**Multispectral Data Analysis: A Signal Theory Perspective©**

by David Landgrebe

School of Electrical Engineering

Purdue University

West Lafayette IN 47907-1285

landgreb@ecn.purdue.edu

*Preface*

> This document is intended to be a "How to\..." document. How does one
> analyze multispectral data in a fundamentally sound, multivariant
> fashion. The audience in mind is an Earth science researcher who must
> conduct such analyses to obtain information needed for his/her
> research. After a brief introduction, there are two major parts, the
> first having to do with multispectral data of conventional
> dimensionality and the second on analysis of hyperspectral data. Each
> part begins with a brief discussion of the basic concepts to be
> applied. Example analyses using the MultiSpec data analysis software
> system are given to lead one through applying the theory covered in
> the concepts section to actually analyze an example data set. Copies
> of MultiSpec and the data are available upon request without charge.
>
> The problem of analyzing airborne or spaceborne multispectral data is
> obviously a broad one, and clearly not all of the possible facets can
> be covered in such a brief document as this. The intent here is to
> pursue one particular facet of such analysis from beginning to end
> with enough depth to result in an analysis process which is well
> founded in sound principles of signal theory and their application.
> The particular facet chosen to be the vehicle here is that of
> analyzing the multispectral data into useful themes or classes by use
> of discriminant analysis. In the process the added potential of
> hyperspectral data over data of conventional dimensionality is
> demonstrated.

Introduction

The analysis of multispectral image data is now a quite common task for
Earth scientists and other users of such data. However, it is not a
trivial task. Extraction of specific information from a multiband data
set, especially as the number of bands becomes large, is a complex
problem requiring the application of techniques based on rigorous
fundamentals of both Earth science and signal processing theory if the
full potential of such data is to be realized. Although many perfectly
useful ad hoc, situation-specific techniques are described in the
literature, in this document, we shall describe rigorously based
procedures which are sufficiently general to be suitable for use in a
broad range of both local and large area problems. The presentation is
divided into two main parts. The first deals with analysis of
multispectral data of conventional dimensionality (of the order of 10 to
20 spectral bands). The second deals with hyperspectral data (100 or
more spectral bands). To clarify the motivation for the procedures, we
shall begin with a very concise statement of the basic principles and
concepts involved.

Analysis of Conventional Multispectral Data

Background and Concepts

Basic Tenant of Remote Sensing

We begin with the basic idea of remote sensing, that is, that
information is available at the aperture of a sensor, based upon the
electromagnetic fields emanating from the surface and arriving at the
sensor, and in particular via the

> • Spectral,
>
> • Spatial, &
>
> • Temporal Variations

of those fields. For simplicity, at this time we shall concentrate upon
the optimal use of spectral and spatial variations, focusing primarily
upon spectral characteristics, since multispectral data provides unique
potential for information extraction in this case. We shall leave use of
temporal ones to another time.

The Broad View and Signal Space Concepts

Figure 1 provides a systems overview of a passive, optical remote
sensing system. It is seen that, using illumination provided by the sun,
the sensor system views the portion of Earth of interest, and the
resulting data are transmitted to the processing station. There, some
type of preprocessing is often performed, followed by the application of
an appropriate data analysis algorithm and delivery of the resulting
information to the user. Use of ancillary data and human interaction
with the data stream typically take place during preprocessing and
information extraction. Thus, rather than being automatic, the
processing in mind is more accurately described as human-assisted
machine processing. This is significant because in a fully automatic
system one gives up the strength that human intelligence and
adaptability can add to the analysis process. We will thus proceed from
the viewpoint that a well-designed analysis scheme should use the
strengths of both human and machine, but do so in a manner that
maintains economy and objectivity while maximally enhancing performance.

![](media/image1.pct){width="5.166666666666667in"
height="2.6944444444444446in"}

*Figure 1. A Remote Sensing System Overview.*

Figure 2 focuses attention more directly on the information extraction
process, and casts it in the framework of a multivariant classifier. The
patterns to be recognized are patterns in the spectral variations which
have been sensed.

![](media/image2.pct){width="5.388888888888889in"
height="1.4305555555555556in"}

*Figure 2. A Functional Diagram for a Remote Sensing System.*

More specifically, Figure 2 draws attention to the fact that the job of
the sensor system is to transform the information contained in the
observation space into a feature space. In the context of a
multispectral remote sensing system, this amounts mathematically to
transforming *continuous* spectral functions into a *finite dimensional*
*discrete* space. As an example, Figure 3 shows sample spectral patterns
from three simple classes in (continuous) spectral space being sampled
at two discrete wavelengths, λ1 and λ2, and they then appear as three
points in a (two dimensional) feature space.

+-------------------------------------+--------------------------------+
| ![](media/image                     | ![](media/image4.pc            |
| 3.pct){width="2.9027777777777777in" | t){width="2.236111111111111in" |
| height="1.9583333333333333in"}      | height="1.8055555555555556in"} |
|                                     |                                |
| *Spectral Space*                    | *Two Dimensional Feature       |
|                                     | Space*                         |
+-------------------------------------+--------------------------------+

*Figure 3. Spectral Space and Feature Space Representations of Spectral
Responses.*

Of course, sampling at additional wavelengths in Spectral Space would be
possible; the result would be to increase the dimensionality of the
resulting feature space. Though spaces of dimensionality greater than
three cannot be diagrammed for human view, there is no fundamental
reason why higher dimensional spaces could not be used and would be
potentially beneficial. It is also the case that transformations more
complex than simple, discrete wavelength sampling are possible, and
frequently desirable. This becomes increasingly important with sensors
with large numbers of bands and will be treated later.

By this transformation, from Spectral Space to N-dimensional feature
space, the problem of recognition of a pattern is thus converted into
one of determining into what part of the N-dimensional feature space a
given unknown falls so that its association with the appropriate class
can thus be determined. There are many ways that this could be
accomplished. Two examples are shown in the following diagrams.

+-----------------------------------+-----------------------------------+
| ![](media/image5.                 | ![](media/image6.                 |
| pct){width="3.1527777777777777in" | pct){width="2.8472222222222223in" |
| h                                 | height="2.361111111111111in"}     |
| eight="2.4027777777777777in"}*(a) |                                   |
| Parallelepiped Classifier*        | *(b) Minimum Distance Classifier* |
+-----------------------------------+-----------------------------------+

*Figure 4. Decision Boundaries for Two Classification Algorithms*

Assume that data from two classes falls in the two areas indicated in
Figure 4 (a) and (b) above. The Parallelepiped Classifier is of a type
known as an "absolute classifier," i.e., it identifies pixels as being
of Class 2 if they fall within the region marked "decision boundary" in
Figure 4(a). Absolute classifiers are ones which decide in favor of a
given class without regard to the location of other classes. It is
generally the case that, though they may be conceptually simplier,
absolute classifiers do not perform as well as "relative classifiers,"
i.e., classifiers that decide in favor of a class after having
considered all possible classes.

An example of a relative classifier is the Minimum Distance Classifier.
In this case, one associates a given feature vector with the class which
has the nearest mean value. In the case of the two classes of Figure 4,
the decision boundary partitioning the feature space into Class 1 and
Class 2 regions would be as shown in Figure 4(b). After locating the
centroid or conditional class means for the two classes, the decision
boundary becomes the perpendicular bisector of the line connecting these
mean values.

These two algorithms may provide satisfactory performance in simple
cases, but most often algorithms which result in nonlinear decision
boundaries are required. For example, consider the case of the classes
shown in Figure 5.

![](media/image7.pct){width="2.8472222222222223in"
height="2.2222222222222223in"}

*Figure 5. A nonlinear decision boundary.*

In this case, the decision boundary is again a relative classifier, but
with a second order decision surface. It is located not with respect to
the class means alone, but taking into account the spread or
distribution and orientation of the two classes. In particular, it is
located halfway between the nearest points of the two classes, and its
curvature accounts for the fact that with actual data, the points are
not necessarily confined to a tightly defined region, but may scatter
into outlying regions. This curvature accounts for the fact that Class 2
with its larger variance may be expected to scatter more than Class 1.

A classifier with the characteristics of this latter algorithm has been
found in practice to be very practical, i.e., powerful enough to deal
with typical data complexity, but not so complex as to require excessive
calculation or training. Indeed, the concepts described here in rather
simplistic terms in 2-dimensional space have been rigorously studied and
derived based upon fundamentally sound principles of signal processing
in noisy environments. This is certainly the situation one finds in
practical remote sensing circumstances. Indeed, while it is customary
when discussing classification algorithms, to show the classes as quite
separable as was done above, in practice, the data usually occupy a
continuum with no clumping or clustering into discrete units. Thus, one
must choose classifiers and determine their parameters in some optimal
way, recognizing that 100% separability is not usually possible.

Optimal System Design

It is, thus, desirable that the procedure to be used be quantitatively
optimal in some known and appropriate sense. What must one have in order
to carry out an optimal system design? The requirements are:

1\. A Quantitative System Model. In this case this implies

• Scene Model: (We will use Stochastic Process via Training Samples)

• Sensor Model: (We will use a generalized sampling model)

• Processing Model: (We will use a pattern classifier model)

2\. An Optimality Criterion, objectively expressed. (We will use
classification accuracy)

We have already described our means of sensor modeling, that of
generalized sampling and transformation to N-dimensional space, as
above. The Processing Model and Optimality Criterion were then defined
in terms of the partitioning of that N-dimensional space into
non-overlapping regions, one each for each class present in the data.
The matter of Scene Modeling bears some further comment.

A scene model describes the manner in which the continuous spectra
(i.e., before the above transformation) is modeled. There are a number
of theoretical approaches upon which to base such a model of the scene.
Possibilities include

> • A Deterministic Model
>
> • Fuzzy Set Theory
>
> • A number of others which are less well developed theoretically,
> e.g., Interval Valued Probabilities, Dempster-Shafer Theory of
> Evidence, Chaos and Fractal Technology.
>
> • The Theory of Stochastic or Random Processes

Each of these have desirable features. A deterministic model has the
advantage of conceptual simplicity, but does not take advantage of all
information available from the spectra. Fuzzy Set Theory is focused on
situations where knowledge of circumstances and parameters is not
precise. The same is true for the third set, which also deal with cases
when partially conflicting evidence about a given parameter may be
present, characteristics which are certainly present in aspects of
remote sensing.

However, these are not the key characteristics of such data. Rather, our
focus is upon models which maximally convey information of interest.
Thus, we choose to focus upon the high degree of the scene complexity.
Even the noise-free spectral response of a given material of interest
does not occur in practice as a deterministic response, but has a degree
of variability about it. Added to this variability, to be sure, is noise
or corrupting influences. However, since a significant part of this
overall variability is diagnostic of the material itself, it is
important to include it in the model. This is precisely the circumstance
for which the *random process model* was designed. The theory of Random
Processes[^1] is very extensively and rigorously developed in the
literature.[^2],[^3],[^4],[^5] It has found its greatest use, for
example, in the design of modern communication systems, due to its
ability to deal with complex signals, containing complex mixtures of
families of signals embedded in noise. As a result of its wide use, many
theoretical tools and techniques are available for it.

The formal definition of a Random Process is given as consisting of

> • the ensemble of random functions, plus
>
> • a probabilistic description about them.

In this case, what makes up the ensemble of the random functions is the
list of all spectral responses over which the system is to be optimized,
or at least a representative subset of them. What this amounts to, of
course, depends upon the application context.

To make the implications of this theory more clear, Figure 6 shows a
series of spectral response functions of a certain soil type, together
with the corresponding feature

space diagram for a particular two dimensional space. Parts (a & b) of
the figure show a deterministic model, as might be obtained by averaging
several observations. Parts (c & d) show a "signal + noise" model of
this soil. Part (c) was actually derived by displaying the maximum
variation present between five samples of the same soil at each
wavelength.

However, in order to fully describe this soil spectrally from the five
observations, the data of parts (e & f) would be necessary.

+-----------------------------------+-----------------------------------+
| Spectral Space                    | Feature Space                     |
|                                   |                                   |
| ![](media/image8                  | ![](media/image9.                 |
| .pct){width="3.013888888888889in" | pct){width="2.2916666666666665in" |
| height="1.9305555555555556in"}    | height="1.8472222222222223in"}    |
|                                   |                                   |
| *(a) The spectral response of a   | *(b) The 2-D feature space        |
| specific soil.*                   | equivalent for (a).*              |
+-----------------------------------+-----------------------------------+

+-----------------------------------+----------------------------------+
| ![](media/image10.                | ![](media/image11.p              |
| pct){width="2.9722222222222223in" | ct){width="2.2916666666666665in" |
| height="1.8611111111111112in"}    | height="1.8472222222222223in"}   |
|                                   |                                  |
| *(c) The range of responses for   | *(d) The 2-D feature space       |
| five samples of the same soils.   | equivalent for (c).*             |
| This amounts to a signal + noise  |                                  |
| model.*                           |                                  |
+-----------------------------------+----------------------------------+

+-----------------------------------+----------------------------------+
| ![](media/image12                 | ![](media/image13.p              |
| .pct){width="3.013888888888889in" | ct){width="2.2916666666666665in" |
| height="1.8888888888888888in"}    | height="1.8472222222222223in"}   |
|                                   |                                  |
| *(e) The actual spectra of the    | *(f) The 2-D feature space       |
| five samples of the same soil.    | equivalent for (e).*             |
| Compared to (b), this shows how   |                                  |
| the spectra vary about the mean.* |                                  |
+-----------------------------------+----------------------------------+

*Figure 6. Various models of the spectral response for a specific soil
type.*

This example helps to display why the random process model and the kind
of transformation used is significant to the preservation of information
in spectral features. Note that,

> • In the deterministic case, only the location of the centroid of the
> spectral response is maintained. No information about how the spectra
> vary about their mean in spectral space or how the class distributes
> itself over the feature space is provided.
>
> • In the signal + noise case, the centroid and some of the information
> of the shape of the distribution, that having to do with the range of
> the variation, are modeled, but not other characteristics of that
> variation.
>
> • In the last of the three, if each of the five (and perhaps a lot
> more) 2-dimensional data points were to be plotted in the
> 2-dimensional space, they might only occupy the area indicated in
> black. If there were another class in the near vicinity of this class,
> there is less likelihood of overlap if the more complete
> representation of case 3 is used than that of case 2.

