Hyperspectral Image Processing for Automatic Target Detection Applications

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MANOLAKIS, MARDEN, AND SHAWHyperspectral Image Processing for Automatic Target Detection Applications

V O LU ME 14 , NU MBE R 1 , 2003 L INCO LN LA BORATO RY JO URNAL

79

Hyperspectral Image Processing

for Automatic Target DetectionApplicationsDimitris Manolakis, David Marden, and Gary A. Shaw

This article presents an overview of the theoretical and practical issues

associated with the development, analysis, and application of detection

algorithms to exploit hyperspectral imaging data. We focus on techniques that

exploit spectral information exclusively to make decisions regarding the type of

each pixeltarget or nontargeton a pixel-by-pixel basis in an image. First wedescribe the fundamental structure of the hyperspectral data and explain how

these data influence the signal models used for the development and theoretical

analysis of detection algorithms. Next we discuss the approach used to derive

detection algorithms, the performance metrics necessary for the evaluation of

these algorithms, and a taxonomy that presents the various algorithms in a

systematic manner. We derive the basic algorithms in each family, explain how

they work, and provide results for their theoretical performance. We conclude

with empirical results that use hyperspectral imaging data from the HYDICE

and Hyperion sensors to illustrate the operation and performance of various

detectors.

M applications in-volve the detection of an object or activitysuch as a military vehicle or vehicle tracks.

Hyperspectral imaging sensors, such as those illus-trated in Figure 1, provide image data containingboth spatial and spectral information, and this infor-mation can be used to address such detection tasks.The basic idea for hyperspectral imaging stems from

the fact that, for any given material, the amount of ra-diation that is reflected, absorbed, or emittedi.e.,the radiancevaries with wavelength. Hyperspectralimaging sensors measure the radiance of the materials

within each pixel area at a very large number of con-tiguous spectral wavelength bands.

A hyperspectral remote sensing system has four ba-sic parts: the radiation (or illuminating) source, theatmospheric path, the imaged surface, and the sensor.

In a passive remote sensing system the primary sourceof illumination is the sun. The distribution of thesuns emitted energy, as a function of wavelengththroughout the electromagnetic spectrum, is knownas the solar spectrum. The solar energy propagatesthrough the atmosphere, and its intensity and spectraldistributions are modified by the atmosphere. Theenergy then interacts with the imaged surface materi-

als and is reflected, transmitted, and/or absorbed bythese materials. The reflected/emitted energy thenpasses back through the atmosphere, where it is sub-

jected to additional intensity modifications and spec-tral changes. Finally, the energy reaches the sensor,

where it is measured and converted into digital formfor further processing and exploitation.

The signal of interest, that is, the information-bearing signal, is the reflectance spectrum defined by


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80L INCOL N L ABO RA TORY J O URNAL V OL UME 14 , NUMBER 1, 2003

reflectance spectrum

reflected radiation at band

incident radiation at band

( )

( )

( ).

=

The reflectance spectrum, or spectral signature, showsthe fraction of incident energy, typically sunlight, thatis reflected by a material as a function of the wave-

length of the energy.To understand hyperspectral imaging data exploi-

tation, it is important to realize how the presence ofthe atmosphere and the nature of the solar spectrumaffect the relationship between the observed radiancespectra and the associated reflectance spectra. Basi-cally, the observed spectrum would have the sameshape as the reflectance spectrum if the solar spec-trum were flat and the atmosphere had the same

transmittance at all wavelengths. In reality, the mea-sured radiance spectrum is the solar spectrum modi-fied by the transmittance function of the atmosphereand the reflectance spectrum of the imaged groundresolution cell (see Figure 1). The atmosphere restricts

where we can look spectrally because it selectively ab-sorbs radiation at different wavelengths, due to thepresence of oxygen and water vapor. The signal-to-

noise ratio at these absorption bands is very low; as aresult, any useful information about the reflectancespectrum is lost. For all practical purposes these spec-tral bands are ignored during subsequent hyperspec-tral data analysis.

