endmember classes [151]. Mathematically, thisis formulated as W λ = arg min W W (cid:62) S within WW (cid:62) S between W (12)where S within is the weighted sum of the within-class covari-ance matrices, and S between is the covariance matrix of themean endmember spectra.The first approaches applied FDA to SU directly by eitherusing spectral libraries known a priori [151] or constructedusing pure pixels extracted from the observed hyperspectralimage [152]. Another work also considered the augmentationof the spectral library with pure pixels extracted from theimage to improve the discrimination among spectrally similarvegetation species [153]. Later approaches considered othervariations, such as the iterative addition of more columnvectors to W λ using a Gram–Schmidt orthonormalization procedure to increase the dimensionality of its output spacefor multispectral images with a small number of bands [154].Another work proposed to make the spectral signatures of dif-ferent endmembers orthogonal to each other, and the spectralsignatures of the same endmember all unitary and collinear toimprove the numerical conditioning of the SU problem [155].The FDA was also successfully used to improve the perfor-mance of MESMA when unmixing urban surfaces (containingvegetation, soil, water and manmade materials) using image-extracted spectral libraries [156].In contrast with its improved flexibility, the FDA has asa downside its dependence on a good estimation of thecovariance matrices to be used in (12). Thus, the FDA maynot perform well if the amount of samples in the libraries isnot statistically representative [157].IV. U NMIXING M ETHODS T HAT E STIMATE THE E NDMEMBERS FROM THE I MAGE
In more recent years, a large number of works proposedto address spectral variability in SU without relying on priorknowledge about spectral libraries. Different strategies havebeen proposed to this end, which we divide into four groups.Local unmixing methods are both computationally and con-ceptually simple but require significant user supervision. Para-metric endmember models provide more flexibility to representEM spectral variability but make the SU problem harder tosolve. EM-model-free methods address spectral variability byusing different modifications to the SU cost function. Bayesianmethods use statistical representations for the endmembers,which leads to a smaller amount of user supervision at theprice of a high computational complexity. We will review eachof these approaches in the following. A. Local unmixing methods
Local Unmixing: + Conceptually simple and physically motivated + Computationally efficient − Usually requires a significant amount of usersupervision − The selection of the local image regions has asignificant impact on the results − Local EM extraction can be difficult − Grouping the local estimates into global results isalso challengingA conceptually simple and efficient method to deal withspectral variability is to perform both endmember extractionand spectral unmixing locally for small, non-overlapping re-gions of the hyperspectral image. This approach, called localunmixing , assumes the endmember signatures to be constantin each region of the image, benefiting from the knowledgethat spectral variability is often negligible in small regions.The basic framework of local unmixing can be summarizedinto the following steps:1) Divide the observed image into a set of regions; Local SU addresses the variability ofendmember signatures across space byperforming SU on small, compact spatialregions of the image in which the EMscan be assumed to be approximatelyconstant. The local SU results for eachimage region are afterwards clustered inorder to assemble the global abundancemaps and sets of EM spectra. Local SUoffers a lot of flexibility in the choiceof the segmentation of the image and ofthe local EM extraction and clusteringstrategies, which can have a significantimpact on the global SU results.
Figure 12. Illustrative description of local SU techniques.
2) Estimate the number of spectral signatures and extractthe endmembers in each region;3) Perform SU with the local endmember signatures;4) Combine all the local SU results into global sets of end-member signatures and global abundance maps using,e.g., clustering procedures.Although local unmixing methods proposed up to date sharesimilar overall methodologies, there are important differencesin the way the hyperspectral image is partitioned (e.g., usingsimple square tiles or more advanced image segmentation) andhow the endmembers are extracted from each region. This canhave a significant impact on the results.The first approaches for local unmixing required com-plete user supervision. For instance, the variable MESMA(VMESMA) algorithm proposed in [10] used manual imagesegmentation to divide the image into local regions. SU wasthen performed iteratively, updating the segmentation mapsand manually including additional endmembers in the processuntil a satisfying result was obtained. Later approaches at-tempted to reduce the need for user supervision in the process.For instance, endmember extraction and SU were performedindividually in local (square) image tiles in [158], [159].Afterwards, the locally extracted endmembers and abundancemaps were then merged into the global endmember sets andabundance maps using clustering algorithms.Image segmentation methods were later used to providemore flexibility when dividing the hyperspectral image intolocal regions. For instance, in [160] manual endmember ex-traction and spectral unmixing (using the FCLS algorithm)were performed individually in each image region defined by asegmentation algorithm. Another work considered a superpixeldecomposition of the image aided by external map metadatain order to compute a more accurate segmentation [161]. Amore sophisticated method was proposed in [162], [163] byusing a binary partition tree to divide the image into differentregions from a coarse to a fine spatial scale. Local unmixingwas then performed at the scale of the partition tree yieldingthe smallest reconstruction errors.Besides the choice of the segmentation procedure, endmem-ber extraction is also a challenging part of local unmixingand has a great impact on the performance of these algo- rithms. A spatially adaptive unmixing method was proposedin [164] to estimate the distribution of different surfaces inurban environments. Endmember spectra for each pixel weresynthesized as a weighted average of pure pixels extracted in aspatial neighborhood specified by the user, with weights givenas a function of their distance to each mixed pixel at hand.A similar approach used as endmembers the mean values ofpure pixels extracted within each (square) image region, whichwere identified using a classification strategy [165]. Theseapproaches can positively weight pure pixels that are spatiallyclose to each pixel being unmixed. This idea was also exploredin other works such as in [166], which performed SU using avariant of the MESMA algorithm, or in [167], which used onlythe spatially closest pure pixels to process each mixed pixel.Other local unmixing approaches considered hierarchicalsegmentation approaches in which the hyperspectral imagewas divided into two spatial scales, a coarse one where un-mixing was performed with MESMA, and a fine spatial scalein which the spectral libraries were extracted using either thespectral signatures of small and homogeneous objects [168]or a priori knowledge about the abundances obtained fromexternal high resolution classification maps [169].An important issue of local unmixing algorithms is thedetermination of the number of endmembers contained in eachlocal image region. While in most experimental works thiswas performed empirically or even manually, it is desirableto have automated methodologies to estimate the number oflocal endmembers and their spectral signatures. This usuallyinvolves the estimation of the intrinsic dimensionality ofthe local subset of the hyperspectral image [170]. However,the performance of intrinsic dimensionality estimators is of-ten negatively impacted when the size of the data set issmall [171]. This strongly limits the characteristics (i.e., size)of the subsets or segmentation procedures that are selected forunsupervised local unmixing. Collaborative sparse regressionapproaches [110] were proposed to deal with the shortcomingsof intrinsic dimensionality estimation by avoiding the selectionof repeated or mixed signatures during unmixing [172]. Thesparsity level was selected using a Bayesian information crite-rion in order to obtain a good compromise between small re-construction errors and a small number of selected signatures. A different line of work attempted to relax the assumptionof connectedness of the local spatial regions, performing SUin different subsets of the hyperspectral image which arenot necessarily spatially adjacent. For instance, the piecewiseconvex model proposed in [173] considered a set of differentendmember matrices, all estimated from the entire image. Eachpixel was then assigned to one of these EM matrices usinga (fuzzy) membership function, which was estimated alongwith the other variables in a non-convex matrix factorizationproblem. Other works extended this approach by consideringcluster validity indices [174] or sparsity promoting priors [175]to estimate parameters such as the number of EM matricesand the number of material classes in each segment, or usingspatial constraints to encourage neighboring pixels to havesimilar membership values [176].A similar work considered the estimation of multiple EMmatrices in a non-negative matrix factorization frameworkby using abundance sparsity constraints instead of employ-ing (fuzzy) membership functions, while also penalizing themutual coherence between the signatures of different materialclasses to improve inter class separability [177]. A relatedstrategy considered a self-dictionary model where the multipleEM signatures are selected directly as the hyperspectral imagepixels that can best reconstruct most of the remaining pixelsin the scene as a sparse linear combination [178]. Anotherapproach with even more flexibility considered an individualEM matrix for each image pixel in a non-negative matrix fac-torization formulation [179]. A regularization term penalizingthe trace of the covariance matrix of the estimated spectralsignatures for each class was also considered to reduce theill-posedness of the estimation problem. B. Parametric models