Thus, case 3 provides a more complete representation of the data. It
shows not only how the data vary in each band, but how they co-vary
between bands, something that cannot be readily visualized in spectral
space. In the feature space on the right, this characterization shows up
as not only where in the space the centroid is, but the shape of the
variation about that centroid. Indeed, if the spectral space on the left
were sampled at enough wavelengths and these samples expressed in all N
of these dimensions, this would provide a complete characterization of
how the data from this family of spectra vary, complete in the sense
that the spectral space representation of each individual sample could
be completely reconstructed from the feature space representation. Thus,
one could be sure that no information had been lost in transforming from
spectral space to feature space[^6].

We note in passing that it has been found that the shape of the
distribution becomes increasingly important as the dimensionality of the
representation increases[^7]. For hyperspectral data, the shape of the
distribution becomes even more important than its location as determined
by its centroid. Indeed, for high dimensional cases such as for
hyperspectral data, one must avoid using two or three dimensional
concepts, as the rules of geometry in high dimensional space are quite
different than those of the two or three dimensional spaces which have
formed our intuitive understandings. One must also avoid any type of
preprocessing which, while it may adjust the location of the centroid in
an assumed useful way, it also unknowingly distorts the shape of the
distribution. We shall develop this idea more fully later.

Under various circumstances, any of these models might be used. In fact,
all are found or implied by the various analysis methods found in the
literature. Generally speaking, simpler classification tasks can be done
with simpler data (fewer bands, fewer gray-values per band) using
simpler models and correspondingly simpler classification algorithms.
For example, a deterministically based decision algorithm might work
satisfactorily for easily identifiable classes modeled in a
deterministic fashion. One might also choose a linear classifier where
somewhat more difficult classes have been modeled in signal-plus-noise
fashion. However, to extract the most information (most or most detailed
classes, highest accuracy), one will require more complex data and will
want the most complete model. In this case, one would also want to use a
decision algorithm which can make full use of the model complexity.
Hereafter we will focus upon this latter possibility, using a full
random process model and a classification scheme which can take
advantage of both the location and shape of the class distributions in
N-dimensional feature space.

Specifying the Classes

Clearly there must be some way for the analyst to specify to the
analysis process what information is desired. That is, "How does one
specify to the analysis algorithm the classes of materials desired to be
identified?" There are a number of possible approaches to this part of
the problem. For example, if one were to proceed with a deterministic
model in mind, one might approach this part of the problem via
previously stored "spectral signatures," i.e., previously measured
spectral responses from the various classes of material of interest.
While straightforward in concept, this approach has several significant
drawbacks, two of which are the following.

> \(1\) Extensive preprocessing is required. This is the case because
> one must reconcile the observation conditions present when the data
> were collected (atmosphere, sun angle, view angle, terrain effects,
> etc.) with the conditions under which the "spectral signatures" were
> measured. This requires a great deal of additional data, quite complex
> calculations, and is usually difficult to accomplish to precision
> approximating the precision of the original data.
>
> \(2\) One must have already on hand the spectral responses of the
> materials to be identified. Further, since it is fundamentally true
> that relative decisions can be made more accurately than absolute
> ones, one must have spectral responses of other materials which may
> occur in the same scene as well. This makes the approach less robust
> against problems involving large areas and arbitrary locations over
> the Earth.

Instead, a simpler procedure from an operational standpoint is to label
samples within the data set to be analyzed which display the
characteristics of the desired classes. These samples are referred to as
design samples or training samples, and in effect form the model of the
classes to be identified. This allows the analysis process to be more
robust in the sense that it can be applied to data collected from
arbitrary sites. Further, because the design samples are from within the
data set to be analyzed, many of the observation variables are accounted
for, thus there is a reduced need for complex preprocessing. This is the
approach in mind in the following.

The actual means by which labels for the design samples can be
accumulated varies greatly with the situation. Often ancillary
information or knowledge of the scene can be used to interpret the
needed labels from imagery of the scene. In some large area
classifications, low altitude photographic missions, which are
relatively inexpensive to carry out, may provide the needed
information[^8]. In some cases, deterministic knowledge of specific
spectral characteristics may be used[^9]. Suffice it to say that it may
well not be necessary to be able to collect "ground truth" from the
ground in order to be able to accumulate an appropriate set of design
samples.

The Gaussian Assumption

In the remote sensing situation, there is often an additional factor to
the class modeling process. There is usually a paucity of these design
samples by which to estimate the distribution in N-dimensional space for
any given class. Thus, from a limited number of samples, one must
determine as accurately as possible what the location and shape of the
distribution of each class of samples is likely to be in feature space.

Experience has shown that, properly used, the assumption that each of
the class data subsets may be modeled in terms of one or a combination
of Gaussian distributions is a quite practical and powerful way to
proceed. In feature space this is seen to amount to fitting one or a
small number of Gaussian hypershapes to the actual distribution of each
class. Among the advantages of using the Gaussian model is the
mitigation of the need for large training sets to adequately define the
desired classes, especially when the spectral dimensionality is large.
However, use of this assumption does impose upon the analysis process
some means for identifying the various modes of each desired information
class and the fitting of densities to each of these modes. [The analysis
process then consists of finding the parameters for a complete set of
Gaussian distributions which fit the data set and for which subsets of
the distributions correspond to the classes of interest.]{.underline}

Basic Requirement of Feature Space Analysis

In summary, to properly train a classifier one must establish a list of
classes in terms of their distributions in feature space, including
training samples for each, which satisfy the following.

> • The list of classes must be **exhaustive**, in the sense that there
> is a logical class to which to assign every pixel in the scene,
>
> • The classes must be adequately **separable** using the available
> features, and
>
> • The classes must be **of informational value**, i.e., they must be
> classes of interest to the user.

An equivalent statement to this is that a well trained classifier must
have successfully modeled the distribution of the entire data set, but
it must be done in such a way that the different classes of interest to
the user are as distinct from one another as possible. What is desired
in mathematical terms is to have the density function of the entire data
set modeled as a mixture of class densities, i.e.,

p(**x**\|θ) = \\i\\su(i=1,m, αipi(**x**\|φi)) and \\i\\su(i=1,m, αi = 1)
(1)

where p is the probability density function describing the entire data
set to be analyzed, **x** is the measured feature (vector) value, pi is
the density function of class i desired by the user, αi is the
probability of class i, m is the number of classes, and φi denotes the
parameters necessary to define class i. Given the Gaussian assumption,
these parameters are one or a small set of \\x\\to(**x**i) and **Σ**i,
the mean vectors and covariance matrices for class i. Thus the task of
analyzing a given data set reduces to finding the set of {φi:
\\x\\to(**x**i),**Σ**i} which satisfy the exhaustive, separable, and
informational value criteria.

Note again that even when classes are not Gaussian, the above equation
is general enough to handle this situation, in that one may model a
given non-Gaussian class as a linear combination of Gaussian subclasses.
In theory, by using enough such Gaussian subclasses to model a single
class, one can assure that the model can fit any class to any given
precision, e.g., by using a Parzen density approximation with a Gaussian
kernel to fit an arbitrary nonparametric density. In practice, however,
it is usually the case that only a small number (one or two) of such
subclasses are necessary in actual cases.

Summary of the Concepts

The following is a brief summary of the concepts which have been
presented as a sound basis for the analysis of multispectral data:

> • The spectral variations measured at a pixel location play the major
> role in conveying extractable information via multispectral data.
> Spatial variations play a lesser but still significant role.
>
> • Rather than achieving a fully automated analysis method, a more
> powerful goal is to achieve an analysis method which effectively
> combines human interaction with the quantitative strengths of machine
> methods.
>
> • The role of the multispectral sensor is to transform the continuous
> spectral functions emanating from a pixel area to a finite dimensional
> discrete space. In the limit as N, the number of spectral samples
> sensed, increases, this N-dimensional space becomes complete in the
> sense that one could precisely recreate the original continuous
> spectral function from the coordinates of a sample in this space, thus
> implying that no information is lost in undergoing the transformation
> which the multispectral sensor accomplishes.
>
> • The role of the analysis algorithm is to partition the resulting
> N-dimensional feature space into M mutually exclusive subregions, each
> one of which being a class of surface cover of interest to the
> analyst. Optimally locating the boundaries of these subregions, then,
> is the central task of any analysis process, and, indeed, is a
> perfectly general way of defining the analysis process, however
> implemented.
>
> • Relative classifiers, which consider all possible classes in
> determining the optimal one, may generally be expected to outperform
> absolute classifiers, which reach a decision based upon response
> characteristics defined by an absolute basis.
>
> • A key concept is the manner in which the spectral response is viewed
> or modeled. Though any of a number of theoretically based approaches
> to this modeling could be taken, the most general and most thoroughly
> developed theoretically is the random process model. It has the
> advantage of dealing effectively with very complex classes in a noisy
> signal environment, but it can also be gracefully simplified to deal
> with less complex problem situations.
>
> • The defining of the boundaries of the subregions of interest
> involves bringing together or reconciling the data collected and the
> conditions under which it was collected with pre-knowledge which
> delineates the information desired from the analysis. A useful way of
> accomplishing this reconciliation is to select samples from within the
> data set which exemplify the classes of interest, for this mitigates
> many of the non-information bearing variations present in the data. It
> thus reduces the need for preprocessing adjustments of the data,
> adjustments which many times can themselves unknowingly limit the
> information which can be extracted from the data.
>
> • The use of the multivariate Gaussian model to model the
> distributions of the classes in N-dimensional feature space is an
> especially useful way to proceed. It is effective when there is the
> usual paucity of design samples by which to estimate class
> distributions, and by using more than one Gaussian distribution per
> class, it can be effective even when a class, itself, is not Gaussian.
>
> • The goal of the analyst is thus to define a list of classes which is
> exhaustive, separable, and of informational value. Mathematically,
> this means that the density function of the entire data set must be
> partitioned as:

p(**x**\|θ) = \\i\\su(i=1,m, αipi(**x**\|φi)) and \\i\\su(i=1,m, αi = 1)

> where p is the probability density function describing the entire data
> set to be analyzed, **x** is the measured feature (vector) value, pi
> is the density function of class i desired by the user, αi is the
> probability of class i, m is the number of classes, and φi denotes the
> parameters necessary to define class i.

Turning Theory to Practice: How to Analyze a Data Set with
MultiSpec[^10]**.**

We shall next proceed to give an example of analysis of a data set which
makes use of the concepts described above. In doing so, we shall use the
facilities of the MultiSpec data analysis software system. This system
and the data set to be used are available without charge to anyone
requesting them, thus one is encouraged to follow along the analysis
steps in parallel with their description below. It is assumed at this
point that the reader has reviewed the documentation of MultiSpec and
thus is familiar with the various algorithms implemented in it.

Before undertaking the example analysis, we will begin to relate the
previously stated principles and concepts to the various processors
contained in MultiSpec. Please note that the approach which has been
described is quite general, and the applications to which it must
successfully apply are quite varied. There are many specific procedures
to analyzing multispectral image data, almost as many as there are
analysts. The steps used in any given analysis must necessarily be based
upon the scene, the information desired, and the data analyst\'s initial
assumptions about the data characteristics. In general terms, to meet
the requirements of a sound analysis as previously described, there are
several helpful tools in MultiSpec:

> • *Clustering* (sometimes called unsupervised classification)
> algorithms are focused upon finding classes of data that are
> **separable**. However, the resulting clusters do not necessarily have
> any relationship with the classes of informational value.
>
> • *Training samples* focus the analysis upon classes of
> **informational value**. However, the classes initially so defined are
> not necessarily separable. Thus some combined or iterative use of
> clustering and training sample selection is usually required.
>
> • *Clustering* can also initially assist in obtaining a list of
> classes which are **exhaustive**. Later in the process a feature in
> MultiSpec called the *Probability Map* becomes valuable in determining
> any needed additional classes.

The problem is that data do not ordinarily occur in N-dimensional space
as separable "clumps," let alone clumps of the desired informational
classes. Rather, data in N-dimensional space usually exist as a
continuum. Thus it is not possible for a clustering algorithm by itself
to divide data into separable classes of interest.