Any effort to measure the spectral properties of amaterial through the atmosphere must consider theabsorption and scattering of the atmosphere, thesubtle effects of illumination, and the spectral re-

FIGURE 1. Hyperspectral imaging sensors measure the spectral radiance information in a scene to detect tar-

get objects, vehicles, and camouflage in open areas, shadows, and tree lines. Imaging sensors on satellites or

aircraft gather this spectral information, which is a combination of sunlight (the most common form of illumi-

nation in the visible and near infrared), atmospheric attenuation, and object spectral signature. The sensor de-

tects the energy reflected by surface materials and measures the intensity of the energy in different parts of the

spectrum. This information is then processed to form a hyperspectral data set. Each pixel in this data set con-tains a high-resolution spectrum, which is used to identify the materials present in the pixel by an analysis of

reflectance or emissivity.

Solar spectrum

Atmosphere Atmosphere

Sensor-measuredradiance

Signature


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81

sponse of the sensor. The recovery of the reflectancespectrum of each pixel from the observed radiancespectrum is facilitated with the use of sophisticatedatmospheric compensation codes.

The resulting reflectance representation, termedthe spectral signature, if sufficiently characterized,can be used to identify specific materials in a scene.Figure 2, for example, illustrates the different reflec-

tance spectra for naturally occurring materials such assoil, green vegetation, and dry vegetation.

The hyperspectral sensors discussed in this articlemeasure radiation in the region of the electromag-netic spectrum dominated by solar illumination (0.4m to 2.5 m), where they acquire a complete reflec-tance spectrum. Unfortunately, such sensors are inef-fective at night because of the absence of solar illumi-nation. The problem can be avoided by extending therange of hyperspectral sensors into the thermal infra-red region (8 m to 14 m), where materials emit

more radiation than they reflect from the sun. Thisextension allows sensors to operate effectively duringboth day and night.

Hyperspectral imaging sensors are basically ad-vanced digital color cameras with fine spectral resolu-tion at given wavelengths of illumination. Instead ofmeasuring three primary colorsred, green, andbluethese sensors measure the radiation reflectedby each pixel at a large number of visible or invisible

frequency (or wavelength) bands. Such an instrumentis called an imaging spectrometer. Spectrometers areinstruments that divide the impinging electromag-netic radiation into a number of finite bands and

measure the energy in each band. (In this sense, thehuman eye can be considered a crude imaging spec-trometer.) Depending upon the design of the spec-trometer, the bands may or may not be contiguousand of equal width. Therefore, the measured discretespectrum is usually not a uniformly sampled versionof the continuous one.

The optical system divides the imaged ground areainto a number of contiguous pixels. The ground areacovered by each pixel (called the ground resolutioncell) is determined by the instantaneous field of view

(IFOV) of the sensor system. The IFOV is a functionof the optics of the sensor, the size of the detector ele-ments, and the altitude. The rectangular arrangementof the pixels builds an image, which shows spatialvariation, whereas the spectral measurements at eachpixel provide the integrated spectrum of the materials

within the pixel.Scene information is contained in the observed ra-

diance as a function of continuous space, wavelength,and time variables. However, all practical sensors havelimited spatial, spectral, radiometric, and temporal

resolution, which results in finite-resolution record-ings of the scene radiance. The spatial resolution ofthe sensor determines the size of the smallest objectthat can be seen on the surface of the earth by the sen-sor as a distinct object separate from its surroundings.Spatial resolution also relates to how well a sensor canrecord spatial detail. The spectral resolution is deter-mined by the width of the spectral bands used tomeasure the radiance at different wavelengths . Theradiometric resolution is determined by the numberof bits used to describe the radiance value measured

by the sensor at each spectral band. Finally, the tem-poral resolution is related to how often the sensor re-visits a scene to obtain a new set of data.

Spectral and radiometric resolution determine howfaithfully the recorded spectrum resembles the actualcontinuous spectrum of a pixel. The position, num-ber, and width of spectral bands depend upon thesensor used to gather the information. Simple opticalsensors may have one or two wide bands, multispec-

FIGURE 2. Different materials produce different electromag-

netic radiation spectra, as shown in this plot of reflectance

for soil, dry vegetation, and green vegetation. The spectral

information contained in a hyperspectral image pixel cantherefore indicate the various materials present in a scene.

