An Information Theoretic-Based Measure for Spectral Similarity and Discriminability
Chein-I Chang
Remote Sensing Signal and Image Processing Laboratory
Department of Computer Science and Electrical Engineering
University of Maryland Baltimore County
Baltimore, MD 21250
Abstract
In this paper, we propose an information theoretic criterion,
called Spectral Information Divergence (SID) for spectral
similarity and discriminability. It is derived from the concept
of divergence arising in information theory and can be used
to describe the statistics of a spectrum. Unlike Spectral Angle
Mapper (SAM) which extracts geometric features between
two spectra, SID views each pixel spectrum as a random
variable and then measures the discrepancy of probabilistic
behaviors between two spectra. In order to evaluate SID,
SAM is used for comparison via hyperspectral data.
Experimental results show that SID can characterize spectral
similarity and variability more effectively than SAM.
I. INTRODUCTION
Due to the fact that a multispectral/hyperspectral pixel is
generally mixed by materials present in the pixel with various
abundance fractions, many pure pixel-based similarity
measures may not be effective for mixed pixel classification
since they do not take advantage of spectral correlation. One
of popular spectral metrics used in remote sensing is Spectral
Angle Mapper (SAM) [1] which measures spectral similarity
by finding the angle between two spectra. Another common
measure used in pattern classification is Euclidean Distance
(ED) which calculates the distance between two samples. In
this paper, we present an information theoretic spectral
metric, called Spectral Information Divergence (SID) which
is derived from the concept of divergence in information
theory [3-4]. SID considers each pixel as a random variable
and uses its spectral histogram to define a probability
distribution. The spectral similarity between two pixels is
then measured by the discrepancy of probabilistic behaviors
between their spectra. Such advantage cannot be achieved by
any deterministic metric. ED is a common practice in
classical pattern classification to measure the spatial distance
between two data samples and so is SAM for remote sensing
images. They both are deterministic in the sense that each
data sample itself is a deterministic data vector not a random
variable as considered by SID. Therefore, SID can be
regarded as a random (stochastic) or probabilistic approach as
opposed to a deterministic approach such as ED and SAM.
One of advantages yielded by SID is that it measures
spectral variability of a single mixed pixel from a
probabilistic point of view. It can be used as a single-pixel
measure. This is particularly useful for hyperspectral data
since each hyperspectral image pixel vector is acquired by
hundreds of spectral bands and provides very valuable
spectral information in material discrimination, detection,
classification and identification. However, this advantage is
also offset by many unknown interferers that may be also
extracted from hyperspectral sensors. The unpredictability
caused by such interference can only be described by
randomness. SID arises from this need and is able to capture
the uncertainty created by unpredictable interference. From
this aspect, SID is a more appropriate measure than other
deterministic metrics such as SAM or ED. Additionally, with
the help of a spectral data base or a spectral library SID can
be also used for spectral characterization, e.g., material
identification. Through experiments we can show that SID
seems to have advantages over SAM in characterizing
spectral similarity and discriminability.
II. INFORMATION SPECTRAL DIVERGENCE (SID)
For a given multispectral/hyperspectral pixel vector
( )TLxx ,,1 ê=x , each component lx is a pixel of band
image lB . Then x can be modeled as a random variable by
defining an appropriate probability distibution. We first
assume that all component lx 's in x are nonnegative due to
the nature of radiance or reflectance. With this assumption,
we can normalize lx 's to the range ]1,0[ by defining
∑=
=
L
l
ljj xxp
1
/ so that
L
llp 1}{ ==p is the desired probability
vector resulting from the pixel vector x. In order to further
study how to use concepts arising from information theory to
capture relationship and correlation between two
multispectral/hyperspectral pixel vectors, assume that there is
another pixel vector ( )TLyy ,,1 ê=y with the probability
distribution given by Lllq 1}{ ==q and ∑=
=
L
l
ljj yyq
1
/ . Using
p and q we define Spectral Information Divergence (SID) by
)||()||(),(SID xyyxyx DD +=
(1)
where ( )∑=
=
L
l
lll qppD
1
/log)||( yx and
( )∑=
=
L
l
lll pqqD
1
/log)( x||y . It should noted that )||( yxD
is called the relative entropy of y with respect to x which is
also known as Kullack-Leibler information function, directed
0-7803-5207-6/99/$10.00 (c) 1999 IEEE

divergence or cross entropy [2]. The SID defined by Eq. (1)
can be used to measure the spectral similarity between two
pixel vectors 1x and 2x .
III. SPECTRAL DISCRIMINABILITY MEASURES
USING SID
In many applications spectral discriminability may be
important. For example, we may want to know the
discriminability of one signature against another signature
relative to a reference signature. Or we may be interested in
the discriminatory probabilities of identifying one signature
using all signatures in a reference spectral library or data
base. For this purpose, three discriminatory measures are
introduced in this section. Assume that )( , kjm ss is any
spectral metric between js and ks and d is a reference or
desired signature selected for comparison against between js
and ks .
A. Relative Spectral Discriminability Power (RSDP)
First of all, we define the discriminability of js against
ks relative to d, called Relative Spectral Discriminability
Power (RSDP), denoted by );,(RSDP dss kj as the spectral
similarity ratio of between d and js to between d and ks by