There are two additional matters to be mentioned before turning to
specific steps useful in analyzing data. The first has to do with
choosing or calculating the best set of features to use for a given
classification. There are at least two reasons why one might not want to
use all of the features available in the data set to do a
classification. One is that if adequate performance can be obtained with
a subset of the features smaller than the full set, much computation
time can be saved.

However, perhaps a more compelling reason is that one may well expect
higher accuracy if a smaller set of features are used. This is due to
what is known as the Hughes effect[^11]. It is easy to understand that
if one has very precise knowledge of the class distribution functions,
the classification accuracy should be a non-decreasing function of the
number of features used. However, because there are only a finite number
of training samples available, and indeed, that number is often not
large, the precision to which the class distributions can be estimated
is often limited. As a result, the accuracy of the analysis which can be
achieved will usually increase for a while as more features are used,
then peak and begin to decline. A further practical complication is that
it is usually not possible to determine before the fact just what the
best feature dimensionality is for a given classification.

For these reasons, feature selection, (i.e., selecting e.g., the best 4
of the available 10 features) and feature extraction (i.e., calculating,
usually via a linear transformation on the existing features, a new
feature space which is more suited to the specific problem) capabilities
are very important in the analysis process. Several such capabilities
are implemented in MultiSpec. Two, in particular, are to be noted.
Discriminant Analysis Feature Extraction (DAFE) is a method that is well
known in the field of pattern recognition[^12]. It provides an optimal
transformation to a feature space of up to M -- 1 dimensions, where M is
the number of classes. The second, Decision Boundary Feature Extraction
(DBFE), provides an optimal linear transformation to a space of
arbitrary dimensionality and gives information as to how many dimensions
need be used in any given problem. It is predicted by theory and usually
found in practice that DAFE, which is a very fast computation, provides
the best results when the number of features to achieve satisfactory
classification is less than M, while DBFE, a more lengthy calculation,
provides better results when the number of features needed is greater
than M. MultiSpec also implements the principal components
transformation, which may be useful at low dimensionality, but which
becomes undesirable at moderate or high dimensionality, since it tends
to de-emphasize narrow band features which are diagnostic of classes,
often one of the very reasons for gathering high dimensional data.

The final characteristic of an analysis algorithm which needs to be
mentioned at this point is that of its effectiveness at generalization.
That is, given that it can accurately classify its training samples, how
well can it classify other members of the same classes in the data set.
This is closely related to how well the set of classes defined by the
training samples result in equation (1) above being an equality. An
algorithm in MultiSpec called "Statistics Enhancement" is directed at
improving the fit of the classes as indicated by equation (1). The
algorithm is more fully described in the reference[^13], however,
basically it is a technique for using a combination of the training
statistics and a portion of all other samples (unlabeled) in the data
set in an iterative fashion to improve the degree of equality of
equations (1).

Typical Analysis Steps using MultiSpec

The steps for accomplishing the analysis task according to the discussed
approach using MultiSpec are:

1\. Familiarization with the data set.

> • Display the data using the Display Image processor. Compare the
> displayed image with any ground reference information about the site
> which may be available. Compose a tentative list of classes which is
> adequately (but not excessively) exhaustive for this data set.

2\. Preliminary selection of the classes and their training sets.

> • Using the Cluster processor, cluster the area from which training
> fields are to be selected, saving the results to disk file. Display
> the resulting thematic map for use in marking training areas.
>
> • Using either the display of the original data or that of the
> thematic cluster map, make a preliminary selection of training fields
> which adequately represent the selected classes.
>
> • Use the Feature Selection Processor to determine how separable the
> tentative classes are. It may then be necessary to iterate between the
> previous steps.
>
> • Cluster the training fields to check the modality. This will also be
> useful in identifying the need for subclasses.

3\. Determination of the spectral features to be used.

> • Use the Feature Selection processor to choose the best subset of
> features for carrying out the classification for a given training set.
> The Feature Extraction processor is also available for this task.

4\. Preliminary Classification of the data.

> • Classify the training fields only, using the spectral bands or
> features selected to verify their purity and separability.

5\. Classification, Evaluation of the classification and Extraction of
the desired information.

> • Review the results of training field classification. Make
> modifications to the training as required to obtain satisfactory
> results at this point. Mark as many fields as possible as Test Fields
> using the Statistics Processor. Classify these fields and examine the
> accuracies obtained to determine how well the classifier training
> generalizes beyond the training set.
>
> • Depending upon the results of these evaluations it may be necessary
> to repeat previous steps after modifying the class definitions and
> training. After becoming satisfied with the results, classify the
> entire data set, perhaps setting a modest threshold value, saving the
> classification results to a disk file, and creating a Probability
> Results file.
>
> • Use the Display Image processor to generate thematic map versions of
> the results and the probability results files for subjective
> evaluation purposes. The results file display is useful in determining
> that the classification results are appropriate and consistent from a
> spatial distribution standpoint. The portion of points thresholded in
> the results display, together with the probability map helps to
> determine if any important modes in the data have been missed in the
> class definition process. Depending on the outcome, it may again be
> necessary to iterate using some of the previous steps. The List
> Results processor can be used to provide a quantitative evaluation of
> the results based upon the accuracy figures of the training and test
> fields classification.
>
> • From the Probability Results file displayed as a thematic map,
> determine if there are areas with very low likelihood of membership to
> any of the existing classes. For any that are found, create
> appropriate new classes. Also, determine how well the classifier
> generalizes to areas outside the training sets. Use the Enhance
> Statistics processor to improve the fit of the composite class
> statistics to the data set and reclassify.

Again, substantial variance from these steps might be appropriate,
depending upon circumstances, however, these steps provide a reasonable
overall picture of how to proceed.

An Example Analysis

The specific procedure to be used in analyzing a data set is controlled
by what information one has about the scene before the fact and what
specific information one desires as a result of the analysis. For
example,

> • Urban Land Use Example. One might desire to obtain a map of an urban
> area which delineates areas of the city that are in such land use
> classes as high density housing, low density housing, commercial,
> industrial, etc. One might expect to know, before the fact, where in
> the city, examples of each of these classes are found.
>
> • Geology Example. One might have example spectra of specific minerals
> that are expected to be expressed on the surface and one might wish to
> determine the general layout and shape of the expression of these
> minerals for purposes of estimating the manner, in a geologic sense,
> in which they came to be distributed as they are.
>
> • Soil or Vegetation Example. One might have a general knowledge of
> the location of various soil categories and plant species in the scene
> and from
>
> that one might wish to determine tabular information giving the
> portion of the scene each occupies.
>
> • Soil or Vegetation Example. Again from general knowledge of the
> location of various soil and plant classes in the scene, one might
> desire a thematic map showing not only how much of each is in the
> scene, but where each is located.

Note that the goal in the above cases varies somewhat. If the results
are to be in the form of a thematic map, then accuracy of the
classification becomes of greater importance. On the other hand, if only
tabular information is required, then accuracy decreases in importance
in favor of obtaining unbiased results with regard to the classes, i.e.
it is not as important that each pixel be correctly classified as it is
that the proper number of pixels be assigned to each class.

We shall illustrate an analysis of a data set of the fourth case
indicated above. In order to simplify the logistics of making this
example analysis available to others, only a small portion of a data set
was chosen for this example. It contains 145 lines by 145 pixels (21,025
pixels) and 9 spectral bands selected from a June 1992 AVIRIS data
set[^14] of a mixed agriculture/forestry landscape in the Indian Pine
Test Site in Northwestern Indiana. The data set is designated 92AV3C9.
Each of the nine bands are approximately 10 nm wide centered at the
following wavelengths.

+--------------------+---------------------+--------------------------+
| Band No.           | AVIRIS              | Wavelength               |
|                    |                     |                          |
|                    | Band No.            | center, µm               |
+--------------------+---------------------+--------------------------+

  --------------------- ---------------------- ---------------------------
  1                     8                      0.4795

  --------------------- ---------------------- ---------------------------

  --------------------- ---------------------- ---------------------------
  2                     16                     0.5584

  --------------------- ---------------------- ---------------------------

  --------------------- ---------------------- ---------------------------
  3                     27                     0.6675

  --------------------- ---------------------- ---------------------------

  --------------------- ---------------------- ---------------------------
  4                     39                     0.7560

  --------------------- ---------------------- ---------------------------

  --------------------- ---------------------- ---------------------------
  5                     46                     0.8235

  --------------------- ---------------------- ---------------------------

  --------------------- ---------------------- ---------------------------
  6                     70                     1.0550

  --------------------- ---------------------- ---------------------------

  --------------------- ---------------------- ---------------------------
  7                     86                     1.2092

  --------------------- ---------------------- ---------------------------

  --------------------- ---------------------- ---------------------------
  8                     136                    1.6589

  --------------------- ---------------------- ---------------------------

  --------------------- ---------------------- ---------------------------
  9                     186                    2.2186

  --------------------- ---------------------- ---------------------------

In carrying out the analysis of the data set, we will generally follow
through the numbered steps above, although, due to the relatively small
size of the data set and the small number of spectral bands, all of the
steps may not be needed. We will indicate specific choices of MultiSpec
options and parameter values in **boldface** type and,

> • Ancillary comments not in the mainstream of this processing task but
> which may be relevant in other circumstances will be indicated in
> indented, smaller type.

Whenever a given step involves a significant amount of computation, we
will indicate the number of seconds needed for execution by MultiSpec on
a Macintosh Quadra 840av. It is noted that the native code PowerPC
version of MultiSpec executes 3 to 10 times as fast on a Macintosh
8100/80 depending on the amount of floating point processing involved.
Classification, for example, runs 10 times as fast. On a Macintosh

6100/60, processing is about 75% the speed of an 8100/80. MultiSpec
processing speeds on Sun or HP workstations are somewhat slower.

*1. Familiarization with the data set and related information.* Begin by
choosing **Open Image** from the **File** menu and use the default
options to display the data in simulated color infrared image form.[^15]
The result is shown in Figure 7.

![](media/image14.pct){width="0.3333333333333333in"
height="0.7361111111111112in"}![](media/image15.pct){width="4.027777777777778in"
height="4.027777777777778in"}

Figure 7. Data Set 92AV3C9 displayed in simulated color infrared
form[^16].

It is seen that the area appears to be about 2/3 agriculture and 1/3
forest or other natural perennial vegetation. However, due to the early
season date of data collection, the cultivated land appears to have very
little canopy cover as yet. There is a major dual lane highway (U.S. 52
& U.S. 231) and a rail line crossing near the top and a major secondary
road (Jackson Highway) near the middle, both in a NW-SE direction.
Several other county roads are also somewhat apparent, and show more
clearly on a USGS quadrangle map, (Figure 8), as does the terrain relief
of the area.

![](media/image16.pct){width="4.625in" height="4.819444444444445in"}

Figure 8. Portion of a USGS Quadrangle map of the test area.

The small rectangular dots on the quadrangle map indicate houses or
other buildings existing at the time the quadrangle map was made. Thus a
number of low density housing areas can be found in the data. Additional
information useful in deriving training samples for the ground cover of
the area is contained on the following generalized reconnaissance map.

![](media/image17.pct){width="6.513888888888889in"
height="4.041666666666667in"}

Figure 9. A generalized reconnaissance map of the test area, designated
file 92AV3gt.GIS (in color).

A comparison of this reconnaissance map with the data in image form
(Figure 7) shows that the reconnaissance map is highly generalized. Many
small variations within fields which can be seen in the image data are
ignored in the reconnaissance map. Thus, not every pixel inside of a
designated area on the reconnaissance map should be expected to belong
to the class associated with that area. Further, the map tends to give
land use classes, rather than land cover classes. Thus, an area marked
"Corn-notill" on the map may really have a land cover of bare soil and
residue from previous vegetation, with only a small percent of corn
vegetation as land cover.

> • In the case of any given data set, various kinds of additional
> ancillary information might be available. Examples include soil maps,
> imagery from other sensors, airphotos, etc. Assembling this kind of
> information is an important part of a data analysis process, but
> obviously varies substantially for each analysis situation.

*2. Preliminary selection of classes and their training sets.* Next we
consider how detailed a thematic map it might be both possible and
desirable to develop. For this purpose, it is useful to consider
possible classes in a hierarchical form. Besides the cultural features,
there are several species of annual and perennial vegetation. Further
subdivision possible within the annual vegetation category would be
corn, oats, soybeans, and wheat. Still further subdivision within these
might be to subdivide the corn and soybean classes by tillage practice,
since the amount of residue from previous vegetation varies in this
case. Within the category of perennial vegetation, classes such as
alfalfa/hay, grass, and trees/woods would be possible categories. Due to
the early season date of data collection, several species of annual
vegetation may have so limited a canopy (\< 5%) that these areas may not
be separable in terms of

plant species. Rather, the soil type or the tillage practice as
manifested by the amount of surface residue from previous vegetation may
be more appropriate classes.

All of the previous information will be useful in selecting classes and
training sets. A very useful additional display is a map in which the
scene has been subdivided into objects, i.e., areas of contiguous pixels
which are spectrally similar to one another but spectrally separable
from one another. Such a presentation can be obtained by selecting
**Cluster** from the **Processor** menu. It is useful to adjust the
parameters of the clustering so that one achieves roughly the same
number of clusters as the number of classes desired, about 15 in this
case. Choose **Isodata** with **one pass for initial centers**. After a
little experimentation, the following parameter values would be found to
lead to about 15 clusters.