Wavelength ( m)

Dry vegetation

Soil

Green vegetation

Reflectance

0.6

0.4

0.2

00.5 1.5 2.52.01.0


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tral sensors a few wide bands, and hyperspectral sen-sors a few hundred narrow bands. Table 1 summarizessome of the distinctions among remote sensing sys-tems that exploit spatial information and spectral in-formation.

As noted in the introductory article to this special

issue of the Journal, entitled Spectral Imaging forRemote Sensing, by Gary A. Shaw and Hsiao-hua K.Burke, multispectral sensing and, later, hyperspectralsensing were developed for remote sensing of thenatural environment, including such applications asmineral exploration, characterization of groundcover, and characterization of crop health. In theseapplications, morphological (or shape) information(which is an important prerequisite for remote sens-ing of man-made objects) is of minimal utility in thedetection process, since the various natural materials

of interest do not have predefined shapes. For ex-ample, an area of diseased crops will have an unpre-dictable size and shape within a larger field of healthycrops. Similarly, mineral deposits will have unpredict-able shapes and may even be intimately mixed withother types of materials.

Another important aspect of traditional applica-tions for hyperspectral imaging is that the applica-tions are oriented toward classifying or grouping

similar pixels, when there are typically many instancesof each class of pixel, and when occasional errors inpixel classification are not significant, since interpre-tation of the scene is based on the clustering of themajority of pixels. For example, in an image of crops,if one in every one thousand pixels is misclassified,

the isolated errors will not change the overall under-standing of whether the crops are healthy or diseased.

In comparison, any target detection applicationseeks to identify a relatively small number of objects

with fixed shape or spectrum in a scene. The process-ing methods developed for environmental classifica-tion are not applicable to target detection for two rea-sons. First, the number of targets in a scene istypically too small to support estimation of statisticalproperties of the target class from the scene. Second,depending upon the spatial resolution of the sensor,

targets of interest may not be clearly resolved, andhence they appear in only a few pixels or even a singlepixel. The isolated character of the targets makes con-firmation by means of clustering of like samples prob-lematic. Target detection is still possible, however, inthese situations. For example, hyperspectral sensingcould be used to increase the search rate and probabil-ity of detection for search and rescue operations atsea. A life jacket, which may be unresolved spatially,

Table 1. Comparison of Spatial Processing and Spectral Processing in Remote Sensing

Spatial Processing Spectral Processing

Information is embedded in the spatial arrangement Each pixel has an associated spectrum that can be

of pixels in every spectral band (two-dimensional image) used to identify the materials in the corresponding

ground-resolution cell

Image processing exploits geometrical shape information Processing can be done one pixel at a time

Very high spatial resolution required to No need for high spatial resolution

identify objects by shape (many pixels on the target) (one pixel on the target)

High spatial resolution requires large apertures Spectral resolution more important than

and leads to low signal-to-noise ratio spatial resolution

Data volume grows with the square of the Data volume increases linearly

spatial resolution with the number of spectral bands

Limited success in developing fully automated Fully automated algorithms for spectral-featurespatial-feature exploitation algorithms exploitation have been successfully developed

for selected applications


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can still be detected and identified on the basis of itsextreme color contrast with the ocean background.

In practice, hyperspectral sensors represent a delib-erate trade-off in which spatial resolution is degraded

in favor of improved spectral resolution. By implica-tion, hyperspectral imaging is therefore best matchedto applications in which spectral information is morereliable or measurable than morphology or shape in-formation. For example, traditional analysis of passiveimagery of military vehicles depends strongly uponbeing able to determine the length, width, and distin-guishing features of a vehicle. If the vehicles are par-tially hidden under foliage or camouflaged nets, themorphological information is unreliable and alternatemeans of detection and identification are needed.

Since hyperspectral imaging does not rely on shapeinformation, it is less impacted by attempts at con-cealment and deception.

Historically, interpretation of passive imagery hasgenerally depended upon either human analysis of vi-sual information in a live video or a still image, or elseupon machine measurement and analysis of param-eters that distinguish a man-made object (i.e., a tar-get) from the natural surroundings. As described inthe next section, the high dimensionality of hyper-spectral imagery precludes full exploitation of the in-

formation through simple visual analysis of the data.In fact, visual representation of the high-dimensionaldata is itself a challenge, and the essence of hyperspec-tral detection algorithms is the extraction of informa-tion of interest from the high-dimensional data.