Parametric Endmember Models: + The SU algorithms are computationally efficient + Very flexible and physically motivated models torepresent any kind of variability + Easy to incorporate prior information − Determining a good EM model might requiresome degree of expert knowledge − Require significant user supervision for tuningfree model parameters − Estimating the parameters of the EM models(along with the abundances) can be challengingdue to the presence of non-convex optimizationproblems and sensitivity to parameter choice orinitializationA flexible and physically reasonable way to address spectralvariability consists of employing parametric models to repre-sent the endmember spectra. These strategies allow for greatfreedom to incorporate constraints and information from theunderlying application. They are generally based on represent-ing the EM spectra as M n = f ( M , θ n ) , (13) where f ( · ) is a function of an average or reference EM matrix M and of a vector of parameters θ n . The number of param-eters in θ n is usually small, which allows one to confine theendmember spectra to a low-dimensional manifold. The SUproblem is then formulated as the recovery of the abundancesand of the parameters θ n for all pixels of the image.The model in (13) can be defined either based on theunderlying physics describing material spectra as a functionof numerous geometric and photometric parameters, such asHapke’s [48], [49] or Shkuratov’s [180] for packed particlesand the PROSTECT or PROSAIL models for vegetation [68],[69]. However, the model (13) can also be inspired byphysics but chosen in order to allow for more flexibility andmathematical tractability. We will review these approachesin the following. Physics-based methods
The first SU approaches using parametric models aimed toobtain fractional abundances from intimate mineral mixturesby inverting the Hapke model [181]. With perfect knowledgeabout the viewing geometry, the scattering properties of thedifferent materials and the single scattering albedo of theEMs, the SU problem using Hapke’s model becomes linearin the albedo domain [182]. However, since these variablesare hardly available in practice, many works attempted toinvert Hapke and related models blindly. This inversion ismathematically and computationally very difficult in generaland requires hyperspectral images acquired at multiple viewinggeometries [181]. Thus, subsequent works proposed simpli-fications of the scattering characteristics of the materials inthe model (13) to improve its mathematical tractability [183].These methods have been successfully applied to estimateabundance maps at different scenes, including the Cupritemining district at Nevada [184] and the Moon [185], [186].This approach has also been applied to SU of vegetationmixtures based on the inversion of radiative transfer models.The first works simplified the problem by assuming externalknowledge of biophysical parameters. For instance, a modelfor mixtures of vegetation, shadowed and illuminated soil wasproposed for SU by approximating plant geometry with spa-tially distributed cylinders containing layers of leaves [187].Although spectral variability was allowed by means of changesin biophysical parameters, these were assumed to be known apriori to solve the SU problem. Another approach consideredSU of soil and vegetation using a simplified mixing modelas a function of NDVI values instead of the full spectralsignatures [188]. In this case, a physical model was used torepresent the variability of the NDVI values as a functionof parameters such as the viewing geometry, leaf densityand clumping effect. However, the NDVI “endmembers” foreach pixel had to be estimated before SU by using multi-angled observations and assuming prior knowledge of theleaf biophysical parameters. A later approach for SU of soiland vegetation mixtures proposed to estimate the biophysicalparameters blindly from the hyperspectral image using thePROSAIL model for vegetation spectra [189]. The SU problemwas formulated as the recovery of both the abundances andthe two parameters of the PROSAIL model, and solved usingan alternating optimization procedure. Note that the other Parametric EM models represent the(variable) signatures of the EMs as afunction of a low-dimensional vector ofparameters. The abundances and the vec-tor of EM parameters for each pixel arethen recovered by solving an optimiza-tion problem.
EM-model-free methods,on the other hand, generally attemptto mitigate spectral variability indirectlythrough the design of robust cost func-tions using, e.g., additive residual terms.The use of regularization terms is impor-tant in both cases to incorporate a priori knowledge about the problem.
Figure 13. Illustrative description of SU techniques based on parametric EM models and of EM-model-free SU approaches. parameters of the PROSAIL model had to be fixed a priori .Although these models carry a strong physical motiva-tion, their use in SU leads to computationally intensive andmathematically challenging (i.e., non-convex, significantly ill-posed [190]) problems. This occurs because physics modelswere originally devised as forward models that accuratelydescribe the reflectance spectra based on a set of parameters,and were not originally designed to be inverted, which limitstheir use for SU in practical problems [191]. Physically motivated and non-physics-based methods
The low mathematical tractability of physics-based modelshas motivated recent studies leading to more flexible orparsimonious models that are only inspired by the underlyingphysics. Although these models are not as precise as thosepresented in Section IV-B1 when representing physical phe-nomena underlying spectral variability, they allow for moreefficient SU algorithms estimating the involved parameters θ n from the observed image. Moreover, although models inspiredby physics can be ill-posed, the EM spectra are often confinedto a low-dimensional manifold since they only depend on asmall number of physico-chemical variables. This propertycan be exploited to design parsimonious models with possibleconstraints and reduce the ill-posedness of the SU problem.Several parametric models have been recently proposed withthese objectives. One of the resulting SU algorithms is thescaled constrained least squares method [18], which attemptsto represent uniform illumination variations in each pixel byintroducing an additional scaling factor ψ n ∈ R + in the EMmatrices as M n = ψ n M . (14)SU can be performed using model (14) by solving a simplenon-negative least squares problem, which is convex andcomputationally efficient. However, this model lacks capabilityto represent more complex spectral variability that have beenobserved in practical scenes, motivating the search for moreflexible models.An extended version of the LMM (ELMM) was laterproposed in [51], [192] by allowing each endmember in a pixel to be individually scaled by a constant factor, resultingin the following representation for the EM matrices: M n = M diag( ψ n ) , (15)where vector ψ n ∈ R P + contains the scaling factors for eachof the P materials. The ELMM can represent more complexvariability originated from variations in both illumination andtopography, which can affect each material in the hyperspectralimage differently. Furthermore, the ELMM can be obtainedfrom successive physical approximations of the Hapke modelfor small-albedo materials [52]. Based on an estimate of M obtained from the observed image, SU under the ELMMwas formulated as a non-convex matrix factorization problemin which the model (15) was enforced by means of anadditive penalty in the cost function [51]. A regularizationpromoting spatial homogeneity of the scaling factors ψ n was also considered to reduce the ill-posedness of the SUproblem [51]. The ELMM has also shown good performancefor multitemporal data [193] and has been used to facilitatethe interpretation of local unmixing results [194]. Moreover,the ELMM can be derived from a Taylor series expansionof a general nonlinear mixture model [195], what introducesSU with spectral variability (viewed as a locally linear SUproblem) as a direct way to address the general nonlinearSU problem. This shows that some mixture models originallydevised to represent spectral variability (such as the ELMM)can achieve good performance in nonlinear SU.Despite its physical motivation, the ELMM model lacksflexibility to represent more complex spectral variability, e.g.,affecting the spectra non-uniformly. To address this limitation,the generalized LMM (GLMM) was later proposed in [196] byintroducing an individual scaling factor for each band, leadingto the following EM model M n = Ψ n ◦ M , (16)where the matrix Ψ n ∈ R L × P contains the scaling factorsfor each element of M and ◦ denotes the Hadamard (ele-mentwise) matrix product. Note that the amount of spectralvariability brought by the GLMM is proportional to theamplitude of the reference spectra M in each band. However,the larger number of parameters makes the SU problem resulting from (16) more ill-posed with challenging estimationproblems. This motivated the development of a tensor inter-polation framework to estimate the matrices Ψ n from traininghyperspectral data obtained based on prior knowledge aboutthe positions of pure pixels in the hyperspectral image [197].However, the performance of the method proposed in [197]depends strongly on the amount of pure pixels available inthe image. The GLMM has also been successfully used inmultitemporal SU [198] and to represent spectral variabilitywhen fusing hyperspectral with multispectral images acquiredat different time instants [199].Note that the performance of unmixing methods basedon the ELMM and GLMM depends strongly on the qualityof the reference EM matrix M , which must be estimatedfrom the observed image. In order to reduce the dependenceof the ELMM on M , the authors of [200] proposed toestimate M jointly with the remaining variables during SU.Each column of M was also constrained to have a unit normin order to obtain EMs as directional data in the spectralspace. Moreover, M was initialized using a simple cosine-based k-means clustering of the observed data-cube, whichimproved the robustness of the method to the presence ofshadowed pixels.A different EM model was proposed in [201] by consideringan additive term to the mean EM matrix, resulting in thefollowing EM representation M n = M + dM n , (17)where the matrix dM n ∈ R L × P is an additive perturbationrepresenting spectral variability. In this case, both the referenceEM matrix M and the pixel-dependent additive perturbationterms dM n were estimated blindly from the hyperspectralimage. However, this model has a large number of parameters.Thus, to mitigate the ill-posedness of the SU