**Min. initial cluster size of 50**

**First line distance of 500**

**Final distance of 1000**

**Threshold level of 256**

**Use All 9 bands.**

**Classify the Image Area**

**Save to Disk File**

Use the default values for all other parameters. The result of the
clustering may be displayed using the **Open Image** selection of the
**File** menu, as follows.

![](media/image18.pct){width="5.416666666666667in"
height="4.027777777777778in"}

Figure 10. Cluster Classification Map (in color). 38 seconds

It is seen that, in most cases, even the agricultural fields are not
uniform in their association with a single cluster, but are
significantly mottled in appearance. This

characteristic can be mitigated somewhat and a clearer association with
individual clusters obtained by use of the ECHO spectral/spatial
classifier using the cluster statistics. Thus select **Classify** from
the **Processor** menu and choose the **ECHO** procedure, classify the
**Image Area**, and save the results to **Disk File.** Continue to use
all bands since their number is not large. The result will be as
follows.

![](media/image19.pct){width="5.388888888888889in"
height="4.027777777777778in"}

Figure 11. ECHO classification of Cluster Statistics (in color). 19
seconds.

It is seen that this display clarifies the association of areas with
spectral characteristics to some degree, although most areas are still a
mixture of more than one cluster in spectral space.

> • Note that this two-step procedure, clustering followed by ECHO
> classification, provides an effective unsupervised multivariate image
> partitioning capability.

By now referring to the image of the data (Figure 7), the ECHO
classification (Figure 11) and the generalized reconnaissance map
(Figure 9), one can make a first draft at selecting classes and training
fields. A possible choice will be illustrated. At this point we are
assuming that all of the classes of the reconnaissance map will be
separable and thus training samples have been established for each. We
show in Figures 12 and 13 the training fields outlined on both the ECHO
classification map and the color infrared image, because the process is
one of selecting classes and their training fields which satisfy both
criteria, i.e., classes which are separable and of informational value.
The ECHO map shows classes which are separable, but classes must be
combined in such a way that they are also of informational value.

![](media/image20.pct){width="5.388888888888889in"
height="4.027777777777778in"}

Figure 12. Unsupervised Scene Segmentation showing the training fields
selected (in color).

![](media/image21.pct){width="4.027777777777778in"
height="4.027777777777778in"}

Figure 13. Data displays showing training fields (in color).

In order to allow for precise duplication of the results which follow,
the addresses of the training fields selected are provided in Table 1.
Note that only one field has been selected for each class. This will
likely not be adequate, since one sample from a single area is usually
not adequate to characterize a class in the entire data set, however, it
is deemed adequate for a first trial.

  ------------------------- ---------------- ------------- ---------------
  Class                     Line             Column        No. of Samples

  ------------------------- ---------------- ------------- ---------------

+------------------------+----------------+------------+--------------+
| alfalfa                | 67             | 98         | 18           |
|                        |                |            |              |
|                        | 73             | 98         |              |
|                        |                |            |              |
|                        | 75             | 101        |              |
|                        |                |            |              |
|                        | 70             | 101        |              |
|                        |                |            |              |
|                        | 67             | 98         |              |
+------------------------+----------------+------------+--------------+

  ------------------------- ---------------- ------------- ---------------
  Corn-notill               33 to 41         31 to 56      234

  ------------------------- ---------------- ------------- ---------------

  ------------------------- ---------------- ------------- ---------------
  Corn-min                  63 to 71         6 to 21       144

  ------------------------- ---------------- ------------- ---------------

+------------------------+----------------+------------+--------------+
| Corn                   | 35             | 7          | 108          |
|                        |                |            |              |
|                        | 35             | 5          |              |
|                        |                |            |              |
|                        | 48             | 10         |              |
|                        |                |            |              |
|                        | 48             | 23         |              |
|                        |                |            |              |
|                        | 45             | 22         |              |
|                        |                |            |              |
|                        | 44             | 16         |              |
|                        |                |            |              |
|                        | 35             | 10         |              |
|                        |                |            |              |
|                        | 35             | 5          |              |
+------------------------+----------------+------------+--------------+

  ------------------------- ---------------- ------------- ---------------
  Grass/pasture             75 to 85         4 to 21       198

  ------------------------- ---------------- ------------- ---------------

  ------------------------- ---------------- ------------- ---------------
  Grass/trees               48 to 70         28 to 35      184

  ------------------------- ---------------- ------------- ---------------

  ------------------------- ---------------- ------------- ---------------
  Grass/pasture-mowed       73 to 78         109 to 112    24

  ------------------------- ---------------- ------------- ---------------

  ------------------------- ---------------- ------------- ---------------
  Hay-windrowed             39 to 59         124 to 138    315

  ------------------------- ---------------- ------------- ---------------

  ------------------------- ---------------- ------------- ---------------
  Oats                      63 to 71         23 to 24      18

  ------------------------- ---------------- ------------- ---------------

  ------------------------- ---------------- ------------- ---------------
  Soy-notill                42 to 63         78 to 92      330

  ------------------------- ---------------- ------------- ---------------

  ------------------------- ---------------- ------------- ---------------
  Soy-min till              78 to 111        34 to 45      408

  ------------------------- ---------------- ------------- ---------------

  ------------------------- ---------------- ------------- ---------------
  Soy-clean                 52 to 58         5 to 24       140

  ------------------------- ---------------- ------------- ---------------

  ------------------------- ---------------- ------------- ---------------
  Wheat                     119 to 124       26 to 46      126

  ------------------------- ---------------- ------------- ---------------

  ------------------------- ---------------- ------------- ---------------
  Woods                     121 to 137       91 to 121     527

  ------------------------- ---------------- ------------- ---------------

  ------------------------- ---------------- ------------- ---------------
  Bldg-grass-trees-drives   18 to 27         27 to 34      80

  ------------------------- ---------------- ------------- ---------------

+------------------------+----------------+------------+--------------+
| Stone-steel towers     | 14             | 47         | 45           |
|                        |                |            |              |
|                        | 23             | 44         |              |
|                        |                |            |              |
|                        | 24             | 49         |              |
|                        |                |            |              |
|                        | 16             | 52         |              |
+------------------------+----------------+------------+--------------+

Table 1. Coordinates for training fields. Those for Alfalfa, Corn, and
Stone-steel

towers were delineated in polygon form. All others are rectangular.

> • An alternate approach to defining training statistics is to set the
> parameters of the initial clustering so as to provide two or three
> times as many clusters as the number of final classes desired, then to
> define classes by studying how to combine two or more clusters into a
> desired information class. In this case, the maximum likelihood or
> ECHO classification would be done using the statistics of all
> clusters, and the final classes achieved by grouping classes together
> after classification.

Next choose **Feature Selection** from the **Processor** menu. Under
**Combination Groups**, choose **9**. Use the default options for all
other parameters. This will list the Bhattacharyya distance between all
class pairs to indicate the relative separability of each class pair.
There are 120 possible such class pairs in this case. The results are
given in terms of the following symbols:

  ------------------------------------------------- ---------------------
  Classes used:                                     Symbol

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  1: Alfalfa                                        1

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  2: Corn-notill                                    2

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  3: Corn-min                                       3

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  4: Corn                                           4

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  5: Grass/pasture                                  5

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  6: Grass/trees                                    6

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  7: Grass/pasture-mowed                            7

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  8: Hay-windrowed                                  8

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  9: Oats                                           9

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  10: Soy-notill                                    A

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  11: Soy-mintill                                   B

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  12: Soy-clean                                     C

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  13: Wheat                                         D

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  14: Woods                                         E

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  15: Bldg-grass-trees-drives                       F

  ------------------------------------------------- ---------------------

  ------------------------------------------------- ---------------------
  16: Stone-steel towers                            G

  ------------------------------------------------- ---------------------

Table 2. Classes and their Symbols used in Table 3.

The class pairs and their corresponding B-distance are given in Table 3
in order of ascending B-distance size. The sorting was done by copying
the row of class pair symbols and the row of their corresponding
B-distances from the MultiSpec text window to a spread sheet and sorting
them there.

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
  3C   AB   4B   6F   3B   2B   2A   AC   18   24   DF   BC   3A   78   69   34   9F   5E

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----

  ------ ----- ------ ------ ------ ------ ----- ------ ------ ------ ------ ------ ------ ------ ------ ------ ----- -----
  1.48   2.8   2.91   2.94   2.95   3.02   3.1   3.41   3.45   3.74   4.05   4.08   4.33   4.37   4.62   4.64   5.3   5.4

  ------ ----- ------ ------ ------ ------ ----- ------ ------ ------ ------ ------ ------ ------ ------ ------ ----- -----

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
  23   2C   46   17   7F   EF   4A   6E   9D   4G   4C   AG   8F   BF   5F   6D   CG   49

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----

  ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ---- ---- ------
  5.64   6.56   6.63   6.79   6.82   6.98   7.63   7.95   8.15   8.17   8.74   9.31   9.58   9.61   9.73   10   10   10.4

  ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ---- ---- ------

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
  48   1F   2G   6B   BG   4F   5D   68   7B   BD   DE   28   9B   3G   1B   67   56   6A

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----

  ------ ---- ------ ------ ------ ------ ------ ------ ------ ------ ---- ------ ---- ------ ------ ------ ------ ------
  10.9   11   11.2   11.2   11.2   11.7   12.1   12.1   12.9   12.9   14   14.9   15   15.1   15.2   15.5   15.9   16.4

  ------ ---- ------ ------ ------ ------ ------ ------ ------ ------ ---- ------ ---- ------ ------ ------ ------ ------

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
  26   FG   9E   47   8B   79   59   16   BE   AF   6C   3F   4D   5B   14   2F   7D   8C

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----

  ------ ---- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
  17.1   20   20.9   21.2   21.8   23.5   24.5   25.3   26.1   26.8   27.6   28.5   28.7   28.7   29.3   29.5   31.8   31.9

  ------ ---- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
  CF   8A   7A   89   27   36   19   29   8D   2D   7C   3D   AD   1D   38   4E   CD   9C

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----

  ------ ---- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
  31.9   32   34.2   35.1   35.7   36.1   38.6   38.9   40.8   41.8   41.9   42.8   46.2   46.6   47.4   47.6   48.1   48.8

  ------ ---- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
  6G   39   37   12   9A   1C   8E   3E   57   58   8G   7E   1A   13   DG   7G   9G   45

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----

  ------ ------ ------ ------ ------ ---- ------ ------ ------ ---- ------ ------ ---- ------ ------ ---- ---- ----
  55.3   57.3   58.1   61.7   64.5   66   68.6   70.7   72.8   74   75.1   76.7   77   79.6   82.4   86   93   94

  ------ ------ ------ ------ ------ ---- ------ ------ ------ ---- ------ ------ ---- ------ ------ ---- ---- ----

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- --- ---- --- ---- --- ---
  5G   EG   35   2E   1E   CE   5C   1G   25   15   AE   5A                         

  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- --- ---- --- ---- --- ---

  ------ ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- --- ---- --- ---- --- ---
  99.9   102   112   121   135   149   156   162   162   170   171   194                         

  ------ ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- --- ---- --- ---- --- ---

Table 3. Bhattacharyya interclass distances in order of ascending
distance.

It is seen that some of the distances are quite small, implying that
they may be only marginally separable. Those class pairs less than 3
indicate that it may not be possible with only the 9 bands available to
separate all classes of corn and soybeans from each other, or if so, the
separations may not be particularly robust in generalization to a larger
area. It appears only a little more likely that it will be possible to
separate tillage classes.

*3. Determination of the spectral features to be used.* Under some
circumstances it would be appropriate to use the Feature Selection
processor at this point to select a subset of features to use for
initial classification. This might be called for in cases where the
number of available spectral bands was large or the number of training

samples available for one or more classes was small (e.g., ≤ the number
of spectral bands). However, in the present case, since neither of these
conditions are present, we shall proceed directly to a preliminary
classification.

*4. Preliminary Classification of the data.* Choose **Classify** under
the **Processor** menu and use the default options to classify the
training fields. This will help to verify their purity and the
separability of at least the training fields. (13 seconds) The results
in terms of alphanumeric field maps and tables of accuracy appear in the
MultiSpec text window. Inspection of these results indicate good
accuracy (98.2% overall) with no fields showing unreasonable or
unexplainable errors.

*5. Classification, Evaluation of the Classification and Extraction of
the desired information.* Choose **Classify** from the **Processor**
menu and select **Image File** for classification. Also save the results
to **Disc File** and deselect **Text Window**. This will result in a
maximum likelihood pixel classification of the data set which can be
displayed in thematic map form, as in Figure 14.

![](media/image22.pct){width="6.611111111111111in"
height="4.027777777777778in"}

Figure 14. Maximum Likelihood Classification of original training
statistics (in color). 62 seconds.