Data Representation and Spectral

Information Exploitation

In many applications, the basic automated processing

problem is to identify pixels whose spectra have aspecified spectral shape. This problem raises the fol-lowing fundamental issues: (1) does a spectrumuniquely specify a material, (2) are the spectra of anymaterial the same when observed at different times,(3) how is the spectrum of a ground material relatedto the spectrum observed by the sensor, and (4) howshould we compare two spectra to decide if they arethe same? Answering these questions is a challengingundertaking, involving resources from the areas ofspectroscopy, remote sensing, high-dimensional Eu-

clidean geometry, statistics, and signal processing. Inthis section we make an initial attempt to addressthese issues by introducing the topics of vector repre-sentation of spectra, spectral variability, mixed-pixelspectra, and correlation among spectral bands.

Data Cubes and Spectral Vectors

As a result of spatial and spectral sampling, airbornehyperspectral imaging (HSI) sensors produce a three-dimensional (3D) data structure (with spatial-spatial-spectral components), referred to as a data cube. Fig-

ure 3 shows an example of such a data cube. If weextract all pixels in the same spatial location and plottheir spectral values as a function of wavelength, theresult is the average spectrum of all the materials inthe corresponding ground resolution cell. In contrast,

FIGURE 3. Basic data-cube structure (center) in hyperspectral imaging, illustrating the simultaneous spatial and spec-

tral character of the data. The data cube can be visualized as a set of spectra (left), each for a single pixel, or as a stack

of images (right), each for a single spectral channel.

Spe

ctrald

imen

sion

Spatial dimension

Spatialdimens

ion

Reflectance

Wavelength

Spectra for asingle pixel

Image at asingle wavelength


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the values of all pixels in the same spectral band, plot-ted in spatial coordinates, result in a grayscale imagedepicting the spatial distribution of the reflectance ofthe scene in the corresponding spectral wavelength.

The observed spectral radiance data, or derived ap-parent surface reflectance data, can be viewed as ascattering of points in an K-dimensional Euclideanspace, denoted byK, where Kis the number of spec-tral bands. Each spectral band is assigned to one axisof the space, all axes being mutually orthogonal.Therefore, the spectrum of each pixel can be viewedas a vectorx= [x1 ,x2 , ,xK]

T, where Tdenotes ma-trix transposition. The tip of this vector correspondsto an K-dimensional point whose Cartesian coordi-natesxi are the radiance or reflectance values at each

spectral band. Since each componentxi 0, spectralvectors lie inside the positive cone ofK. Notice thatchanges in the level of illumination can change thelength of a spectral vector but not its orientation,

which is related to the shape of the spectrum. If eachmaterial is characterized by a unique deterministicspectrum, which can serve as a spectral fingerprint,

we can measure the similarity between two spectraxandyby using the Euclidean distance measure

x y = ( )= x yk kk

K2

1

or the spectral angle map (SAM) measure

=( , ) arccos ,x yx y

x y

T

wherexTyis the dot product of vectorsxandy. Thesemetrics are widely used in hyperspectral imaging dataexploitation, even if there are no optimality proper-ties associated with them.

Exploiting Spectral Correlation

When targets are spatially resolved and have knownor expected shapes, the spectral information availablein a hyperspectral image may serve to confirm the na-ture of the target and perhaps provide additional in-formation not available from morphological analysisalone. When targets are too small to be resolved spa-tially, however, or when they are partially obscured or

of unknown shape (e.g., an oil slick), detection mustbe performed solely on the basis of the spectral infor-mation. The discrimination and detection of differ-ent materials in a scene exclusively through the use of

spectral information is sometimes termed nonliteralexploitation, in reference to the fact that this discrimi-nation and detection does not rely on literal interpre-tation of an image by means of morphological analy-sis and inference.

If every material had a unique fixed spectrum andwe could accurately measure its value at a fine grid ofwavelengths, then we could discriminate between dif-ferent materialsat least in theoryby comparingtheir spectra. In this sense, a unique fixed spectrumcould be used as a spectral signature of the material.