problem, aregularization term consisting of the Frobenius norm of dM n , n = 1 , . . . , N was included in the unmixing cost function.Besides the simplicity and mathematical tractability, the useof an additive perturbation in (17) also makes the problemamenable to an interesting interpretation when only a singleadditive perturbation matrix is considered for all image pixels.In this case, the SU problem becomes equivalent to a totalleast squares problem with constraints [202]. Furthermore,the Perturbed Linear Mixing Model (PLMM) has also beenconsidered for robust SU using an outlier-insensitive recon-struction error metric with an L p -quasi norm [203] and formultitemporal and distributed SU [204], [205].One difficulty of parametric EM models is to constructfunctions f ( · ) that are parsimonious but still flexible enoughto represent complex spectral variations. To circumvent thisissue, a deep generative EM model was proposed in [206]based on the hypothesis that the EMs lie on low-dimensionalmanifolds. Instead of fixing the EM model a priori , variationalautoencoders with neural networks were used to learn the para-metric function f ( · ) in (13) using pure pixels extracted fromthe observed hyperspectral image. SU was then formulated asthe recovery of the abundances and of the representations ofthe EMs in the learned manifold, which can be of very small dimension. Despite making SU more well-posed, the resultingcost functions are non-convex and can be difficult to optimize.A different work proposed to exploit the spatial correlationof the endmembers and abundance maps by proposing a gen-eral multiscale mixing model addressing EM variability [207].The SU problem was solved using a multiscale representationof the mixing model, which allowed for the use of anyparametric EM models as in (13). This resulted in improvedresults when compared to standard spatial regularization strate-gies. Although the formulation was algebraically involved,an approximate algorithm with small complexity was derivedunder some simplifying assumptions. C. EM-model-free methods
EM-model-free unmixing: + Algorithms are usually computationally efficient + Involves different strategies with a wide range ofmodel complexity or user supervision + Methods usually make few or restrained as-sumptions about the endmember models (unlikeBayesian or parametric models) − Some approaches have a more limited modelingcapabilitySome methods have proposed to mitigate the effects of spec-tral variability blindly without assuming any specific modelto represent the endmember signatures. One simple approachconsists of using a metric depending on the reconstructionerror in the SU cost function in order to improve the robustnessof SU to endmember variability. It can be motivated by the factthat the commonly used Euclidean distance is very sensitiveto variations in the amplitude of the pixel spectra, being thussignificantly influenced by illumination variations [208]. Thismotivated the consideration of the spectral angle mapper, spec-tral correlation and spectral information divergence metricsdue to their insensitiveness to scaling variations [208], [209].The downside of this approach lies in the nonlinear andpossibly non-convex nature of the resulting SU optimizationproblem, which becomes harder to solve. An efficient strategybased on the projected gradient descent algorithm was pro-posed to optimize the SU cost function when using the spectralangle mapper metric [210]. Although conceptually simple,these approaches focus on specific effects such as brightnessvariations and it is not clear how they can be generalized toaddress more complex spectral variability.More recent SU methods consider more general modelsin order to deal with complex intrinsic variability effects.For instance, an additive residual term in the LMM (1) wasconsidered in [211] in order to account for spectral variabilityand other unmodelled effects. This term was represented asthe product of two matrices. The first matrix correspondedto the first columns of the discrete cosine transform, forcingthe additive terms to be spectrally smooth. The second matrixwas defined for the pixel-dependent coefficients, which wereforced to be spatially sparse and were estimated by solving aconvex optimization problem. A similar approach included ideas from physically moti-vated parametric models by considering the LMM with aconstant scaling factor for each pixel to account for roughillumination variations and an additional non-parametric ad-ditive term to account for other types of spectral variabil-ity [212]. This additive term was defined as the product be-tween an approximately orthonormal basis matrix having lowcoherence with the endmember signatures, and a coefficientmatrix representing the variability contribution to each pixel.However, these constraints make the resulting optimizationproblem non-convex.A different idea was to estimate a subspace projection ofthe observed hyperspectral image that minimizes the effectof spectral variability in SU [213]. This strategy allows SUto be performed by minimizing the reconstruction error inthe projected space. This subspace is forced to be of low-dimension by penalizing the nuclear norm of the projectionoperator in the cost function, which is estimated jointly withthe abundances during SU.A more recent method considered the multidimensionalrepresentation of the pixel-dependent EM matrices and abun-dance vectors by employing mathematical tools from ten-sor decomposition [214]. By assuming that the endmembersand abundance tensors are approximately low-rank, the SUproblem was formulated as a non-convex non-negative tensorfactorization problem. This led to a parsimonious modelwithout the need for explicit parametric representations of theendmembers that are tied to specific applications. D. Bayesian Methods
Bayesian Methods: + Benefit from well-developed statistical estimationtools to derive the SU methods + Can have a very low degree of user supervisiononce the statistical distributions are selected − Can use unrealistic distributions (e.g., isotropicGaussians) to represent the EMs for mathematicaltractability − Generally do not return the specific spectral sig-natures at each image pixel − Suffer from a very high computational cost − Hyperparameters may need to be set by the user,and specifying hyperprior distributions for hierar-chical Bayesian models may not be trivialAnother set of methods considers endmembers to be randomvectors, following multivariate statistical distributions, i.e., m n,p ∼ D ( θ n,p ) , (18)where θ n,p encodes parameters of a distribution D . The spec-tral signatures actually present in each pixel are realizations ofthis random vector, and SU is then formulated as the problemof finding a statistical estimator for the abundances and forthe endmembers.These approaches depend on the statistical distribution D employed to represent EM spectra, on the amount of user supervision that is required and on the computational al-gorithm used to solve the problem. Some methods requirethe parameters of the distribution θ n,p to be set a priori ,which might be difficult in the absence of a large spectrallibrary. Other works reduce user supervision by employinghierarchical Bayesian methods to estimate θ n,p jointly withthe remaining parameters at the cost of a higher computationalcost [215], [216]. The different Bayesian methods addressingspectral variability can be classified according to the statisticaldistribution used to represent the EMs: a Gaussian distribution,which provides mathematical tractability or more complexdistributions providing a more physically reasonable represen-tation. We will discuss both cases in the following. The Normal Compositional Model
The first statistical model that has been considered torepresent endmember spectra was a multivariate Gaussiandistribution, in the so-called Normal Compositional Model(NCM), given by m p,n ∼ N ( θ n,p ) , (19)where D ≡ N and θ n,p = { mean , covariance } contains themean vector and covariance matrix for the p th endmember ofthe n th pixel. The NCM has been widely used due to its mathe-matical tractability [217], [218]. The first works employing theNCM for SU considered expectation-maximization strategiesin which the abundances, the mean endmember values andtheir covariance matrices were estimated iteratively [217].However, due to the non-convexity of the estimation problem,the direct application of expectation maximization approachesis unable to decide whether variations observed in the mixedpixel spectra y n are due to different abundances or to theendmember variability. This might result in the endmembersabsorbing all variation in the observed scene with nearly con-stant abundances [218]. Some approaches proposed to addressthis problem by considering the use of diagonal covariance ma-trices and empirical strategies to estimate the endmember datamore easily from the observed mixed pixels. For instance, end-member means and covariances were estimated both a priori using pure pixels selected from the hyperspectral image [219],and iteratively based on large regions of observed pixels withhomogeneous abundances (obtained from the segmentation ofestimates of the abundances available a priori ) [220].Other works attempted to improve different aspects of thismethod, by using a particle swarm optimization algorithmto solve the (usually intractable) integrals involved in theestimation of the abundances in the “expectation” step of thealgorithm [221], or by incorporating a priori information inthe form of additional constraints penalizing the nuclear normof the abundances in groups of pixels determined throughimage segmentation methods (in order to promote spatialhomogeneity) [222].Despite these advances, the susceptibility of expectation-maximization-based methods to converge to poor local minimaof the non-convex cost function prevented their large-scaleapplicability for this problem. Instead, most recent approachesrely on more robust (although costly) techniques based onMarkov chain Monte Carlo methods to sample the posteriordistribution. Although the works that adopt this approach share Bayesian SU methods represent the EMsignatures at each pixel as a realizationof a statistical distribution. Statistical dis-tributions are first attributed to the EMsand to the abundances and, possibly, toother variables or to hyperparameters ofthese distributions. Using the Bayes rule,the SU results are then derived fromthe posterior distribution in a Bayesianinference problem. The abundances andEM distributions can be computed as,e.g., the mean or as the mode of theposterior distribution.