A superficial evaluation of this result reveals that it shows some
promise, but needs improvement. For example, a large portion of the
large field a the center has been classified as Corn-notil when in fact
it should be Soy-min till. A Corn-min till field near the lower left
corner is classified partly as Soy-min till and partly as corn-notill.
The improvement process can be begun by adding training fields from
these three areas, as follows:

  ------------------------ ----------------------- -----------------------
  Class                    Lines                   Columns

  ------------------------ ----------------------- -----------------------

  ------------------------ ----------------------- -----------------------
  Soy-min till             3 to 17                 102 to 117

  ------------------------ ----------------------- -----------------------

  ------------------------ ----------------------- -----------------------
  Soy-min till             80 to 95                51 to 71

  ------------------------ ----------------------- -----------------------

  ------------------------ ----------------------- -----------------------
  Corn-min till            128 to 134              20 to 46

  ------------------------ ----------------------- -----------------------

A new classification, this time using the ECHO classifier, but otherwise
with default options results in significant improvement, as can be seen
in Figure 15.

![](media/image23.pct){width="6.597222222222222in"
height="4.027777777777778in"}

Figure 15. ECHO classification with 3 new training fields added (in
color). 56 seconds

In order to obtain a more quantitative estimate of the accuracy of the
results, the fields defined by the generalized reconnaissance map can be
used as test fields. To do so, **Close** the current **Project**, being
careful to save the project for future reference, and **Open** the
**Project**, 92AV3.Project, containing the test field boundaries. Then,
with the above ECHO classification map active, choose **List Results**
from the **Processor** menu, and request a **Class Type Map**, and
**Summarize by Class**. The result is as follows.

  --------------- ------------- -----------------------------------------
  Project                       Number of Samples in Thematic Image Class

  --------------- ------------- -----------------------------------------

  ------- ------- --------- -------- --- --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---- ---- ----
  Class   Class   Percent   Number   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16

  ------- ------- --------- -------- --- --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---- ---- ----

Name

No.

Correct

Samples

Alfalf

Corn-n

Corn-m

Corn

Grass/

Grass/

Grass/

Hay-wi

Oats

Soy-no

Soy-mi

Soy-cl

Wheat

Woods

Bldg-g

Stone

Alfalfa

1

66.7

54

36

0

0

0

0

0

0

11

0

0

7

0

0

0

0

0

  ------------- --- ------ ------ --- ----- ---- ---- --- --- --- --- --- ----- ----- ---- --- --- --- ---
  Corn-notill   2   49.6   1434   0   711   19   70   0   2   0   0   0   119   493   11   0   0   9   0

  ------------- --- ------ ------ --- ----- ---- ---- --- --- --- --- --- ----- ----- ---- --- --- --- ---

  ---------- --- ------ ----- --- --- ----- ---- --- --- --- --- --- --- ----- ---- --- --- --- ---
  Corn-min   3   61.6   834   0   6   514   80   0   0   0   0   1   0   183   49   0   0   1   0

  ---------- --- ------ ----- --- --- ----- ---- --- --- --- --- --- --- ----- ---- --- --- --- ---

  ------- --- ---- ----- --- --- --- ----- --- --- --- --- --- --- ---- --- --- --- --- ---
  Corn    4   91   234   0   0   0   213   0   0   0   0   0   0   16   1   0   0   4   0

  ------- --- ---- ----- --- --- --- ----- --- --- --- --- --- --- ---- --- --- --- --- ---

  --------------- --- ------ ----- --- --- --- ---- ----- --- --- --- --- --- ---- --- --- --- ----- ---
  Grass/Pasture   5   64.8   497   0   0   0   12   322   4   2   4   0   0   19   1   0   1   132   0

  --------------- --- ------ ----- --- --- --- ---- ----- --- --- --- --- --- ---- --- --- --- ----- ---

  ------------- --- ------ ----- --- --- --- ---- --- ----- --- --- --- --- --- --- --- --- ---- ---
  Grass/Trees   6   90.2   747   0   0   0   25   0   674   0   0   1   0   1   0   0   1   45   0

  ------------- --- ------ ----- --- --- --- ---- --- ----- --- --- --- --- --- --- --- --- ---- ---

  -------------- --- ----- ---- --- --- --- --- --- --- ---- --- --- --- --- --- --- --- --- ---
  Grass/pas-mo   7   100   26   0   0   0   0   0   0   26   0   0   0   0   0   0   0   0   0

  -------------- --- ----- ---- --- --- --- --- --- --- ---- --- --- --- --- --- --- --- --- ---

  --------------- --- ------ ----- --- --- --- --- --- --- --- ----- --- --- --- --- --- --- --- ---
  Hay-windrowed   8   98.6   489   1   0   0   0   0   0   2   482   0   0   0   0   0   0   4   0

  --------------- --- ------ ----- --- --- --- --- --- --- --- ----- --- --- --- --- --- --- --- ---

  ------- --- ---- ---- --- --- --- --- --- --- --- --- ---- --- --- --- --- --- --- ---
  Oats    9   95   20   0   0   0   0   0   0   0   0   19   0   0   0   0   0   1   0

  ------- --- ---- ---- --- --- --- --- --- --- --- --- ---- --- --- --- --- --- --- ---

  ----------------- ---- ------ ----- --- ---- --- ---- --- --- --- --- --- ----- ----- --- --- --- --- ---
  Soybeans-notill   10   68.5   968   0   10   3   32   0   2   0   0   1   663   249   6   0   0   2   0

  ----------------- ---- ------ ----- --- ---- --- ---- --- --- --- --- --- ----- ----- --- --- --- --- ---

  -------------- ---- ------ ------ --- ---- ---- ---- --- --- --- --- --- ---- ------ ---- --- --- ---- ---
  Soybeans-min   11   90.6   2468   0   65   49   40   0   2   0   0   0   40   2236   23   0   0   13   0

  -------------- ---- ------ ------ --- ---- ---- ---- --- --- --- --- --- ---- ------ ---- --- --- ---- ---

  --------------- ---- ------ ----- --- --- --- ---- --- --- --- --- --- --- ----- ----- --- --- --- ---
  Soybean-clean   12   22.5   614   0   1   5   24   0   5   2   0   1   2   433   138   0   0   3   0

  --------------- ---- ------ ----- --- --- --- ---- --- --- --- --- --- --- ----- ----- --- --- --- ---

  ------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- ----- --- --- ---
  Wheat   13   96.7   212   0   0   0   0   0   0   0   0   0   0   1   0   205   0   6   0

  ------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- ----- --- --- ---

  ------- ---- ------ ------ --- --- --- --- ---- --- --- --- --- --- --- --- --- ------ ---- ---
  Woods   14   94.9   1294   0   0   0   0   13   4   0   0   0   0   0   0   2   1228   47   0

  ------- ---- ------ ------ --- --- --- --- ---- --- --- --- --- --- --- --- --- ------ ---- ---

  ------------------ ---- ------ ----- --- --- --- --- --- ---- --- --- --- --- --- --- --- ---- ----- ---
  Bldg-Grass-Tree-   15   70.5   380   0   0   0   0   2   21   0   0   1   0   4   1   0   83   268   0

  ------------------ ---- ------ ----- --- --- --- --- --- ---- --- --- --- --- --- --- --- ---- ----- ---

  ------------- ---- ------ ---- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----
  Stone-steel   16   93.7   95   0   0   2   0   0   0   0   0   0   0   0   0   0   0   4   89
  towe                                                                                       

  ------------- ---- ------ ---- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----

TOTAL

10366

37

793

592

496

337

714

32

497

24

824

3642

230

207

1313

539

89

  -----------------------------------------------------------------------
  OVERALL PERFORMANCE(7824/10366)=75.5

  -----------------------------------------------------------------------

Table 4. Quantitative classification summary for the fields of Figure 9.

The overall accuracy appears to be 75.5, however, note that many of the
"errors" are not significant. For example, class 5, Grass/Trees had many
of the test samples assigned to the Building-Grass-Trees-Drives, which,
depending on the specific objective of the analysis, may not represent
errors at all. Indeed, if the objective of the analysis was directed
primarily at soils, tillage, or crop classification, the classes 5, 6,
7, and 15 could be combined into a single group, substantially raising
the calculated accuracy.

Similarly, if the interest is primarily in crop classification, classes
2, 3, and 4 could be combined, as could classes 10, 11, and 12. On the
other hand, if erosion related to tillage practice is the primary
interest, classes 2 and 10, 3 and 11, and 4 and 12 could be combined,
again increasing the calculated accuracy.

There are several additional possibilities which could be investigated
as well. These include the following.

> • As previously indicated, defining a class by continuous pixels from
> a single area is usually not adequate as it is unlikely that pixels
> from a single area show all of the characteristic variability that
> will exist over a whole scene. Thus additional training areas over
> those already added would no doubt improve the characterization of the
> classes.
>
> • The extent to which the list of classes is exhaustive relative to
> the data. If a **Probability Map** had been requested and **Saved to
> Disk** at the initial classification, the following map (Figure 16)
> could have been viewed. This is a map showing in color coded fashion
> the degree of membership of each
>
> pixel to the class to which it has been assigned. The dark blue areas
> of the map indicate low probability of class membership for the class
> to which those pixels have been assigned, the yellow or light areas,
> high probability. The legend quantifies this probability in terms of a
> Gaussian distribution. In this case, this map points up several cases
> where additional classes might be established. For example, it would
> appear useful to define a class called Roads.

![](media/image24.pct){width="5.666666666666667in"
height="4.041666666666667in"}

Figure 16. Probability Map from Initial Classification (in color).

> • The use of feature selection or feature extraction. For example, one
> might choose **Feature Extraction** from the **Processor** menu and
> generate a new feature space via an optimal linear transformation of
> the spectral bands. Though a classification using all of the new
> features would give the same performance as the complete set in the
> old space, an appropriately sized subset of the new features might
> improve the performance.
>
> • Once an adequately exhaustive set of classes have been appropriately
> defined use of the **Enhance Statistics** processor under the
> **Processor** menu should make the class statistics more robust in the
> sense of providing for better generalization to samples other than the
> training samples.

These are some of the additional possibilities. Obviously, there are
many parameter settings, permutations and combinations of these which
could be explored, too many to test here. However, in order to obtain
some idea of how well this data set could be classified into the classes
contained in the generalized reconnaissance map, we will

use all of the fields defined in that map as training fields for one
final classification. The result is shown in Figure 17 and was obtained
by using the Discriminant Analysis option of the Feature Extraction
processor. The classification was done using the ECHO algorithm with the
first 5 of the transformed features. Table 5 shows the accuracy summary
based upon the generalized reconnaissance map fields and indicates an
accuracy above 80%. As can be seen, there remains some confusion between
soybeans and corn, and given the limited canopy development (≤ 5%) there
appears little likelihood that this confusion can be eliminated using
this data set.

![](media/image25.pct){width="6.486111111111111in"
height="4.027777777777778in"}

> Figure 17. ECHO Classification of first 5 Decision Boundary Feature
> Extraction (DBFE) transformed features of recon. map training
> statistics (in color). 30 sec.

  --------------- ------------- -----------------------------------------
  Project                       Number of Samples in Thematic Image Class

  --------------- ------------- -----------------------------------------

  ------- ------- --------- -------- --- --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---- ---- ----
  Class   Class   Percent   Number   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16

  ------- ------- --------- -------- --- --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---- ---- ----

Name

No.

Correct

Samples

Alfalf

Corn-n

Corn-m

Corn

Grass/

Grass/

Grass/

Hay-wi

Oats

Soybea

Soybea

Soybea

Wheat

Woods

Bldg-G

Stone

Alfalfa

1

83.3

54

45

0

0

0

1

0

0

3

0

5

0

0

0

0

0

0

  ------------- --- ------ ------ --- ------ ---- ---- --- --- --- --- --- ---- ---- ----- --- --- --- ---
  Corn-notill   2   78.9   1434   0   1132   56   16   2   0   0   0   1   31   82   110   0   0   1   3

  ------------- --- ------ ------ --- ------ ---- ---- --- --- --- --- --- ---- ---- ----- --- --- --- ---

  ---------- --- ------ ----- --- ---- ----- ---- --- --- --- --- --- --- --- ---- --- --- --- ---
  Corn-min   3   78.3   834   1   81   653   15   0   0   0   0   1   2   5   75   0   0   1   0

  ---------- --- ------ ----- --- ---- ----- ---- --- --- --- --- --- --- --- ---- --- --- --- ---

  ------- --- ------ ----- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- --- ---
  Corn    4   91.5   234   1   3   1   214   1   5   0   0   1   1   1   3   0   0   3   0

  ------- --- ------ ----- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- --- ---

  --------------- --- ------ ----- --- --- --- --- ----- --- --- --- --- --- --- ---- --- --- ---- ---
  Grass/Pasture   5   83.3   497   3   0   2   3   414   4   0   0   1   0   1   11   0   0   57   1

  --------------- --- ------ ----- --- --- --- --- ----- --- --- --- --- --- --- ---- --- --- ---- ---

  ------------- --- ---- ----- --- --- --- --- --- ----- --- --- --- --- --- --- --- --- ---- ---
  Grass/Trees   6   94   747   0   0   0   4   0   702   0   0   0   0   0   0   0   3   38   0

  ------------- --- ---- ----- --- --- --- --- --- ----- --- --- --- --- --- --- --- --- ---- ---

  -------------- --- ------ ---- --- --- --- --- --- --- ---- --- --- --- --- --- --- --- --- ---
  Grass/pas-mo   7   92.3   26   0   0   0   0   1   0   24   1   0   0   0   0   0   0   0   0

  -------------- --- ------ ---- --- --- --- --- --- --- ---- --- --- --- --- --- --- --- --- ---