For example, the three types of materials in Figure 2could be easily discriminated by measuring the spec-tra in a narrow band around the 1.0-m wavelength.

The situation is very different in practice, however.Figure 4 shows the range of variation of spectra, from37 pixels for a vehicle to 8232 pixels for a tree-coveredarea. Simple inspection of this plot leads to some cru-cial observations. If we observe a spectrum residing inthe red region of the vehicle class where it overlaps

with the green spectrum of the tree class, it is almost

FIGURE 4. Illustration of spectra variability for the tree class

and the vehicle class, where the green and red shaded areas

represent the range of observed spectra for trees and ve-

hicles, respectively. At wavelengths where the vehicle spec-

trum (in red) overlaps the tree spectrum (in green), such as

at 0.7 m and 1.25 m, it is almost impossible to discriminate

between the two classes by using observations from a

single spectral band.

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

0

0.2

0.4

0.6

0.8

1.0

Wavelength ( m)

Reflectance

0.7 1.25 Region o

overlap


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FIGURE 5. Simultaneous exploitation of the spectral bands at 0.7 m and 1.25 min the spectra in Figure 4 makes discrimination possible because of a clearly de-

fined decision boundary between the tree class and the vehicle class.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

051015202530354045

0.1 0 0.1 0.2 0.3 0.40

10

20

30

40

50

60

0.1 0 0.1 0.2 0.3 0.4

Decisionboundary

Vehicle class

Tree class

0

0.1

0.2

0.3

0.5

0.6

0.7

0.8

0.4

Histogramo

fbandat1.2538

m

Histogram of band at 0.7087 m

impossible to say with certainty if that spectrum cor-responds to a vehicle or tree pixel. In actual systems,

where we obtain measurements only in certain spec-tral bands, we would expect discrimination to be evenmore difficult.

The key question now is thiscan we discriminatebetween vehicle and tree pixels based on spectral ob-

servations alone? To answer this question, we selecttwo narrow spectral bands, centered on 0.7 m and1.25 m. Figure 4 shows that it is not possible, usingany one band alone, to discriminate between the twoclasses because they overlap completely. Fortunately,this observation is not the final answer. If we plot thetwo-band data as points in a two-dimensional space

with the reflectance values at the two bands as coordi-nates, we obtain the scatter plot shown in Figure 5.

Surprisingly, it is now possible to separate the twoclasses by using a nonlinear decision boundary, de-spite the complete overlap in each individual band.

The fundamental difference between Figures 4 and5 is that Figure 5 allows for the exploitation of the ex-tra information in the simultaneous use of two bands.This is the essence of multiple spectral-band data ex-

ploitation in remote sensing spectroscopy.

Spectral Variability Models

Unfortunately, a theoretically perfect fixed spectrumfor any given material doesnt exist. The spectra ob-served from samples of the same material are neveridentical, even in laboratory experiments, because ofvariations in the material surface. The amount of vari-ability is even more profound in remote sensing ap-


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plications because of variations in atmospheric condi-tions, sensor noise, material composition, location,surrounding materials, and other factors. As a result,measured spectra corresponding to pixels with the

same surface type exhibit an inherent spectral vari-ability that prevents the characterization of homoge-neous surface materials by unique spectral signatures,as shown in Figure 6(a).

Another significant complication arises from theinterplay between the spatial resolution of the sensorand the spatial variability present in the ground scene.The sensor integrates the radiance from all materials

within the ground surface that are seen by the sen-sor as a single image pixel, as shown in Figure 6(b).Therefore, depending on the spatial resolution of the

sensor and the distribution of surface materials withineach ground resolution cell, the result is a hyperspec-tral data cube comprised of pure and mixed pix-els, where a pure pixel contains a single surface mate-rial and a mixed pixel contains multiple materials.Spectral variability and mixed-pixel interference arethe main obstacles that need to be addressed andovercome by HSI data-exploitation algorithms.