Figure 14. Illustrative description of Bayesian SU techniques. the same underlying idea, they differ significantly in the wayin which the endmembers and abundances are represented andin the amount of user supervision that is required. For instance,different strategies have been proposed to represent the meanand covariance matrices of the endmembers in the NCM. Oneof the first approaches considered the endmember mean valuesto be known a priori and their covariance matrices to be mul-tiples of the identity matrix [223], while employing conjugatedistributions to make the estimation of the parameters easier.Later works attempted to add more flexibility by considering,for instance, a single full covariance matrix shared by allendmembers [224] or a positive definite matrix defined a priori and multiplied by EM-dependent scaling parameters [225].Diagonal covariance matrices were employed in [226], whichalso considered the estimation of the EM mean values in ahierarchical Bayesian framework, using hyperpriors to esti-mate the distribution parameters directly from the observedhyperspectral image. The Bayesian framework has also beenused in [227] to estimate the number of EMs in the sceneblindly using a uniform discrete prior.Other works attempted to address physically motivatedparticular cases of the general NCM. This includes the con-sideration of statistical dependence between different EMsto represent spectral variability that may affect all materialsin the scene equally (e.g., atmospheric effects) [228], andthe explicit representation of the higher correlation betweenadjacent spectral bands to introduce spectral smoothness tothe signatures, leading to a well-posed model that is also fastto compute [229].An alternative approach which has been used to simplifythe unmixing process associated with the NCM is to estimatethe endmember means and covariance matrices a priori basedon spectral libraries extracted from the observed image. Thishas been performed considering libraries obtained both usingpure pixel-based endmember bundle extraction methods [230]and on multiple endmember matrices estimated by a piecewiseconvex blind SU algorithm [175]. However, these methods suf-fer from the limitations of image-based EM bundle extractiontechniques, which will be discussed in detail in Section V-A.Other works also considered a piecewise convex modelwhich uses a set of different Gaussian distributions to model the endmembers. Afterwards, during SU each image pixel isassigned to one of these distributions using a membershipfunction represented by a Dirichlet random variable. Theunmixing problem under this model was solved by consideringboth an alternating optimization method in a maximum aposteriori framework [231] and a Markov chain Monte Carlosampling approach providing an estimate of the posteriordistribution of interest [232].Although the Dirichlet prior distribution is frequently usedto represent the abundances, many works have consideredvariations which incorporate useful information from the un-derlying practical problem. Examples include the enforcementof abundance sparsity using a sparse Dirichlet prior [233],or the encouragement of spatial homogeneity by dividingthe abundance maps into a finite number of classes sharingthe same Dirichlet distribution parameters. This division hasbeen performed either blindly by means of a classificationprior using the Potts model [226] or through an a priori segmentation of the hyperspectral image in a latent Dirichletallocation framework [234].More recently, the NCM has also been applied to problemsother than that of linear unmixing or spectral variability.For instance, the NCM has been considered to represent theuncertainties in EM estimation instead of the intrinsic vari-ability of the material classes, which changes the problem byintroducing statistical dependence between the different imagepixels [235]. Other works also applied the NCM to problemssuch as nonlinear SU with a bilinear mixing model [236], forthe linear unmixing of sediment grain size distribution (wherethe EMs represent the grain sizes of constituent materials)to study transport and deposition of sediments [237], or torepresent the variability of the endmembers across multipleimages in multitemporal SU, using additional spatially sparseterms accounting for potential abrupt spectral changes betweenthe different images [238]. Other Endmember Distributions
Despite its popularity, the NCM does not have a strongphysical motivation, which led to the consideration of moreaccurate distributions to represent the EMs. For instance, aBeta distribution was considered in [239] in order to constrainreflectance values to physically meaningful ranges and to allow for possible skewness in the distribution. Unfortunately, adirect solution to the SU problem cannot be obtained. Thus,a piecewise constant model was assumed for the abundances,which allowed the parameters of the distribution to be esti-mated using a combination of a clustering algorithm and avariant of the method of moments.A Gaussian mixture model has also been considered in [240]in order to allow for possibly multi-modal EM distributions.The SU problem was solved as a maximum a posteriori esti-mation problem using a generalized expectation-maximizationapproach. However, since learning the parameters of Gaussianmixture models can be difficult, they were estimated beforeperforming SU based on spectral libraries assumed to beknown a priori .Another approach proposed to represent EM spectra as asum of an average spectral signature known a priori anda spatially and spectrally smooth function representing EMvariability to provide a model that is physically more reason-able [241]. Bilinear mixing models were also considered alongwith an additive residual term to account for mismodelingeffects or outliers.A different approach has been proposed which does notmake an explicit assumption about the distribution of EMspectra and instead only relies on some of their statistics.This is the case of [242], which formulates the SU problemsimilarly to the method of moments by trying to find the abun-dance values which match the mean and covariances obtainedthrough the LMM to those of the observed mixed pixels. Asimilar work applied the same idea using transformed statisticsconstructed from the ratio between the means and covariancesof the pixels and endmembers in different spectral bands [243].This strategy increases the robustness of the method sinceband ratios are invariant to illumination variations. However,similarly to [239], a piecewise constant abundance model isused to estimate the covariance matrix of the observed pixels.Moreover, the covariance matrices of the EMs are assumed tobe known a priori .V. S PECTRAL L IBRARIES
A large number of SU techniques discussed in Section IIIaddress spectral variability by using spectral libraries or bun-dles known a priori . The performance of these methods isoften heavily impacted by how well the libraries can representthe endmembers actually present in the scene. Moreover, inmany practical situations it is either very costly or evenimpossible to obtain laboratory or in situ measurements ofendmember spectra. Another problem with many methods pre-sented in Section III (like MESMA) is that their computationalcomplexity increases very quickly with the library size, whichcan make the problem intractable for large libraries.Thus, the problems of removing redundant or irrelevantspectra before SU and, especially, of extracting spectral li-braries directly from observed hyperspectral images are ofcentral importance in order to allow the techniques discussedin Section III to be widely applicable. Fortunately, severaltechniques have been proposed to address both of theseproblems, which we will discuss in detail in this section. A. How to construct spectral libraries?