  --------------- --- ------ ----- ---- --- --- --- --- --- --- ----- --- --- --- --- --- --- --- ---
  Hay-windrowed   8   95.7   489   10   0   0   0   0   0   6   468   0   0   0   0   0   0   5   0

  --------------- --- ------ ----- ---- --- --- --- --- --- --- ----- --- --- --- --- --- --- --- ---

  ------- --- ---- ---- --- --- --- --- --- --- --- --- ---- --- --- --- --- --- --- ---
  Oats    9   90   20   0   0   0   0   0   2   0   0   18   0   0   0   0   0   0   0

  ------- --- ---- ---- --- --- --- --- --- --- --- --- ---- --- --- --- --- --- --- ---

  ----------------- ---- ------ ----- --- --- --- ----- --- --- --- --- --- ----- --- ---- --- --- --- ---
  Soybeans-notill   10   74.9   968   0   7   1   153   0   6   0   0   1   725   1   73   0   0   1   0

  ----------------- ---- ------ ----- --- --- --- ----- --- --- --- --- --- ----- --- ---- --- --- --- ---

  -------------- ---- ------ ------ --- ---- ---- ----- ---- --- --- --- --- ---- ------ ---- --- --- --- ---
  Soybeans-min   11   81.2   2468   5   88   51   129   21   4   0   0   0   72   2003   91   0   0   4   0

  -------------- ---- ------ ------ --- ---- ---- ----- ---- --- --- --- --- ---- ------ ---- --- --- --- ---

  --------------- ---- ------ ----- --- --- --- ---- --- --- --- --- --- --- --- ----- --- --- --- ---
  Soybean-clean   12   92.5   614   4   0   5   11   0   3   0   0   0   9   5   568   0   0   5   4

  --------------- ---- ------ ----- --- --- --- ---- --- --- --- --- --- --- --- ----- --- --- --- ---

  ------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- ----- --- ---- ---
  Wheat   13   87.7   212   0   0   0   0   0   0   0   0   0   0   1   0   186   0   25   0

  ------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- ----- --- ---- ---

  ------- ---- ------ ------ --- --- --- --- --- --- --- --- --- --- --- --- --- ----- ----- ---
  Woods   14   70.4   1294   0   0   0   0   2   1   0   0   0   0   0   0   0   911   380   0

  ------- ---- ------ ------ --- --- --- --- --- --- --- --- --- --- --- --- --- ----- ----- ---

  ------------------ ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- --- ---- ----- ---
  Bldg-Grass-Tree-   15   73.9   380   0   0   1   3   1   3   0   0   0   0   0   1   1   88   281   1

  ------------------ ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- --- ---- ----- ---

  ----------------- ---- ------ ---- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----
  Stone-steeltowe   16   98.9   95   0   0   0   0   0   0   0   0   0   0   0   1   0   0   0   94

  ----------------- ---- ------ ---- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----

TOTAL

10366

69

1311

770

548

443

730

30

472

23

845

2099

933

187

1002

801

103

  -----------------------------------------------------------------------
  OVERALL PERFORMANCE( 8438/10366) = 81.4

  -----------------------------------------------------------------------

Table 5. Quantitative Classification summary based upon the fields of
Figure 9.

Analysis of Hyperspectral Data

More Background and Concepts

For some years, the availability of multispectral data has been limited
to data of relatively low dimensionality (\< 10 to 20 bands). Recent
advances in sensor technology, making possible data with several hundred
bands, has provided a significant step function in the complexity of
such data, and was motivated by the hypothesis that this new type of
data, commonly referred to has hyperspectral data, provides the
potential for gathering much more detailed information than previously
possible, and doing so with greater accuracy. However, the significant
increase in data complexity, quite aside from its increased volume,
means that more than simply scaling up of computer speed and storage
allocation will be required to realize the potential which this data
provides. New analysis techniques will be required.

The techniques described in this paper and implemented in MultiSpec
result from a substantial effort[^17] to devise suitable means to
analyze hyperspectral data. The techniques resulting (as well as those
described above) are based upon principles having their origin in
communication systems research of the last half dozen decades. Basically
the perspective is that of modeling both the signal (variations
diagnostic of the materials of current interest) and the noise
(variations not currently of interest) in order to optimally
discriminate among the expected list of signals. In the field of
communication engineering, this approach has made possible the
transmission of

information even when the signal-to-noise ratio is very much less than
one, e.g., the transmission of scientific data from deep space probes
using very low power transmitters.

The hypothesis followed here is that following these communication
system principles in the case of multispectral remote sensing data
analysis should make possible the extraction of information from data to
a similarly unexpected degree. The advent of hyperspectral data makes
practical a realistic test of this hypothesis. Using hyperspectral data
from the site and time of the previous section but using all 220
spectral bands instead of the nine used previously, we will focus upon
the six classes of soybeans and corn defined previously. These six
classes represent two different plant species which have been grown
under three different tillage practices, no-till, minimum-till, and
clean till. Crops grown under no-till, minimum-till and clean till
practices result in, respectively, a substantial amount, a moderate
amount, or almost no residue from last season's crops being present on
the surface. At the time of collection, the corn and soybean plants had
only recently emerged and had achieved a canopy of only very limited
ground cover (≈ 5%). Thus the variation in spectral response due to (a)
the soil type variations, and (b) the varying amount of residue from
last season's crop, would provide a much greater influence upon the
pixel spectral response than would the vegetative species variation. In
this sense, then, the problem of species identification represents a
specific example of a generic low signal-to-noise information extraction
problem. It is seen in the previous section that, even following sound
practices, it was not really possible to discriminate between these two
current plant species with high accuracy using the nine well-positioned
spectral bands of the data of conventional dimensionality. In the
following section we will lead the reader through analysis steps that
allow for discriminating between these two plant species to essentially
arbitrarily high accuracy.

In addressing the problem of high dimensional data analysis, it is
important to recognize the differences that working in high dimensional
space makes. One must understand as thoroughly as possible, based upon
currently available fundamental knowledge about high dimensional space,
what may be different about the information extraction characteristics
of such spaces. This is an area in which the background fundamental
knowledge is still not very far advanced, but there are some
characteristics which are known to be different than intuition might
suggest. For example, here are two conjectures about geometrical
properties whose validity would at first seem clear[^18].

> • Borsuk's Conjecture: If you break a stick in two, both pieces are
> shorter than the original.
>
> • Keller's Conjecture: It is possible to use cubes (hypercubes) of
> equal size to fill an N-dimensional space, leaving no overlaps nor
> underlaps.

As it turns out, counter-examples to both have been found for higher
dimensional spaces. Thus, one must be careful about using two or three
dimensional conceptual truths as a basis for conclusions in higher
dimensional spaces.

Other aspects which are out of the ordinary are statements such as, "as
the dimensionality increases, the amount of volume increases enormously
fast, and most of it is away from the origin." Indeed, it may be this
enormous volume that provides the potential for discriminating between
the spectral responses of many more materials, for there is volume
available within which many more spectral responses can reside without
overlapping others. However, even there it is the case that what seems
obviously true is not quite as one would expect. As an example, consider
the following graph of results[^19].

![](media/image26.pct){width="6.333333333333333in"
height="5.041666666666667in"}

Figure 18. Classification accuracy using first order statistics only,
second order statistics only, and both.

The top curve shows classification accuracy vs. dimensionality for a
conventional maximum likelihood Gaussian classification. Such a
classifier uses both first and second order statistics, both the mean
vector and the correlation or covariance matrix. The class mean vector
defines the location of the centroid of the class in N-dimensional
space. The second order statistics define the shape and orientation of
the class distribution. The lower two curves of Figure 18 show the
performance using the mean vector information only and the covariance
information only. Note that at low

dimensionality, N \< 6, the mean-only classification provides higher
accuracy than the covariance-only information, as one might expect. That
is, the location of the distribution apparently contributes more to
class separability than does the shape and orientation of the
distribution. However, in this case, with increased dimensionality this
performance soon saturates and improves no further. On the other hand,
the covariance-only result continues to increase with dimensionality,
indicating that for high dimensionality the shape and orientation are
more significant to separation than is the location information. The
implication of this is that, for hyperspectral data, one must pay close
attention to the estimation of the second order statistics.

Some implications of this to the analysis of hyperspectral data are the
following:

> • Larger numbers of training samples will be needed to properly
> characterize classes than is the case with multispectral data of
> conventional dimensionality. In general, estimates of second order
> statistics converge more slowly to their final values than do first
> order statistics as the number of samples is increased.
>
> • Care must be exercised that, preprocessing procedures which are
> intended to improve the first order characteristics do not
> inadvertently negatively impact the second order characteristics and
> thus, rather than improve, actually diminish the information content
> of the data. Examples of where this could occur are in attempts to
> "calibrate" the data or to "correct" for various observational or
> environmental effects.

Other differences arise for more straightforward reasons. For example,
while the principal component transformation is quite commonly used with
low dimensional data, the fact of now having a large number of spectral
bands change the implications of its use. The principle component
transformation basically focuses upon the signal variation *of the data
set as a whole,* not of the separation between classes*.* It attempts to
maximize the variation contained in the low order transformed
components, relegating variations of less significant size to higher
order components. Thus in 4-band data, a modest variation occurring in a
single band is likely to have a significant effect, showing up in
low-ordered principal components. However, the same sized variation
occurring in one of 220 bands might not, and yet it could be just as
diagnostic of a class of interest.

The matter of the best way to analyze hyperspectral data is far from a
mature technology. Indeed, it will be appropriate for this to be the
subject of research for some time to come, as more hyperspectral data
from different sensors over different types of scenes for different
applications come under study. At present, it seems inappropriate to
attempt to analyze such high dimensional data in a single step. Rather
we shall follow a process as diagrammed in Figure 19. We shall introduce
a preliminary or pre-processing step which is intended to re-orient the
feature space toward the specific set of classes of current interest.
This will allow for the selection of a lower dimensional set of features
which can be effective with a conventional classification algorithm.

![](media/image27.pct){width="5.638888888888889in"
height="2.5277777777777777in"}

Figure 19. The concept to be followed in analyzing a hyperspectral data
set.

As previously indicated, the data set to be used is a small segment of
an AVIRIS data set collected in June 1992 over the Indian Pine Test
site, a 100 square mile area in northwestern Indiana. The data set is
the same 145 by 145 set of pixels used previously, but this time the
full 220 bands will be used. The data set is designated 92AV3C, and is
9.04 Megabytes in size. The small size (21,025 pixels) of this data set,
in fact, make it too small to be of practical significance. The limited
size was chosen as a compromise toward making the data set easily
portable so that the reader would find it easier to acquire remotely.
Due to the need to have more extensive sized training sets for the 220
band data, we will use the entire set of fields defined by the
generalized reconnaissance map, file 92AV3gt.GIS, as the training set.
Since this will mean that the training set will be the set used for
accuracy evaluation, it will not be possible to test the ability of the
classifier to generalize beyond its training set. This question will be
left to another occasion. The file 92AV3.Project contains the fields of
the generalized reconnaissance map as training fields and may be used as
the MultiSpec project file.

Example Analysis Steps Using MultiSpec[^20]

Begin the analysis process by **Open**ing the project file
**92AV3.Project**, and **Open** the project image using the default
bands 50, 27, and 27 for red, green, and blue respectively. The fields
of the generalized reconnaissance map should show outlined as training
fields. The first step will be to do the class-conditional preprocessing
step indicated above, to focus the needed dimensionality based upon the
now available training statistics to a level that will enable a
classifier to function effectively.

> • There are several possibilities for this within MultiSpec. The most
> conventional might be to use the Feature Selection processor. One
> cannot simply search, for example, for the best 50 bands out of 220,
> as the number of possibilities which must be tested is too large.
> However, some suboptimal options are available in the Feature
> Selection processor to bring the number
>
> of search combinations within practical size. For example, one could
> set the **Number of Contiguous Channels** to **5** and set the **Step
> Search** option to on. This will provide a subset of bands that could
> be considered further until a final subset is chosen. Even so, several
> classes will need to be excluded from consideration, since they have
> too small a number of training samples to allow for matrix inversion.
> Another possibility will be raised later in this discussion.
>
> • At a number of points in processing hyperspectral data, exercising
> good judgment can reduce the amount of processing time or effect a
> tradeoff between processing and storage requirements. For example, the
> project file 92AV3.Project, has been provided without the class
> statistics stored with it. This significantly reduces the size of the
> file and speeds the process of opening the project substantially.
> However, it means that the project statistics must be recalculated
> before any processor requiring them can be used. (To do so, open the
> **Statistics** processor, click on the **Project** button and then the
> **Update P. Statistics** button.) Saving and loading a project file
> with the project statistics included may be faster than recalculating
> them each time the file is opened, but substantially more storage
> space is required.

We will use the **Feature Extraction** processor under the **Processor**
menu, and we will choose the **Decision Boundary** option of this
processor. It will be necessary to use a subset of classes which
excludes all classes that have less than 221 training samples, to avoid
singularity problems with matrix inversion. Classes which must thus be
excluded are Alfalfa, Grass/pasture-mowed, Oats, Wheat, and Stone-steel
towers. This is a very long calculation (9745 seconds ≈ 2.7 hours on a
Power Macintosh 8100/80). It determines a linear transformation which is
optimal for the set of classes involved, and orders the features
determined in order of their value in discriminating between the
classes. One may then use the first N of these features to classify the
data to the level of accuracy desired.