In the spectral band space, spectra without vari-ability (known as deterministic spectra) correspond toa single fixed point, whereas the tips of vectors corre-

sponding to spectra with variability (known as ran-dom spectra) produce a cloud of points. Determinis-

tic spectra are used in spectral libraries as idealizedprototype spectral signatures associated with differentmaterials. However, most spectra appearing in realapplications are random, and their statistical variabil-

ity is better described by using probabilistic models.We next present three families of mathematical

models that can be used to characterize the spectralvariability of the pixels comprising a hyperspectraldata cube. To understand the basic ideas for the dif-ferent models, we use as an example two spectralbands from a Hyperspectral Digital Imagery Collec-tion Experiment (HYDICE) cube.

Probability Density Models

Figure 7 shows a picture of the ground area imaged by

the HYDICE sensor and a scatter plot of the reflec-tance values for two spectral bands. The scatter plotcan help determine areas on the ground that havesimilar spectral reflectance characteristics. The resultshown in the scatter plot is a set of spectral classes thatmay or may not correspond to distinct well-definedground-cover classes. This distinction between spec-tral classes is illustrated in Figure 8, which uses colorcoding to show several spectral classes in the scatterplot and the corresponding ground-cover areas in theimage (these results were obtained by using the k-

means clustering algorithm with an angular distancemetric) [1].

Subpixeltarget

(mixed pixel)

Full-pixeltarget

(pure pixel)

Wavelength ( m)

Reflectance

(a)(b)

FIGURE 6. Illustration of spectral variability and mixed-pixel interference. (a) Measured spectra corresponding to pixels with

the same surface type exhibit an inherent spectral variability that prevents the characterization of homogeneous surface materi-

als with unique spectral signatures. The gaps in the spectra correspond to wavelengths near water absorption bands where the

data has been discarded because of low signal-to-noise ratio. (b) Radiance from all materials within a ground resolution cell is

seen by the sensor as a single image pixel. Therefore, depending on the spatial resolution of the sensor and the spatial distribu-

tion of surface materials within each ground resolution cell, the result is a hyperspectral data cube comprised of pure and

mixed pixels, where a pure pixel contains a single surface material and a mixed pixel contains multiple materials. Spectral vari-

ability and mixed-pixel interference are the main obstacles that need to be addressed and overcome by hyperspectral data-ex-

ploitation algorithms.


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87

A reasonable probabilistic model for such spectraldata is provided by the following probability mixturedensity:

f fk kk

M

( ) ( ; ) ,x x==

1

where we assume that each observed spectrum x isgenerated by first selecting a spectrally homogeneousclass kwith probabilityk, where k 0 and

k

k

M

=

=1

1,

and then selecting a spectrumxaccording to the con-ditional probability lawf(x; k) specified by the pa-rameter vectors k.

The component densitiesf(x; k) are usually cho-

sen as multivariate normal distributions defined by

f eK

T

( )( )

,/ /

( ) ( )x

x x=

1

2 21 2

1

21

where E{ }x is the mean vector,

E ( )( )x x T

FIGURE 7. Ground image from the Hyperspectral Digital Im-

agery Collection Experiment (HYDICE) sensor (top), and a

two-dimensional scatter diagram (bottom) illustrating joint

reflectance statistics for two spectral bands in the scene.

FIGURE 8. Color-coded example of the variety of spectral

classes obtained by clustering in the full-dimensional spec-

tral space (top), but illustrated in the scatter plot (bottom)

with only two spectral bands.

Reflectance(960-

mb

and)

Reflectance (620- m band) Reflectance (620- m band)

Reflectance(960-

mb

and)


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is the covariance matrix, and | | represents the de-terminant of matrix . We use the notation x ~N( , ) to denote a multivariate normal distribu-tion with mean vector and covariance matrix .

The quadratic expression in the exponent of the mul-tivariate normal distribution,

2 1= ( ) ( ) ,x x T

is a widely used statistical distance measure, known asthe Mahalanobis distance [2]. This quantity plays amajor role in the design and evaluation of target de-tection algorithms.

Subspace Models

The point specified by the tip of a spectrum vectorcan be anywhere in the K-dimensional band space.The direction of this vector is specified by the shapeof the spectrum and the length of this vector is speci-fied by the amplitude of the spectrum. Therefore,changes in the illumination of a pixel alter the lengthof its spectral vector but not its direction. In thissense, we can say that the illumination-induced vari-ability is confined into a one-dimensional subspace ofthe band space.