Many library-based SU works assume that spectral librariesare manually obtained from in situ or through controlled lab-oratory measurements [37], [244], which may be complicatedin practical applications. Moreover, existing libraries may havebeen acquired at conditions which do not reflect those actuallyobserved in the scene, which introduces errors in the SUprocess [37], [103], [245]. Even the spatial resolution at whichthe hyperspectral image is acquired was found to have aconsiderable impact on the results of SU with MESMA inurban environments when the library was fixed a priori [246].Traditional endmember extraction algorithms (EEAs), onthe other hand, typically consider only a single spectralsignature per material and are thus unable to appropriatelyaddress spectral variability [33], [247]. These shortcomingsmake the construction of spectral libraries one of the mainchallenges of library-based SU methods [244]. A simple andreliable method that has been employed to construct spectrallibraries in practice depends on expert knowledge to manuallyselect pure pixels of each material from the hyperspectralimage [246], [248]. However, there has been a growing interestin developing methods that can reduce the amount of usersupervision and automatically extract libraries directly fromobserved hyperspectral images. Three main general lines ofresearch can be identified in this direction:a) extract multiple pure pixels from the observed hyperspec-tral image to generate a candidate library, and then clusterthe extracted signatures into their respective materialclasses;b) generate libraries using radiative transfer models thatrepresent endmember variability mathematically;c) extract pure pixels while keeping information about theirspatial locations, and apply an interpolation algorithm togenerate endmember signatures for each image pixel.The diagram in Fig. 15 gives an illustrative overview of the keyideas underlying each of these approaches, which are reviewedin the following. Image based library construction
Image-based Library Extraction: + Allows spectral libraries to be extracted withsignatures that are at the same conditions of theimage pixels + Can benefit from expert knowledge to reliablyidentify pure pixels in the image − Depends strongly on the presence of pure pixels − The observed image should not be too small − Mixed pixels may be included in the library bymistake − Clustering the extracted signatures into their cor-rect material classes is challengingThe simplest approaches for the construction of image-based spectral libraries are completely supervised. Image pix-els are included in the library either based on their correlationto some initial endmembers manually selected as the extreme Extract a set ofpure pixels or
Extract purepixels and theirspatial locationsSynthesize EMsfor the otherspatial locations Physical model(expertknowledge) candidate EMsCluster theextractedsignatures
Select di ff erentparameterssets of physical e.g., chlorophyllconcentration Synthesize EMsignatures usingthe parametricmodel (Section V-A1) (Section V-A2)(Section V-A3)
Observed Image Spectral libraries
Figure 15. Illustrative diagram depicting existing approaches to generate spectral libraries: image based library generation using endmember extraction (left,discussed in Section V-A1), spatial interpolation of pure pixels extracted from the image at known locations (center, discussed in Section V-A3), and thegeneration of synthetic signatures from physics-based models (right, discussed in Section V-A2). points of the PCA of the observed image [97], [249], orsimply by manually screening a large number of pure pixelsextracted from the image using expert knowledge about thespectral characteristic of the materials in the scene [248]. Purepixels were also extracted from multiple hyperspectral imagesof the same scene acquired at different spatial resolutions toincrease the diversity of the resulting spectral library in urbanenvironments [246]. Other work used only partially labeleddata in order to reduce the amount of domain knowledgethat is required [250]. Recent strategies attempted to automatethis process by extending EEAs for the extraction of multiplesignatures of each material in the observed image. The firstwork in this direction proposed to apply traditional EEAs torandom subsets of pixels that are sampled from the hyper-spectral image (with or without replacement) [251]. Differentsets of EM signatures are generated using this method. Allthe extracted signatures are then grouped into different setscorresponding to the material classes by using a clusteringalgorithm (e.g., k-means). The size of the image subsets,however, must be selected with great care in order for EEAsto work satisfactorily [171], and the clustering step can be challenging.Later works proposed different strategies for the extractionor selection of multiple pure pixels or endmember candidatesfrom the observed image. One simple iterative strategy consistsof including in the library all pixels that are within a givenspectral distance of some reference EMs [252]. This process isperformed iteratively, with the reference EMs initialized usinga standard EEA and then updated as the mean values of thelibrary signatures at the previous iteration. Besides being verysimple, this procedure does not require the library signaturesto be clustered afterwards. A related strategy worked in areverse way, by iteratively removing pure pixels from a largeinitial set of candidate signatures in order to obtain the finalspectral library [253]. A pixel candidate is removed if it canbe represented with small error as a convex combination of theremaining signatures in the library. A clustering procedure isthen performed to group the selected spectra into EM classes.Recent works have proposed more involved empirical ap-proaches to differentiate between spectrally similar materialswhen extracting or clustering the EM signatures, or to re-move mixed pixels from the constructed library. For instance, in [254] EM extraction was performed multiple times fordifferent subsets of the spectral bands constructed at multiplespectral scales and intervals. These signatures were afterwardsclustered into EM classes based on a metric constructed fromfeatures derived from applying clustering algorithms individ-ually to the spectral scales and intervals used previously.A related strategy considered both the extraction and clus-tering of the library signatures based on subsets of the wavelettransform coefficients of the reflectance spectra that are robustto spectral variability [255]. These subsets were selected basedon how much their empirical statistical distribution deviatesfrom an uncorrelated Gaussian distribution. The hyperspectralimage was also partitioned into spatial segments using a hier-archical clustering algorithm, and only one signature for eachspatial segment was considered to be included in the library.Another strategy proposed to extract spectral signatures asimage pixels which can best represent all other pixels in theobserved image as a sparse linear combination [256]. After-wards, these signatures were grouped into material classesusing spectral features derived from the slopes of a piecewiselinear approximation of each signature.Auxiliary libraries available a priori have also been usedto aid in the extraction of image-based spectral librariesin [156]. The k -nearest neighbor algorithm was first used toclassify the image pixels in the different material classes, usinglibrary spectra known a priori as training data. This led toa set of candidate EMs for each material class. Based onthe classification results, the image-extracted library was thendefined as the average spectra of those candidate EMs of eachclass that were contained in a spectral neighborhood of eachof the training samples (from its corresponding class) [156].Another group of approaches makes use of the empiricalobservation that pure pixels are more likely to be containedin spatially homogeneous regions. Spectral libraries can beconstructed either by restricting EM candidates to be containedin sufficiently homogeneous regions [257], [258], by applyingan image over-segmentation strategy before pure pixel extrac-tion [259], or by considering EM candidates as the averageof homogeneous regions obtained from a coarse spatial scaleselected from a multiscale image decomposition [260]. Thesestrategies should be applied with care to avoid the inclusionof pixels extracted from mixed, homogeneous regions intothe library.Some alternatives tried to build spectral libraries by usingdifferent forms of matrix factorization of the hyperspectralimage. For instance, spectral libraries for each material classare constructed in [177] by learning sparse representationsof sets of pure pixels of each material, which are extractedfrom the observed image. More precisely, dictionary learningis applied to the pure pixels of each material, from whichthe resulting basis matrices are used to construct the spectrallibrary. Another approach proposed to extract the spectrallibrary using the results of an SU procedure using a matrixfactorization approach which does not accounts for spectralvariability [261]. However, besides depending on the resultsof another SU algorithm, there is no guarantee that the selectedsignatures are pure pixels. Generating spectral libraries from physics models
Physics-based Library Synthesis: + Can generate libraries independently of the ob-served image + Can represent a wide range of spectral variabilityif more complex models are employed − Depends on the availability of an accurate phys-ical model for the spectra of the EMsAn alternative approach to generate spectral libraries whichdoes not depend on the observed hyperspectral image is to em-ploy a physics-based (i.e., radiative transfer) model describingthe reflectance of the EMs as a function of physico-chemicalparameters. This allows us to generate different instancesof the material spectra to constitute a synthetic library bysampling the free parameters of the model. Examples of suchmodels include the PROSPECT model [69] for vegetationor Hapke’s [48] and Shkuratov’s models [262] for packedparticle spectra.Different models inspired by physics have been employed togenerate or augment spectral spectral libraries for SU in manyapplications. These applications include models for canopyas a function of its height and canopy radius [263], firetemperature radiance as a function of view and solar geometryand atmospheric conditions [264] and soil reflectance as afunction of moisture content [73]. This strategy has also beenapplied to generate training data for SU of binary mixtures ofvegetation and impervious materials using machine learningalgorithms [265]–[268]. Note, however, that directly samplingall parameters of complex models such as PROSPECT mightlead to a very large number of signatures. This has motivatedstrategies to sample the parametric models more efficiently orto remove redundant spectra from the generated library [269].In spite of their advantages, a significant drawback of thesemethods is the requirement of accurate knowledge of thephysical process governing the observation of the reflectanceof the materials by the sensor. A different approach attemptedto circumvent this issue by proposing a data augmentationstrategy, where one wishes to synthesize additional signaturesto be included in small, pre-existing libraries [270]. Thespectral signatures in the library are used as training data inorder to learn the statistical distribution of the EMs using deepgenerative models such as variational autoencoders and deepneural networks. This allows one to sample new signaturesfrom the learned distributions to augment the existing library. Spatial interpolation of endmember signatures