Figure 20 shows the resulting maximum likelihood pixel classification
for N = 19, the largest value of N that can be used and still include
all of the classes, since Oats has only 20 training pixels. Table 6
gives the quantitative evaluation of the classification. It is seen that
the overall accuracy is 86.3% at this dimensionality.

> • If the classification is done directly after the Decision Boundary
> Feature Extraction using the **Use Transformation** option of the
> Classifier processor, note that a 19 feature classification will still
> take as long (29.9 minutes) as a 220 feature classification would,
> because the classifier must use all 220 bands to calculate the linear
> combination defining each feature. A possibly faster method would be
> to use the **Reformat** processor of the **Processor** menu to create
> a transformed data set with the transform just calculated. Then
> classifications can be done without the **Use Transformation** option
> of the Classifier processor set, and only the first 19 of the features
> will actually be used for the 19 feature classification. That is what
> was done for the following examples.

![](media/image28.pct){width="6.513888888888889in"
height="4.027777777777778in"}

Figure 20. 19 DBFE Feature Maximum Likelihood Pixel Classification (in
color). 21 Seconds

  --------------- ------------- -----------------------------------------
  Project                       Number of Samples in Thematic Image Class

  --------------- ------------- -----------------------------------------

  ------- ------- --------- -------- --- --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---- ---- ----
  Class   Class   Percent   Number   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16

  ------- ------- --------- -------- --- --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---- ---- ----

Name

No.

Correct

Samples

Alfalf

Corn-n

Corn-m

Corn

Grass/

Grass/

Grass/

Hay-wi

Oats

Soybea

Soybea

Soybea

Wheat

Woods

Bldg-G

Stone

Alfalfa

1

100.0

54

54

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

  ------------- --- ------ ------ --- ------ ---- --- --- --- --- --- --- ---- ----- --- --- --- --- ---
  Corn-notill   2   83.1   1434   0   1191   46   2   2   2   0   0   0   75   110   5   0   0   1   0

  ------------- --- ------ ------ --- ------ ---- --- --- --- --- --- --- ---- ----- --- --- --- --- ---

  ---------- --- ------ ----- --- ---- ----- --- --- --- --- --- --- --- ---- ---- --- --- --- ---
  Corn-min   3   89.0   834   0   31   742   2   0   0   0   0   0   4   37   18   0   0   0   0

  ---------- --- ------ ----- --- ---- ----- --- --- --- --- --- --- --- ---- ---- --- --- --- ---

  ------- --- ------ ----- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- --- ---
  Corn    4   99.1   234   0   0   0   232   2   0   0   0   0   0   0   0   0   0   0   0

  ------- --- ------ ----- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- --- ---

  --------------- --- ------ ----- --- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- ---
  Grass/Pasture   5   96.6   497   0   1   0   0   480   0   0   0   0   1   3   8   0   0   4   0

  --------------- --- ------ ----- --- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- ---

  ------------- --- ------ ----- --- --- --- --- --- ----- --- --- --- --- --- --- --- --- ---- ---
  Grass/Trees   6   97.7   747   0   0   0   0   0   730   0   0   0   1   0   0   0   0   16   0

  ------------- --- ------ ----- --- --- --- --- --- ----- --- --- --- --- --- --- --- --- ---- ---

  -------------- --- ------- ---- --- --- --- --- --- --- ---- --- --- --- --- --- --- --- --- ---
  Grass/pas-mo   7   100.0   26   0   0   0   0   0   0   26   0   0   0   0   0   0   0   0   0

  -------------- --- ------- ---- --- --- --- --- --- --- ---- --- --- --- --- --- --- --- --- ---

  --------------- --- ------ ----- --- --- --- --- --- --- --- ----- --- --- --- --- --- --- --- ---
  Hay-windrowed   8   98.2   489   9   0   0   0   0   0   0   480   0   0   0   0   0   0   0   0

  --------------- --- ------ ----- --- --- --- --- --- --- --- ----- --- --- --- --- --- --- --- ---

  ------- --- ------- ---- --- --- --- --- --- --- --- --- ---- --- --- --- --- --- --- ---
  Oats    9   100.0   20   0   0   0   0   0   0   0   0   20   0   0   0   0   0   0   0

  ------- --- ------- ---- --- --- --- --- --- --- --- --- ---- --- --- --- --- --- --- ---

  ------------ ---- ------ ----- --- ---- --- --- --- --- --- --- --- ----- ---- --- --- --- --- ---
  Soy-notill   10   87.7   968   0   47   2   0   2   2   0   0   0   849   66   0   0   0   0   0

  ------------ ---- ------ ----- --- ---- --- --- --- --- --- --- --- ----- ---- --- --- --- --- ---

  -------------- ---- ------ ------ --- ----- ----- --- ---- --- --- --- --- ----- ------ ---- --- --- --- ---
  Soybeans-min   11   68.3   2468   2   290   146   1   17   3   0   0   0   234   1685   87   0   0   2   1

  -------------- ---- ------ ------ --- ----- ----- --- ---- --- --- --- --- ----- ------ ---- --- --- --- ---

  --------------- ---- ------ ----- --- --- ---- --- --- --- --- --- --- --- --- ----- --- --- --- ---
  Soybean-clean   12   96.7   614   0   0   10   0   0   0   0   0   0   0   8   594   0   0   0   2

  --------------- ---- ------ ----- --- --- ---- --- --- --- --- --- --- --- --- ----- --- --- --- ---

  ------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- ----- --- --- ---
  Wheat   13   99.1   212   0   0   0   0   0   0   0   0   0   0   1   0   210   0   1   0

  ------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- ----- --- --- ---

  ------- ---- ------ ------ --- --- --- --- --- --- --- --- --- --- --- --- --- ------ ---- ---
  Woods   14   95.7   1294   0   0   0   0   5   2   0   0   0   0   0   0   0   1238   49   0

  ------- ---- ------ ------ --- --- --- --- --- --- --- --- --- --- --- --- --- ------ ---- ---

  ----------------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- --- ---- ----- ---
  Bldg-Grass-Tree   15   84.5   380   0   0   0   0   1   6   0   0   0   0   0   0   0   51   321   1

  ----------------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- --- ---- ----- ---

  ----------------- ---- ------- ---- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----
  Stone-steeltowe   16   100.0   95   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   95

  ----------------- ---- ------- ---- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----

TOTAL

10366

65

1560

946

237

509

745

26

480

20

1164

1910

712

210

1289

394

99

  -----------------------------------------------------------------------
  OVERALL PERFORMANCE (8947/10366 ) = 86.3

  -----------------------------------------------------------------------

Table 6. 19 DBFE Feature Maximum Likelihood Classification Evaluation

It is seen that the classification map appears to have a good many "salt
and pepper" errors to it. A more accurate classification results if the
ECHO spectral/spatial classification algorithm is used on the same
statistics. The results of doing so are shown in Figure 21 and Table 7.

![](media/image29.pct){width="6.569444444444445in"
height="4.027777777777778in"}

Figure 21. 19 DBFE ECHO classification (in color). 21 seconds

  --------------- ------------- -----------------------------------------
  Project                       Number of Samples in Thematic Image Class

  --------------- ------------- -----------------------------------------

  ------- ------- --------- -------- --- --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---- ---- ----
  Class   Class   Percent   Number   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16

  ------- ------- --------- -------- --- --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---- ---- ----

Name

No.

Correct

Samples

Alfalf

Corn-n

Corn-m

Corn

Grass/

Grass/

Grass/

Hay-wi

Oats

Soybea

Soybea

Soybea

Wheat

Woods

Bldg-G

Stone

Alfalfa

1

100.0

54

54

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

  ------------- --- ------ ------ --- ------ --- --- --- --- --- --- --- ---- ---- --- --- --- --- ---
  Corn-notill   2   90.2   1434   0   1294   7   2   2   2   0   0   0   36   85   5   0   0   1   0

  ------------- --- ------ ------ --- ------ --- --- --- --- --- --- --- ---- ---- --- --- --- --- ---

  ---------- --- ------ ----- --- --- ----- --- --- --- --- --- --- --- ---- ---- --- --- --- ---
  Corn-min   3   96.2   834   0   6   802   2   0   0   0   0   0   2   10   12   0   0   0   0

  ---------- --- ------ ----- --- --- ----- --- --- --- --- --- --- --- ---- ---- --- --- --- ---

  ------- --- ------ ----- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- --- ---
  Corn    4   98.3   234   0   0   2   230   2   0   0   0   0   0   0   0   0   0   0   0

  ------- --- ------ ----- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- --- ---

  --------------- --- ------ ----- --- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- ---
  Grass/Pasture   5   97.0   497   0   1   0   0   482   0   0   0   0   1   3   8   0   0   2   0

  --------------- --- ------ ----- --- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- ---

  ------------- --- ------ ----- --- --- --- --- --- ----- --- --- --- --- --- --- --- --- ---- ---
  Grass/Trees   6   97.9   747   0   0   0   0   0   731   0   0   0   1   0   0   0   0   15   0

  ------------- --- ------ ----- --- --- --- --- --- ----- --- --- --- --- --- --- --- --- ---- ---

  -------------- --- ------- ---- --- --- --- --- --- --- ---- --- --- --- --- --- --- --- --- ---
  Grass/pas-mo   7   100.0   26   0   0   0   0   0   0   26   0   0   0   0   0   0   0   0   0

  -------------- --- ------- ---- --- --- --- --- --- --- ---- --- --- --- --- --- --- --- --- ---

  --------------- --- ------ ----- --- --- --- --- --- --- --- ----- --- --- --- --- --- --- --- ---
  Hay-windrowed   8   99.0   489   5   0   0   0   0   0   0   484   0   0   0   0   0   0   0   0

  --------------- --- ------ ----- --- --- --- --- --- --- --- ----- --- --- --- --- --- --- --- ---

  ------- --- ------- ---- --- --- --- --- --- --- --- --- ---- --- --- --- --- --- --- ---
  Oats    9   100.0   20   0   0   0   0   0   0   0   0   20   0   0   0   0   0   0   0

  ------- --- ------- ---- --- --- --- --- --- --- --- --- ---- --- --- --- --- --- --- ---

  ------------ ---- ------ ----- --- ---- --- --- --- --- --- --- --- ----- ---- --- --- --- --- ---
  Soy-notill   10   96.8   968   0   12   1   0   2   2   0   0   0   937   14   0   0   0   0   0

  ------------ ---- ------ ----- --- ---- --- --- --- --- --- --- --- ----- ---- --- --- --- --- ---

  -------------- ---- ------ ------ --- ---- ---- --- ---- --- --- --- --- ----- ------ ---- --- --- --- ---
  Soybeans-min   11   90.2   2468   1   56   16   0   17   3   0   0   0   116   2227   30   0   0   2   0

  -------------- ---- ------ ------ --- ---- ---- --- ---- --- --- --- --- ----- ------ ---- --- --- --- ---

  --------------- ---- ------ ----- --- --- ---- --- --- --- --- --- --- --- --- ----- --- --- --- ---
  Soybean-clean   12   96.4   614   0   0   15   0   0   0   0   0   0   2   3   592   0   0   0   2

  --------------- ---- ------ ----- --- --- ---- --- --- --- --- --- --- --- --- ----- --- --- --- ---

  ------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- ----- --- --- ---
  Wheat   13   99.1   212   0   0   0   0   0   0   0   0   0   0   1   0   210   0   1   0

  ------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- ----- --- --- ---

  ------- ---- ------ ------ --- --- --- --- --- --- --- --- --- --- --- --- --- ------ ---- ---
  Woods   14   95.2   1294   0   0   0   0   5   1   0   0   0   0   0   0   0   1232   56   0

  ------- ---- ------ ------ --- --- --- --- --- --- --- --- --- --- --- --- --- ------ ---- ---

  ---------------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----- ---
  Bldg-Grass-Tre   15   97.4   380   0   0   0   0   1   4   0   0   0   0   0   0   0   4   370   1

  ---------------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----- ---

  ---------------- ---- ------- ---- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----
  Stone-steeltow   16   100.0   95   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   95

  ---------------- ---- ------- ---- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----

TOTAL

10366

60

1369

843

234

511

743

26

484

20

1095

2343

647

210

1236

447

98

  -----------------------------------------------------------------------
  OVERALL PERFORMANCE(9786/10366 ) = 94.4

  -----------------------------------------------------------------------

Table 7. 19 DBFE Feature ECHO Classification Evaluation

Indeed the accuracy can be further improved by using more of the DBFE
features. Figure 22 and Table 8 give the results for the 50 feature
case, again using the ECHO algorithm. Note that the accuracy has now
risen to 97.8 %.

![](media/image30.pct){width="7.069444444444445in"
height="4.027777777777778in"}

Figure 22. 50 DBFE Feature ECHO Classification (in color). 85 Seconds.

  --------------- ------------- -----------------------------------------
  Project                       Number of Samples in Thematic Image Class

  --------------- ------------- -----------------------------------------

  ------- ------- --------- -------- --- --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---- ---- ----
  Class   Class   Percent   Number   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16

  ------- ------- --------- -------- --- --- --- --- --- --- --- --- --- ---- ---- ---- ---- ---- ---- ----

Name

No.