In general, asubspace modelrestricts the spectrum

vector to vary in an M-dimensional subspace of theband space, whereM< K. Clearly, the variability in-creases as Mincreases from one to K. The observedsubspace spectrum is described by

x s Sa= =

=

ak kk

M

1

. (1)

The vectors sk (assumed linearly independent), orequivalently the matrix S (assumed full rank), providea basis for the spectral variability subspace. We can in-

troduce randomness to the subspace model in Equa-tion 1 either by assuming that akare random variablesor by adding a random error vector w to the right-hand side:

x s w Sa w= + = +

=

ak kk

M

1

.

Subspace variability models can be obtained by using

either physical models and the Moderate ResolutionTransmittance (MODTRAN) code [3] or statisticalapproaches such as principal-component analysis [4].

Linear Spectral Mixing ModelsThe most widely used spectral mixing model is thelinear mixing model [5], which assumes that the ob-served reflectance spectrum, for a given pixel, is gen-erated by a linear combination of a small number ofunique constituent deterministic spectral signaturesknown as endmembers. The mathematical equationand the constraints defining the model are

x s w Sa w= + = +

=

=

=

a

a

a

k k

k

M

k

k

M

k

1

1

1

0

(additivity constraint)

(positivity constraint),

where s1, s2, , sM, are theMendmember spectra,assumed linearly independent, a1, a2, , aM, are thecorresponding abundances (cover material fractions),andwis an additive-noise vector. Endmembers maybe obtained from spectral libraries, in-scene spectra,

or geometrical techniques [5].A linear mixture model based on three or four end-

members has a simple geometrical interpretation as atriangle or tetrahedron whose vertices are the tips ofthe endmember vectors. In general, spectra satisfyingthe linear mixture model with both sets of constraintsare confined in an K-dimensional simplex studied bythe mathematical theory of convex sets [6]. If thereare Kbands in the mixture model there can be K+ 1or fewer endmembers. IfM= K+ 1 there is a uniquesolution for the fractional abundances; however, we

usually assume thatMKto allow the use of least-squares techniques for abundance estimation.

Figure 9 uses three-band spectral vectors to illus-trate the basic ideas of the probability density, sub-space, and linear mixture models for the characteriza-tion of spectral variability.

The linear mixture model holds when the materi-als in the field of view are optically separated so thereis no multiple scattering between components. The


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89

combined spectrum is simply the sum of the frac-tional area times the spectrum of each component.However, when different materials are in close con-tact or close proximity in a scattering surface, such asthe mineral grains in a soil or rock, the resulting spec-trumdepending on the optical properties of eachcomponentis a highly nonlinear combination ofthe endmember spectra. In such cases, a nonlinearmixing model is needed to describe the spectrum ofmixed pixels.

More elaborate models can be obtained if the end-

member spectra are randomly and independentlydrawn from multivariate normal distributions. Suchstochastic mixing models have been developed else-

where [7], but their applicability is limited becausethe estimation of their parameters leads to highlynonlinear optimization problems.

Dealing with spectral signature variability andspectral compositions in mixed pixels are among themost challenging problems in HSI data exploitation,both theoretically and practically.

Design, Evaluation, and Taxonomyof Target Detectors

In theory, the design and evaluation of detection algo-rithms is facilitated by assuming some meaningfulprobability distributions for the target and non-targetspectra. The key features of the adopted probabilisticsignal models and the used criterion of optimality de-termine the nature of the obtained detectors and theirperformance.

Theoretical Design and Performance Evaluation

The mathematical framework for the design andevaluation of detection algorithms is provided by thearea of statistics known as binary hypothesis testing.There are several approaches for the systematic designof detection algorithms. However, it is well knownthat detectors based on the likelihood ratio (LR) testhave certain advantages. First, in some sense LR testsminimize the risk associated with incorrect decisions.Second, the LR test leads to detectors that are opti-

mum for a wide range of performance criteria, in-cluding the maximization of separation between tar-get and background spectra.