Spatial Endmember Interpolation: + Uses the hypothesis of spatially correlated EMsignatures − Needs knowledge of the spatial position of purepixels in the scene − The amount of pure pixels available can stronglyaffect the performance of the methods A number of approaches based on the assumption thatEMs are spatially correlated proposed to synthesize pixel-dependent EM signatures based on a set of pure pixels atknown spatial locations using interpolation techniques. Manyof these works aim to perform SU of vegetation and soilmixtures by using vegetation indices (i.e., spectral featuresgiven by ratios of band differences, such as the NDVI) in lieu of traditional endmembers.For instance, the spatial interpolation of vegetation and soilNDVIs based on linear regression has been considered for SUof coarse resolution images, where the training samples for theEMs were obtained using using classification maps from com-plementary, high resolution images available a priori [271].A similar strategy considered the use of spatially weightedkriging employing as training samples pure pixels which wereeither manually extracted from the scene [272] or obtained byrandomly sampling the vertices of the simplex obtained by alow-dimensional projection of the hyperspectral image [273].This strategy allowed one to weight the contribution of thetraining samples according to their spatial distance to eachinterpolated signature.Other works also considered the spatial interpolation ofactual spectral signatures instead of just vegetation and soilindices using spatially weighted linear regression or krig-ing. This has been performed considering training data ob-tained both from complementary high resolution classificationmaps [274], or from pure pixels extracted from the imageinside sub-regions appropriately selected with the aid of aclassification algorithm [275]. B. Library pruning techniques
One significant problem with many SU methods based onspectral libraries such as MESMA is that their computationalcomplexity increases quickly with the size of the spectrallibrary. Furthermore, databases containing laboratory acquiredspectra often contains hundreds of different materials. Usinga library of this size can actually decrease the performance ofSU since the problem becomes more and more ill-posed.One solution to this problem consists of removing redundantor irrelevant signatures from large spectral libraries before theSU process. These approaches, also called library pruning,have been largely applied in order to reduce the complexityand improve the accuracy of both MESMA [38] and sparseunmixing algorithms [276]. There are three main groups oflibrary pruning techniques. Library reduction techniques justremove redundant signatures to improve the computation time.Endmember selection techniques identify which materials arepresent in each hyperspectral image pixel to remove absent EMclasses from the library before SU. Same-class library pruningattempt to identify and remove signatures which are acquiredat different conditions from those of the observed image. Theseapproaches will be reviewed in detail in the following. Library reduction techniques
Spectral Library Reduction: + Very simple strategies that do not depend on theobserved hyperspectral image − Only reduces computational complexity, but doesnot improve the quality of the SU resultsLibrary reduction techniques attempt to remove redundantspectral signatures from the library regardless of the observedhyperspectral image, which tends to improve the computa-tional complexity of SU but not necessarily its quality. Acommon idea is to find a small set of signatures which can bestrepresent the remaining spectra of the same EM class in somesense [277] such as the squared error [38], the average spectralangle [89] or the count-based EM selection metric, where onecounts the number of signatures one candidate can representwith error below a threshold [93]. An alternative method alsodivided the library signatures into groups according to theirEuclidean norm, selecting one signature from each group toexplicitly account for brightness variations [278]. Endmember selection methods
Endmember Selection: + Remove only entire material classes from thelibrary for each pixel, and is also effective forvariability-free SU + Leverage information from the observed hyper-spectral image + Can improve the SU quality and reduce the com-putational complexity − Usually depend on some sort of classificationprocedure − Rely on the observation that usually only a fewmaterials are contained in each pixelEndmember selection techniques attempt to identify whichEM classes are present in each pixel using information suchas classification maps [10], [279] to remove entire absent ma-terials from the library and improve the unmixing results [10],[279]. This relies on the observation that hyperspectral imagepixels usually contain only a small number of materials, andhas also been applied to SU without considering spectralvariability [280], [281].The simplest EM selection methods use classification algo-rithms to select the EM classes present in mixed pixels [282],[283]. Another work employed a block sparse unmixing al-gorithm as a preprocessing step in order to remove materialclasses with low abundances values from the library foreach image pixel before applying the MESMA algorithm toobtain the final SU results [284]. A more elaborate approachproposed to semantically organize subsets of material classesin a hierarchical tree, starting from a rough (e.g. pervious andimpervious) up to a fine differentiation between the endmem-bers (e.g., different vegetation species) [59]. Afterwards, SUis performed at each level of the tree, using the abundanceresults in the previous, coarser level to constrain which EMscan be selected at the current one (i.e., a pixel containing only a pervious EM in the coarse scale cannot have concrete EMin the finer one).Some recent approaches have also proposed to use external,complementary data in order to aid in identifying whichmaterials are present in each pixel. For instance, in [285]the hyperspectral image was divided into rural and urbansubsets by using external data of road network density, whichallowed for the use of a separate set of EM classes foreach of the subsets. Another work proposed to use additionalLIDAR data to remove material classes from the library ofeach pixel based on its height distribution (e.g., a “tree” or“building” endmember can be removed from a pixel that haslow height) [286]. Pruning libraries within the same class
Same-Class Endmember Pruning: + Remove spectral signatures from each materialclass that are not representative of the observedhyperspectral image + Can improve the SU quality and reduce thecomputational cost even for libraries with fewmaterials classes − Identify which signatures in the spectral librarydo not share the same acquisition conditions withthe observed image is generally difficultRecent approaches proposed to remove signatures from thelibrary that have been acquired at conditions different fromthose of the hyperspectral image, keeping only signatures thatare representative of the observed image. However, measuringthe representativeness of the EM signatures is a difficulttask. A simple approach proposed to remove signatures thathave a large spectral angle and spectral L distance relativeto the observed pixels [287]. However, this strategy mightdiscard relevant signatures in the presence of many mixedpixels. Another work proposed to compare only pure pixelsextracted from the image with the library spectra in thewavelet domain [288].A different approach proposed to remove library elementsthat have large distances to a small set of the leading eigenvec-tors of the observed hyperspectral image, and are thus unlikelyto be present therein [289]. This strategy eliminates the directneed for pure pixels in the scene. It has also been successfullyapplied for plant production system monitoring [290], andwas later extended to consider a brightness normalization pre-processing step and other strategies from Section V-B1 toadditionally remove redundant spectra [279].Another work also proposed to perform library pruningiteratively in a sparse unmixing formulation by removingsignatures corresponding to low abundance values during theSU process [291]. However, this process depends directly onthe accuracy of the SU process at the first iterations.VI. E XPERIMENTAL E VALUATION
This section presents a brief discussion about the experi-mental evaluation of the unmixing algorithms when spectralvariability is considered. We first discuss the generation of synthetic data in detail. Afterwards, some existing softwarepackages that can be useful for practitioners are presented.Finally, an illustrative, tutorial-style simulation example ispresented in order to demonstrate the use of a few of theSU techniques reviewed in the paper, after selection using thedecision tree in Fig. 2. A. Generating Synthetic Data
One challenge in the evaluation of unmixing methods is thelack of reliable ground truth data for the abundances of realhyperspectral images. The difficulty in collecting ground truthdata is even more pronounced when endmember variability isconsidered. Thus, being able to generate realistic synthetic data(for which the true abundances are available) turns out to beimportant to allow a quantitative evaluation of SU algorithms.More precisely, the generation of synthetic data can beroughly divided into three steps:1) generating synthetic abundances;2) generating endmember signatures for each pixel in theimage;3) applying the mixing model of choice (in our case, theLMM) to generate the mixed image pixels.We discuss each of these steps in the following. Generating synthetic abundance data