= ),(
),(
,),(
),(
max);,(RSDP
ds
ds
ds
ds
dss
j
k
k
j
kj
m
m
m
m
. (2)
In other words, the magnitude of );,(RSDP dss kj is
proportional to the degree of discriminating js from ks
relative to d. The higher is the );,(RSDP dss kj , the
discrimination is better between js and d than between ks
and d. Obviously, );,(RSDP dss kj is symmetric and
bounded below by one, i.e., 1);,(RSDP ≥dss kj with
equality if and only if kj ss = .
B. Relative Spectral Discriminability Rate (RSDR)
In material identification, one may be interested in how
much likelihood of a particular signature d will be identified
by a selective set of p signatures, S, denoted by pjj 1}{ =s . We
define the spectral discriminability probabilities of all js 's in
S relative to d by
∑=
=
p
k
kj mmjp
1
S, ),(/),()( sdsdd (3)
where ∑
=
p
k
km
1
),( sd is a normalization constant determined by
d and the measure m. The probability vector
))(,),2(),1(( S,S,S,S, pppp ddddp ê= in Eq. (10) will be called
Relative Spectral Discriminability Rate (RSDR) of d with
respect to S.
C. Relative Spectral Discriminability Entropy (RSDE)
Using Eq. (3) we can further define the Relative Spectral
Discriminability Entropy (RSDE) of signature d with respect
to the set S, denoted by )S;(RSDE dH by
)(log)()S;( S,
1
S,RSDE kpkpH
p
k
ddd ∑−=
=
. (4)
Eq. (4) provides a measure of the uncertainty of identifying d
with respect to a reference signature set pkk 1}{S == s . A
higher )S;(RSDE dH may have a less chance to identify d.
V. EXPERIMENTS
Since it was shown in [2], ED performed closely to
SAM, we only compare SID against SAM. The data set to be
used for experiments is Airborne Visible/Infrared Imaging
Spectrometer (AVIRIS) reflectance data which comprise
spectra of five material signatures, black brush, creosote
leaves, dry grass, sage brush and red soil shown in Fig. 1.
Tables 1-2 show the spectral similarity among the five
signatures using SAM and SID respectively. As we can see, it
is very difficult to distinguish these five signatures using
SAM. In contrast, SID not only can discriminate red soil
from black brush but also can tell how close sagebrush to
black brush. Tables 3-4 and Tables 5-6 are RSDP and RSDR
which are produced by SAM and SID respectively where the
desired signature d was randomly linearly mixed by 0.1055
black brush, 0.0292 creosote leaves, 0.0272 dry grass, 0.7588
red soil and 0.0974 sage brush. Once again, SID performed
significantly better than SAM. The spectral discriminability
between black brush and redsoil is 16.9736 in Table 4, 16
times that obtained using SAM in Table 3. If we compare the
spectral discriminability between creosote leaves and redsoil,
SID achieved 44.1359 in Table 4 as opposed to 1.1300
obtained by SAM in Table 3. There is 44 times using SID
better than using SAM. Table 7 shows RSDEs of RSDRs in
Tables 5-6 and the one produced by SID yielded the smallest
entropy. This further demonstrates the superiority of SID.
0-7803-5207-6/99/$10.00 (c) 1999 IEEE

VI. CONCLUSION
In this paper, an information theoretic-based spectral
measure is presented for multipsectral/hyperspectral pixel
analysis. It can be used to measure spectral similarity and
discriminability. In order to measure spectral similarity
between two pixel vectors, a new concept, SID was
introduced and shown to be superior to ED and SAM. In
order to further evaluate spectral discriminability among a
set of pixels, three discriminatory measures were also
developed. The experiments demonstrate the potential utility
in hyperspectral analysis.
Acknowledgment
The author would like to thank Dr. Harsanyi for
providing hyperspectral data set used for experiments in this
paper. He also thanks Mr. Shao-Shan Chiang for generating
figures and tables.
References
[1] R.A. Schowengerdt, Remote Sensing: Models and
Methods for Image Processing, 2nd ed., Academic Press,
1997.
[2] C.-I Chang, "A pixel-level quantitative analysis of
hyperspectral measures for spectral similarity and
discriminability," unpublished.
[3] S. Kullback, Information Theory and Statistics, John
Wiley & Sons, N.Y., 1959 or Dover, 1968.
Table 1 − Spectral similarity using SAM
Blackbru Creosote Drygrass Redsoil Sagebrus
Blackbru 1
Creosote 0.9844 1
Drygrass 0.9670 0.9126 1
Redsoil 0.9188 0.8411 0.9764 1
Sagebrus 0.9977 0.9917 0.9563 0.8998 1
Table 2 − Spectral similarity using SID
Blackbru Creosote Drygrass Redsoil Sagebrus
Blackbru 0
Creosote 0.0497 0
Drygrass 0.0766 0.2298 0
Redsoil 0.1861 0.4154 0.0640 0
Sagebrus 0.0063 0.0303 0.0973 0.2340 0
Table 3 − RSDP by SAM
Blackbru Creosote Drygrass Redsoil Sagebrus
Blackbru 1
Creosote 1.0745 1
Drygrass 1.0425 1.1202 1
Redsoil 1.0516 1.1300 1.0087 1
Sagebrus 1.0166 1.0569 1.0598 1.0691 1
Table 4 − RSDP by SID
Blackbru Creosote Drygrass Redsoil Sagebrus
Blackbru 1
Creosote 2.6003 1
Drygrass 3.3999 8.8407 1
Redsoil 16.9736 44.1359 4.9924 1
Sagebrus 1.3176 1.9735 4.4796 22.3639 1
Table 5 − RSDR by SAM
Blackbru Creosote Drygrass Redsoil Sagebrus
0.1997 0.1858 0.2081 0.2100 0.1964
Table 6 − RSDR by SID
Blackbru Creosote Drygrass Redsoil Sagebrus
0.1897 0.4933 0.0558 0.0112 0.2500
Table 7 − RSDE
SAM SID
1.6085 1.2218
Figure 1
0-7803-5207-6/99/$10.00 (c) 1999 IEEE