For the purpose of theoretical analysis, spectra aretreated as random vectors with specific probabilitydistributions. Given an observed spectrumxwe wantto choose between two competing hypotheses:

H

H

0

1

:

:

target absent

target present.

Band 1

Band 2

Band 3

Linear mixing model

Band 1

Band 2

Band 3

Probability density model

Band 1

Band 2

Band 3

Subspace model

Convex hull

Subspace

hyperplane

FIGURE 9. Illustration of models for the description of spectral variability. Probability density models provide the

probability for a spectrum to be at a certain region of the spectral space. Subspace models specify the linear vec-

tor subspace region of the spectral space in which spectral vectors are allowed to reside. Linear mixing models,

with the aid of positivity and additivity constraints, permit spectra only in a convex hull of the spectral space.

Ifp(x|H0) andp(x|H1) are the conditional probabil-ity density functions ofxunder the two hypotheses,the LR is given by

( )( |

( |.x

x

x

p

p

target present)

target absent)(2)

If(x) exceeds a certain threshold , then the targetpresenthypothesis is selected as true. Basically, the LRtest accepts as true the most likely hypothesis. This


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suggests the detector structure shown in Figure 10.The LR testy= (x), or any monotonic function ofit, provides the information that is used to decide

whether a target is present. For this reason, we oftenrefer to this function as adetection statistic.

A practical question of paramount importance to asystem designer is where to set the threshold to keepthe number of detection errors (target misses and falsealarms) small and the number of correct detectionshigh. Indeed, there is always a compromise betweenchoosing a low threshold to increase the probability

of (target) detection PDand a high threshold to keepthe probability of false alarm PFA low. For any givendetector, the trade-off between PD and PFA is de-scribed by the receiver operating characteristic (ROC)curves, which plot PD() versus PFA() as a functionof all possible values of the threshold . Therefore,ROC curves provide the means to evaluate detectorperformance or compare detectors independently ofthreshold selection. Figure 11 illustrates the trade-offsin PDand PFA values for a typical ROC curve. Clearly,any systematic procedure to determine ROC curves

or the threshold requires specifying the distributionof the observed spectrax under each of the twohypotheses.

If the conditional densities in Equation 2 are com-pletely known (in which case the hypotheses arecalled simplehypotheses), we can choose the thresh-old to make the detector specified by the LR an opti-mum detector according to several criteria. The Bayescriterion, used widely in classification applications,

chooses the threshold in a way that leads to the mini-mization of the probability of detection errors (bothmisses and false alarms). In detection applications,

where the probability of occurrence of a target is very

small, minimization of the error probability is not agood criterion of performance, because the probabil-ity of error can be minimized by classifying everypixel as background. For this reason, we typically seekto maximize the target probability of detection whilekeeping the probability of false alarm under a certainpredefined value. This approach is the celebratedNeyman-Pearson criterion, which chooses the thresh-old to obtain the highest possible PD while keepingPFA . Hence the ROC curve of the optimumNeyman-Pearson detector provides an upper bound

for the ROC of any other detector.In almost all practical situations, the conditional

probability densities in Equation 2 depend on someunknown target and background parameters. There-fore, the ROC curves depend on unknown param-eters and it is almost impossible to find a detector

FIGURE 11. The standard performance metric for detection

applications is the receiver operating characteristic (ROC)

curve, which shows the probability of detection PD versus

the probability of false alarm PFA as the detection threshold

takes all possible values. The determination of ROC curves

in practice requires the estimation of PD and PFA. The accu-

racy of estimating probability as a ratio (PA = NA / N, where

NA is the number of occurrences of event A in Ntrials) de-

pends on the size of N. Roughly speaking, the accuracy P

of P is in the range of 10/N.

FIGURE 10. Decomposition of optimum-likelihood-ratio de-

tector. Every detector maps the spectrum x of the test pixel

(a multidimensional vector) into a scalar detection statistic

y = (x). Then the detection statistic is compared to a

threshold to decide whether a target is present.

DetectorDetector

TargetTargetNo targetNo target

Detectionstatistic

Detectionstatistic

ThresholdThresholdx

y>

y

Hyperspectral Image Processing for Automatic Target Detection Applications

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