The generation of synthetic abundance maps can be per-formed in different ways. A simple strategy is to sample theabundance values randomly from a Dirichlet distribution. Thisapproach allows one to control the amount of pure pixels inthe image, and can be useful when performing Monte Carlosimulations in which large amounts of data must be generated.Another approach consists in introducing spatial contextualinformation (i.e., pixels that are close in space tend to havesimilar abundance values) into the generated abundances togenerate more realistic data. Such data can be generatedusing, for instance, piecewise smooth images sampled from aGaussian random field [292]. This approach is able to generateimages containing smooth regions, sharp transitions and finedetails whose spatial composition and regularity characteristicscan be controlled by the user [292]. One software tool that canbe used to generate abundance maps according to Gaussianrandom fields is the Hyperspectral Imagery Synthesis tool forMatlab (available for download here). Another way to obtainrealistic synthetic abundance maps is to consider abundancesobtained by applying an existing spectral unmixing algorithmon a real hyperspectral image [293]. The resulting abundancemaps will have a realistic spatial distribution, and can be usedas ground truth to generate new synthetic datasets. Generating synthetic endmember variability
Generating realistic endmember variability data is not asimple task since, as explained before, the spectral signaturesof the materials present a complex dependence of differentphysico-chemical and environmental parameters. Fortunately,very accurate radiative transfer models have been developedfor many applications. Such models describe the physicalprocesses governing, e.g., vegetation spectra [68], mineralinteractions [48], [262] and atmospheric effects [294]. Well-calibrated radiative transfer models can be used togenerate realistic simulated image scenes that allow one tosimultaneously study nonlinear mixtures and endmember vari-ability effects. Experimental studies have found that the datasimulated using such models show a very strong agreementwith reference ground truth data collected under the samecircumstances using ground-based spectral measurement set-ups [295], [296]. This approach has been already used toevaluate nonlinear unmixing models in [297]. Thus, well-calibrated radiative transfer models can be used to generaterealistic simulated hyperspectral data that allow us to develop,optimize, test and compare different SU techniques consider-ing endmember variability.Although complex ray-tracing simulations can be consid-ered (e.g., [48], [68], [262], [294], [298]), here we presentsome simplified models for illustrative purposes, which de-scribe variability present in vegetation spectra and caused bydifferent viewing geometries.The first model we consider is the PROSPECT-D [76],which represents vegetation leaf spectra as a function of,e.g., the chlorophyll and dry matter content, and of theequivalent water thickness. PROSPECT-D and other relatedmodels for vegetation spectra can be downloaded here, fordifferent software platforms (including Matlab and Python).We also consider a simplification of Hapke’s model [48]by assuming a Lambertian (isotropic) scattering and a denselypacked medium. This simplified model describes variations inthe reflectance spectra of a material y sensor (at each wave-length) as a function of the viewing geometry [50], [52]: y sensor = ω (1 + 2 µ √ − ω )(1 + 2 µ √ − ω ) , (20)where ω is the single scattering albedo of the material, and µ (resp. µ ) is the cosine of the angle between the incoming(resp. outgoing) radiation and the normal to the surface. Thismodel allows us to generate different EM spectra by varyingthe values of µ and µ . While (20) is approximately linear forsmall albedo values, important nonlinearities occur for largealbedo values [52].We also consider variability introduced by errors occur-ring in a simple atmospheric compensation model, wherethe reflectance of each pixel at each wavelength is obtainedby dividing the corresponding pixel’s radiance by the radi-ance observed at a perfectly reflective calibration panel [255,Section IV-A1], [299]. Assuming full visibility and that theadjacency effect is negligible, this model is given by: y sensor = y s E sun − gr µ + E sky E sun − gr µ + E sky , (21)where y s and y sensor denote the reflectance at the ground andat the sensor, E sun − gr denotes the solar radiance observed atthe ground level, and E sky denotes the skylight. Parameters µ and µ are the cosines of the angles between the surfacenormal and the direction of the sun at each pixel and at thecalibration panel, respectively. By fixing µ , E sun − gr and E sky a priori , µ can be varied to simulate spectral signatures atdifferent viewing geometries. Figure 16. Generated spectral signatures used in the synthetic hyperspectralimage, vegetation (left), dirt (center) and water (right). Generating the mixed pixels
Finally, each pixel can be generated according to the spectralvariability accommodating LMM described in (2), with theendmember for each pixel ( M n columns) sampled randomlyfrom the set of synthetically generated signatures . Additivenoise can also be introduced to obtain a desired signal-to-noiseratio. B. Available Software Resources
Several software packages are available to perform SUwith spectral variability. Classical techniques such as MESMAand some of its alternatives (including library pruning andtransformation methods) can be found in the VIPER toolssoftware package [300], which is available both as plug-insfor well-established software such as ENVI (here) and QGIS(here), and also as a standalone Python package (here). Animplementation of the MESMA algorithm is also availablein R in the RStoolbox (here).Algorithms that were developed more recently, on the otherhand, are usually only available as standalone prototypesimplemented in Matlab or Python. A list of software packagesfor some of the papers reviewed in this work (most of whichfound at the authors’ websites) is contained in Table III. Also,the OpenRemoteSensing website, which aims to share anddisseminate codes and papers, also has an increasing numberof SU methods (here), some of which considering spectralvariability. C. Experimental setup and results
We now present a simulation to illustrate the applicationof some of the algorithms reviewed in this work. Note thatthis simulation is merely illustrative, and not a comprehensiveperformance evaluation. We generated a synthetic hyperspec-tral image containing vegetation, dirt and water as constituentmaterials. Spatially correlated abundances with × pixelswere first sampled from a Gaussian random field. Then, wefollowed the procedure described in Section VI-A2 to generatedifferent endmember spectra for each material in the scene.The PROSPECT-D model was used to generate vegetationspectra, while the simplified Hapke and atmospheric modelswere used to generate dirt and water spectra, respectively, at More complex models can also be employed to consider the spatial rela-tionship between the endmembers by, e.g., generating the viewing geometryaccording to a digital terrain model as in [51]. Table IIIL
IST OF COMPUTATIONAL CODES CONTAINING IMPLEMENTATIONS OFSOME OF THE WORKS REVIEWED IN THIS PAPER ( PROVIDED BY THERESPECTIVE AUTHORS )Method Link LanguageMethods that use spectral librariesMESMA [37], AAM [87] link MatlabSUnSAL [103], SUnSAL-TV [112] link MatlabSparse SU with mixed norms [84] link MatlabBayesian methodsBCM [239] link MatlabNCM-E (NCM by Eches et al. ) [223] link MatlabUsGNCM [226] link MatlabBayesian OU [238] link MatlabPCOMMEND [231] link MatlabGMM [240] link MatlabParametric EM modelsELMM [51] link MatlabPLMM [201] link MatlabGLMM [196] link MatlabDeepGUn [206] link MatlabMUA-SV [207] link MatlabOU [205] link MatlabEM-model-free methodsRUSAL [211] link MatlabSULoRa [213] link MatlabALMM [212] link MatlabULTRA-V [214] link Matlab different viewing geometries. The generated synthetic signa-tures, containing L = 198 bands, can be seen in Fig. 16. Theendmembers contained in each pixel were sampled randomlyfrom this set of synthesized signatures, and the pixel spectrawere then generated following the LMM with variability in (2),with white Gaussian noise added to the image to obtain asignal-to-noise ratio of 30 dB.To evaluate the SU results, we considered as quantitativequality measures the Root Mean Squared Error (RMSE) andthe Spectral Angle Mapper (SAM). The RMSE between twogeneric variables X and (cid:98) X is defined as RMSE X = (cid:114) N X (cid:107) X − (cid:98) X (cid:107) F , (22)where (cid:107)·(cid:107) F is the Frobenius norm and N X denotes the numberof elements in X . We used the RMSE to evaluate the estimatedabundances (cid:98) A , the reconstructed images (cid:98) Y and the estimated,pixel dependent endmembers (cid:99) M n (for the cases when thisestimate was available). The SAM was also used to evaluatethe estimated endmembers as: SAM M = 1 LP N N (cid:88) n =1 P (cid:88) p =1 arccos (cid:18) m (cid:62) p,n (cid:99) m p,n (cid:107) m p,n (cid:107)(cid:107) (cid:99) m p,n (cid:107) (cid:19) , (23)where N is the number of pixels and P the number ofmaterials in the hyperspectral image.We also evaluated the complexity of the algorithms throughtheir execution times, measured in an Intel Core I7 processorwith 4.2 GHz and 16 Gb of RAM. Finally, in order to increase Figure 17. Endmember bundles extracted by batch VCA [251], for vegetation(left), dirt (center) and water (right). the reliability of the results, we executed the simulation for tenindependent Monte Carlo realizations and report the averagevalues for all metrics. Algorithm selection and setup
For illustrative purposes, we considered the recovery ofthe abundance maps following four different paths in thedecision tree of Fig. 2 (selected according to the algorithmimplementations available in Table III).1) Small spectral libraries extracted directly from the im-age, no expert knowledge available:a) less user supervision: MESMA and variants [37];b) less computational cost: Sparse unmixing (fractionalsparse SU [84]);2) Spectral libraries not available a priori :a) less user supervision necessary: Bayesian methods(NCM-E [223], BCM [239], [301])b) less computational cost: Parametric models(ELMM [51], DeepGUn [206]); EM-model-freemethods (RUSAL [211])We additionally considered the FCLS solution as a baseline,using a single set of endmembers extracted from the imageusing the Vertex Component Analysis (VCA) algorithm [302].The EMs extracted by VCA were also used as initializationor as reference/mean signatures for some of the algorithms(ELMM, DeepGUn, RUSAL, NCM-E). For MESMA andsparse SU, the spectral libraries were extracted from the ob-served image, as will be described in greater detail in the nextsubsection. The spectral libraries were also used to estimatethe parameters of the beta distribution in the BCM. Theregularization/tuning parameters of the algorithms (fractionalsparse SU, ELMM, DeepGUn, RUSAL) were manually ad-justed to maximize the abundance reconstruction performancemeasured in an independent dataset generated following thesame specifications as in the beginning of Section VI-C. Library extraction