Correct

Samples

Alfalf

Corn-n

Corn-m

Corn

Grass/

Grass/

Grass/

Hay-wi

Oats

Soybea

Soybea

Soybea

Wheat

Woods

Bldg-G

Stone

Alfalfa

1

100.0

54

54

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

  ------------- --- ------ ------ --- ------ --- --- --- --- --- --- --- ---- ---- --- --- --- --- ---
  Corn-notill   2   96.1   1434   0   1378   3   0   2   2   0   0   0   18   28   2   0   0   1   0

  ------------- --- ------ ------ --- ------ --- --- --- --- --- --- --- ---- ---- --- --- --- --- ---

  ---------- --- ------ ----- --- --- ----- --- --- --- --- --- --- --- --- --- --- --- --- ---
  Corn-min   3   99.2   834   0   2   827   0   0   0   0   0   0   0   1   4   0   0   0   0

  ---------- --- ------ ----- --- --- ----- --- --- --- --- --- --- --- --- --- --- --- --- ---

  ------- --- ------ ----- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- --- ---
  Corn    4   99.1   234   0   0   2   232   0   0   0   0   0   0   0   0   0   0   0   0

  ------- --- ------ ----- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- --- ---

  --------------- --- ------ ----- --- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- ---
  Grass/Pasture   5   99.0   497   0   0   0   0   492   0   0   0   0   0   4   1   0   0   0   0

  --------------- --- ------ ----- --- --- --- --- ----- --- --- --- --- --- --- --- --- --- --- ---

  ------------- --- ------ ----- --- --- --- --- --- ----- --- --- --- --- --- --- --- --- --- ---
  Grass/Trees   6   99.9   747   0   0   0   0   0   746   0   0   0   0   0   0   0   0   1   0

  ------------- --- ------ ----- --- --- --- --- --- ----- --- --- --- --- --- --- --- --- --- ---

  ---------------- --- ----- ---- --- --- --- --- --- --- --- ---- --- --- --- --- --- --- --- ---
  Grass/past.-mo   7   0.0   26   0   0   0   0   1   1   0   23   0   0   1   0   0   0   0   0

  ---------------- --- ----- ---- --- --- --- --- --- --- --- ---- --- --- --- --- --- --- --- ---

  --------------- --- ------- ----- --- --- --- --- --- --- --- ----- --- --- --- --- --- --- --- ---
  Hay-windrowed   8   100.0   489   0   0   0   0   0   0   0   489   0   0   0   0   0   0   0   0

  --------------- --- ------- ----- --- --- --- --- --- --- --- ----- --- --- --- --- --- --- --- ---

  ------- --- ----- ---- --- --- --- --- ---- --- --- --- --- --- --- --- --- --- --- ---
  Oats    9   0.0   20   0   4   0   0   11   2   0   0   0   0   1   0   0   0   2   0

  ------- --- ----- ---- --- --- --- --- ---- --- --- --- --- --- --- --- --- --- --- ---

  ---------------- ---- ------ ----- --- --- --- --- --- --- --- --- --- ----- --- --- --- --- --- ---
  Soybeans-notil   10   98.6   968   0   3   0   0   2   2   0   0   0   954   7   0   0   0   0   0

  ---------------- ---- ------ ----- --- --- --- --- --- --- --- --- --- ----- --- --- --- --- --- ---

  -------------- ---- ------ ------ --- --- --- --- ---- --- --- --- --- ---- ------ ---- --- --- --- ---
  Soybeans-min   11   97.6   2468   0   9   5   0   12   6   0   0   0   12   2408   14   0   0   2   0

  -------------- ---- ------ ------ --- --- --- --- ---- --- --- --- --- ---- ------ ---- --- --- --- ---

  --------------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- ----- --- --- --- ---
  Soybean-clean   12   98.5   614   0   0   4   0   0   0   0   0   0   2   2   605   0   0   0   1

  --------------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- ----- --- --- --- ---

  ------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- ----- --- --- ---
  Wheat   13   99.5   212   0   0   0   0   0   0   0   0   0   0   0   0   211   1   0   0

  ------- ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- ----- --- --- ---

  ------- ---- ------ ------ --- --- --- --- --- --- --- --- --- --- --- --- --- ------ ---- ---
  Woods   14   98.5   1294   0   0   0   0   1   2   0   0   0   0   0   0   0   1274   17   0

  ------- ---- ------ ------ --- --- --- --- --- --- --- --- --- --- --- --- --- ------ ---- ---

  ------------------ ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----- ---
  Bldg-Grass-Tree-   15   98.9   380   0   0   0   0   0   0   0   0   0   0   0   0   0   4   376   0

  ------------------ ---- ------ ----- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----- ---

  ----------------- ---- ------- ---- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----
  Stone-steeltowe   16   100.0   95   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   95

  ----------------- ---- ------- ---- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ----

TOTAL

10366

54

1396

841

232

521

761

0

512

0

986

2452

626

211

1279

399

96

  -----------------------------------------------------------------------
  OVERALL PERFORMANCE(10141/10366)=97.8

  -----------------------------------------------------------------------

Table 8. 50 DBFE Feature ECHO Classification Evaluation.

Figure 23 shows a graph of accuracy vs. number of features used for the
DBFE features and shows that at 50 features, the accuracy is still
increasing for the Maximum Likelihood pixel classifier. For ECHO, the
accuracy is slightly higher than for the Maximum Likelihood Pixel
classifier. In spite of the greater complexity of this algorithm, it is
also usually as fast or faster than the pixel classifier.

![](media/image31.pct){width="5.791666666666667in"
height="4.902777777777778in"}

> Figure 23. Accuracy vs. number of features for the Decision Boundary
> and Discriminant Analysis Feature Extraction algorithms with Maximum
> Likelihood Pixel classification. For the DBFE algorithm, the ECHO
> classifier accuracy is also indicated for 19 and 50 features.
>
> • Another possible method for carrying out the Class-Conditional
> Preprocessing calculation is to use the Discriminant Analysis Feature
> Extraction (DAFE) option of the Feature Extraction Processor. This
> option has several desirable features. It is a much shorter
> calculation (158 seconds vs. 9745 seconds for DBFE), and classes may
> be included even though they have a number of training samples fewer
> than the number of features. It also tends to give higher accuracy so
> long as the number of features used is less than M -- 1 where M is the
> number of classes. However, it calculates optimal features only up to
> M -- 1, and the features determined beyond that point are unreliable.
> The performance for the Maximum Likelihood pixel classifier using DAFE
> features is shown in Figure 23. It is seen that the rate of increase
> of performance became small as the number of dimensions, N, approached
> the number of classes, M=15, in the above graph. Thus theory and this
> example suggest that a good practice would be to use DAFE when the
> number of classes to be used is greater than the number of features
> needed, otherwise to use DBFE.

Closing Comments

In this example we have shown one set of steps which can be used to
analyze hyperspectral data. The method produced arbitrarily high
accuracy even for classes which have a signal-to-background ratio which
is much less than one, thus demonstrating the increased value of such
high dimensional data. We note, however, that the generalization
characteristics of the particular example have not been determined, due
to the constraint upon data set size imposed arbitrarily on the example
to hold the data set to a size which can be easily transferred. For this
reason and for reasons having to do with the immature state of the
technology of analysis of such high dimensional data sets, one should
not read too much in the state of maturity of the analysis technology
illustrated here.

Development of these techniques continues. MultiSpec already contains a
processor, the Statistics Enhancement Processor, specifically included
to improve the generalization characteristics of a given set of training
statistics relative to a data set to be analyzed. However, the general
characteristics of this processor have only been measured in a limited
number of circumstances, and a rigorous determination of its proper use
and performance are still to be accomplished.

In considering the acquisition of information by multispectral remote
sensing means, one must ask the following questions.

> 1\. Is the desired information actually present in the electromagnetic
> variations traversing the space between the target and the sensor? If
> so,
>
> 2\. Does the sensor successfully capture these variation with enough
> precision and detail to insure that the desired information is now
> contained in the data stream coming from the sensor to the analyst's
> location.

If the answer to both of these questions is yes, then it is up to the
analyst to find a way to discriminate between the desired diagnostic
variations and all others which are (invariable) present. According to
the framework presented above, the task of the analyst is to find a way
to partition the N-dimensional feature space in such a way that the
feature variations associated with materials of interest are contained
within subregions of this space as uniquely and mutually exclusively as
possible.

The matter that changes substantially with hyperspectral data over that
of conventional dimensionality is the magnitude of the volume in this
space. This is at once both the opportunity that was sought and a curse.
The magnitude of this volume is the very thing that provides the
opportunity for many more materials to be discriminatable, but it is
this very characteristic that adds substantially to the complexity of
doing so. As has been seen, as the dimensionality goes up, the ability
of the analyst to use intuitive concepts which otherwise seem
self-evident, is precluded, and in fact many of these concepts become
misleading. Instead, one must rely upon quantitative tools which
objectively provide information to the analyst about the separability of
spectral responses in feature space, so that the right path through the
analysis process can be quickly found. Indeed, at this stage of
development, it is not so much, "Is it possible to discriminate between
this material and others in the scene," as it is, "Can I find a way to
discriminate between this material and others in the scene."

[^1]: Some texts use the name *Stochastic Process* in place of *Random
    Process.*

[^2]: Davenport, W.B., Jr. and William L. Root, *An Introduction to the
    Theory of Random Signals and Noise,* McGraw-Hill, 1958.

[^3]: Papoulis, A., *Probability, Random Variables, and Stochastic
    Processes,* McGraw-Hill, 1965.

[^4]: Helstrom, Carl W., *Probability and Stochastic Processes for
    Engineers,* 2nd Edition, Macmillan, 1991.

[^5]: Cooper, G.R., and C. D. McGillem, *Probabilistic Methods of Signal
    and System Analysis,* 2nd Edition, Holt, Rinehart & Winston, 1986.

[^6]: Indeed, a much more generalized means for transforming between
    spectral space and feature space exists. Mathematically this is
    expressed by the transformation pair,

    xn = \\i(a,b, X(λ) φn(λ) dλ) X(λ) = \\i\\su(n=1,N, xnφn(λ))

    where X(λ) is the spectral function, xn is the component in feature
    space and the φn are the (generalized) basis functions. The
    instantaneous sampling used above is just a special case of this,
    using a specific set of basis functions. This generalization becomes
    especially important and practical for hyperspectral data, where the
    sensor measures the value of X(λ) at many λ's; It is also important
    for optimal feature extraction.

[^7]: Chulhee Lee and David A. Landgrebe, \"Analyzing High Dimensional
    Multispectral Data, IEEE Transactions on Geoscience and Remote
    Sensing, Volume 31, No. 4, pp 792-800, July 1993.

[^8]: Bauer, M.E., T.E. Burk, A.R. Ek, P.R. Coppin, S.D. Lime, T.A.
    Walsh, D.K. Walters, W. Befort and D. F. Heinze, "Satellite
    Inventory of Minnesota Forest Resources," *Photogrammetric
    Engineering & Remote Sensing, Vol. LX, No. 3, pp 287-298, March
    1994.*

[^9]: Hoffbeck, Joseph P. and David A. Landgrebe, "Classification of
    High Dimensional Multispectral Image Data," Fourth Annual JPL
    Airborne Geoscience Workshop, Arlington, Virginia, October 25-29,
    1993.

[^10]: MultiSpec © Purdue Research Foundation. To obtain a copy with its
    documentation, contact the author.

[^11]: G.F. Hughes, \"On The Mean Accuracy Of Statistical Pattern
    Recognizers,\" *IEEE Trans. Infor. Theory*, Vol. IT-14, No. 1, pp
    55-63, 1968.

[^12]: Fukunaga, K. *Introduction to Statistical Pattern Recognition,*
    Academic Press, 1990.

[^13]: Behzad M. Shahshahani and David A. Landgrebe, "Effect of
    Unlabeled Samples in Reducing the Small Sample Size Problem and
    Mitigating the Hughes Phenomenon," IEEE Transactions on Geoscience
    and Remote Sensing, Vol. 32, No. 5, September 1994.

[^14]: Only 9 bands were selected from the AVIRIS data set to provide an
    analysis problem of conventional dimensionality. An analysis using
    the full 210 bands of the AVIRIS data set, constituting a true
    hyperspectral analysis problem, is given in the following section.

[^15]: Most of the figures which follow are more effective if viewed in
    color.

[^16]: The original for this and several subsequent figures are in
    color. Black & White hard copy versions will thus be difficult to
    interpret. If possible, view these figures in soft copy form on a
    color screen, so that the figures will be more interpretable.

[^17]: D. A. Landgrebe and C. J. Johannsen, Principal Investigators,
    Final Report, NASA grant NAGW-925 "Earth observational research
    using multistage EOS-like data," Purdue University, May, 1994. A
    copy is available upon request.

[^18]: Science, Vol. 259, 1 Jan 1993, pp 26-27

[^19]: Chulhee Lee and David A. Landgrebe, \"Analyzing High Dimensional
    Multispectral Data, IEEE Transactions on Geoscience and Remote
    Sensing, Volume 31, No. 4, pp 792-800, July 1993.

[^20]: Computation times listed in this section will be for a Power
    Macintosh 8100/80. These times will be shorter than for a Quadra
    840av by a factor of 3 to 10, depending on the proportion of
    floating point computation involved.