In order to demonstrate the use of library-based SU methodsin practical settings, the spectral libraries used by MESMA andFractional sparse SU were extracted directly from the observedimage. We used the method described in [251], which consistsof performing endmember extraction (in this case, using theVCA algorithm) in subsets of pixels randomly sampled fromthe image. We extracted five sets of endmembers, using subsetsof 500 pixels each (sampled with replacement).The library was kept small in order to prevent the inclusionof redundant signatures and also to reduce the probability Table IVQ
UANTITATIVE SIMULATION RESULTS (RMSE
RESULTS ARE MULTIPLIEDBY ). RMSE A RMSE M SAM M RMSE Y Time [s]FCLS 9.899 – – 0.239 0.37MESMA 6.083 0.504 0.234 0.159 4.90Fractional 5.993 0.525 0.232 0.159 3.41ELMM 8.695 0.697 0.560 0.127 28.84DeepGUn 7.203 0.447 0.395 0.324 80.42RUSAL 9.509 – – 0.108 1.05NCM-E 9.897 – – 0.239 2482.85BCM 8.105 – – 0.472 468.69 of selecting mixed pixels by mistake. As a byproduct, thisalso keeps the computational complexity of methods such asMESMA very low, while providing good experimental resultsin this example. The estimated signatures can be seen inFig. 17. Although the spectral variability in Fig. 17 is lessaccentuated than that of the true endmembers in Fig. 16, theestimated signatures are good representatives of the materialsin the scene. The good performance of the library extractionmethod can be explained by the presence of multiple purepixels in the synthetically generated abundance maps, whichcan be seen in the first row of Fig. 18. Discussion
The quantitative results are shown in Table IV, while theestimated abundance maps and endmembers are depicted inFigs. 18 and 19, respectively. Note that the
RMSE M and SAM M are not available for the FCLS, RUSAL, NCM-Eand for the BCM, since these algorithms do not estimatethe spectral signatures of the endmembers present in eachpixel of the image. All methods that considered spectralvariability led to better abundance reconstruction results thanthe FCLS baseline. In particular, the library-based methods(MESMA and fractional-based sparse SU) obtained a verygood performance, which likely occurred due to the image-extracted spectral library accurately representing the typicalEM variability contained in this scene. Moreover, sparse SUwith fractional norms performed similarly and slightly betterthan MESMA.The methods based on parametric EM models (ELMMand DeepGUn) also led to considerable improvements whencompared to FCLS, especially considering that the EMs areestimated directly from the image. The EM-model-free method(RUSAL), which takes general variability and mismodelingsinto account, also provided an improvement over FCLS, albeitsmaller when compared to ELMM and DeepGUn. However,the sensitivity of these techniques to the selection of the regu-larization parameters can negatively impact their performancewhen Monte Carlo simulations are considered.Among the Bayesian methods (NCM-E and BCM), BCMprovided a considerable performance improvement over FCLS,especially when taking into account the unsupervised natureof the method (i.e., no parameter has to be adjusted). TheNCM-E results, on the other hand, were virtually identical tothose of the FCLS, which indicates that the isotropic Gaussian Figure 18. Abundance maps estimated by the algorithms (values are mappedto colors ranging from blue ( a = 0 ) to red ( a = 1 ). EM hypothesis may not be appropriate for this dataset.The performance of the different methods can be visuallydistinguished in Fig. 18, especially from the soil endmember,in which the similarity between the reconstructions and thereference abundance maps reflects the general behavior of thequantitative results from Table IV.The endmember reconstruction metrics in Table IV indicatethat the EMs selected by library based approaches (MESMAand fractional sparse SU) are close to the reference ones, espe-cially in terms of SAM M , while the model-based approaches(ELMM and DeepGUn) provided slightly worse results in gen-eral, except for DeepGUn’s RMSE M . The visual assessmentof the estimated signatures in Fig. 19 shows an interestingpattern, since despite the quantitative metrics, the amountof variability (i.e., the variance) estimated by ELMM seemscloser to the reference spectra. This shows that identifyingthe correct spectral signatures present in each pixel is verydifficult.We also note that smaller image reconstruction errorsRMSE Y did not correlate very well with better abundanceestimation results. Since some SU methods that take spectralvariability into account adopt flexible models, they can repre- Figure 19. Spectral signatures returned by the algorithms that estimate theendmember spectra for each image pixel. sent the hyperspectral image pixels in Y very closely withoutnecessarily improving the abundance estimation.The execution times show a considerable difference betweenthe methods. Library-based approaches were able to run veryfast (even for MESMA) since the spectral library containedfew signatures. This shows that the construction of the librarycan significantly impact the run-time performance of thesetechniques. The methods based on parametric models (ELMMand DeepGUn) provided intermediate execution times, whileRUSAL was very fast. Finally, Bayesian methods took thelongest to run, with NCM-E taking significantly more than allthe remaining techniques.Finally, we note that this example is merely illustrativeand not an in depth evaluation of these methods. Thus, theirperformance can be different for other datasets and scenarios.VII. D ISCUSSION , C
ONCLUSIONS AND F UTURE D IRECTIONS
Significant advances have been made to mitigate spectralvariability in SU during the last decade, encompassing con-tributions with both experimental and theoretical motivations.Recent work has, for instance, allowed spectral libraries tobe directly extracted from observed hyperspectral images,provided more accurate or flexible models to represent theendmembers (e.g., in statistical or parametric methods), andincluded different kinds of a priori external information inorder to alleviate the ill-posedness of the problem, such as thelocally correlated characteristics of the EMs and abundances. This was performed either explicitly, by means of regular-ization approaches or in the definition of statistical models,or implicitly in the design of the algorithms (e.g., in localSU). Other methods leveraged the spectral characteristics ofEM variability to design improved algorithms (e.g., in spectraltransformations or robust SU methods).However, there is still a noticeable dependence between thethe quality of the unmixing solutions and the necessary amountof user supervision in the algorithms. Many recent techniquesneed considerable or intricate tuning in order to reach their fullpotential, with a significant portion of algorithm design beingleft to the user. The lack of more extensive data with reliableground truths have also made the evaluation of the algorithmssomewhat difficult. In the following, we detail some aspectswhich we think deserve further consideration. • As discussed above, one important research direction isto improve the robustness of the methods to the selectionof their parameters, or to develop informed adjustmentmethodologies. This could be performed, for instance,by leveraging metadata (e.g., external classification maps)that is available in many applications. This point appliesto the majority of SU algorithms reviewed in this paper,and would make those methods more readily employableas out-of-the-box solutions in practical scenarios. • Most SU algorithms that address spectral variability de-pend strongly on spectral libraries or on reference end-member signatures known a priori or extracted from theobserved hyperspectral image. Improving the robustnessof these methods to the selection of this data is importantto guarantee a more reliable SU performance in practice. • The vast majority of work reviewed in this paper use theLMM to describe the interaction between incident lightand the materials in the scene, even though nonlinear mix-tures are common in many applications [191]. However,as shown in [195], a general nonlinear mixture model isclosely related to a spatially varying version of the LMM,which indicates that linear unmixing with spectral vari-ability is able to address the nonlinear unmixing problemto some extent. Nevertheless, the relationship betweenthese two models deserves to be further investigated.Especially, deciding whether variations in the observedpixel spectra originate from spectral variability, fromnonlinear interactions or from slight abundance variationscan be very difficult. • An aspect that induces difficulties to the evaluation of SUmethods is the lack of more extensive data with reliableground truth. However, there is no clear approach toreliably collect ground truth for abundance values. Thisproblem is more pronounced when spectral variability isconsidered. Particularly, there is not a clearly agreed-uponprotocol to generate realistic synthetic data. A larger,publicly available dataset would strengthen the validationof the methods. • Although many ways have been proposed to modelspectral variability, there is still a distinction betweenrestrained models inspired by specific, concrete appli-cations and mathematically flexible ones that aim for a more generic representation. Combining insight fromthe practical applications with a mathematically thoroughtreatment may lead to improved ways to represent spectralvariability in a given scene. • Many of the methods discussed in this paper rely, ex-plicitly or implicitly, on the solution to complex, non-convex optimization problems which are often solvedonly approximately to achieve a computationally tractablealgorithm. Investigating the use of more reliable ap-proaches to solve those problems can help to evaluatethe potential accuracy of the models by reducing theinfluence from the use of such approximations. • Many algorithms (such as, e.g., MESMA and somestatistical approaches) are computationally expensive anddo not scale very well for large images. Considering thelarge amount of data currently in need of processing, it isimportant to have fast alternatives to solve this problem. • Traditional SU can be readily interpreted as a non-negative matrix factorization problem. This allows us tounderstand many of the limitations of the SU problem,as well as to identify conditions under which it can besolved exactly. However, such understanding is generallynot available when endmember variability is considered,except for the particular case of illumination-based spec-tral variability [200]. A deeper theoretical insight wouldbe valuable to clearly define limiting conditions underwhich this problem can, or cannot be solved.Initially motivated from Earth observation applications,spectral variability is now considered one of the main chal-lenges of SU. Although we have already seen a wealth ofcontributions from both application- and theoretically-orientedresearchers, it is expected that the further exchange of ideasbetween these two areas will help to advance the field evenfurther. R
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