in I. Balderjahn, R. Mathar and M. Schader eds.: Data Highways and Information Flooding, a Challenge
for Classi cation and Data Analysis, Springer 1997.
21st Annual Meeting of the Gesellschaft fur Klassi kation, GfKl'97, Potsdam, March 12 14, 1997.
Unsupervised Fuzzy Classi cation
of Multispectral Imagery
Using Spatial-Spectral Features
Rafael Wiemker
II. Institut fur Experimentalphysik, Universitat Hamburg,
KOGS Informatik, Vogt-Kolln-Str. 30, D-22527 Hamburg, Germany,
http: kogs-www.informatik.uni-hamburg.de wiemker
Abstract: Pixel-wise spectral classi cation is a widely used technique to produce
thematic maps from remotely sensed multispectral imagery. It is commonly based
on purely spectral features. In our approach we additionally consider additional
spatial features in the form of local context information. After all, spatial context
is the de ning property of an image. Markov random eld modeling provides the
assumption that the probability of a certain pixel to belong to a certain class depends
on the pixel's local neighborhood. We enhance the ICM algorithm of Besag
1986 to account for the fuzzy class membership in the fuzzy clustering algorithm
of Bezdek 1973. The algorithm presented here was tested on simulated and real
remotely sensed multispectral imagery. We demonstrate the improvement of the
clustering as achieved by the additional spatial fuzzy neighborhood features.
1 Introduction
Spectral classi cation is a widely used technique to produce thematic maps
from remotely sensed multispectral imagery. The classi cation or labeling
of each pixel relies on its spectrum or spectral signature, which consists of
n spectral values of the spectral bands i = 1:::n of wavelength i. The
similarity between an observed spectral vector x = :::; xi; ::: t 2 lRn
and
a given reference spectrum m is commonly evaluated by the spectral distance
d = kx,mk in the feature space. The n-dimensional spectral feature space
is spanned by the radiance or re ectance signals xi as received in the n
spectral bands of the sensor or camera.
Given the set of all observed spectral vectors x corresponding to the pixels
of a particular multispectral image, unsupervised clustering algorithms are
employed to nd k cluster centers in the spectral feature space lRn
around
which the observed spectra are scattered. Each pixel of the image can then
be classi ed labeled to the class to which its spectral distance is minimal.
Most classi cation techniques applied in multispectral remote sensing Richards
1993 rely on purely spectral features and consider only one pixel at a
time. More recently, a method for utilizing additional contextual information
from neighboring pixels has been derived from Markov random eld
modeling Besag 1986. This `ICM-algorithm' has been shown to improve
classi cation results on multispectral imagery Solberg et al. 1996. So far,
this spectral-spatial labeling approach has been used in conjunction with supervised
classi cation only, i.e., the reference classes were established from
training data by the analyst. In this paper we describe the e ects of incorporating
spatial context information into unsupervised clustering techniques
such as the hard and fuzzy k-means algorithms.
2 Hard and Fuzzy Clustering
with Spectral and Spatial Features
All k-means algorithms or `migrating means'-algorithms work iteratively.
Also, all k-means algorithms, as used in multispectral image classi cation,
determine for each pixel the Euclidean distance dxjmc =
qPixi ,mc;i2
between the spectral vector x and the respective mean spectrum mc of each
class !c c = 1;:::;k in the spectral feature space which is spanned by the
n spectral bands i of the imaging sensor i = 1;:::;n, where xi = xi.
The hard k-means algorithm Ball and Hall 1967 assigns each pixel to the
class !c to which the spectral distance dxjmc is minimal. Then the k
cluster centers mc are recomputed as the means of all pixels which currently
belong to the respective class. The process is repeated until convergence.
The fuzzy k-means algorithm originally `c-means', Bezdek 1973 relies on a
fuzzy membership pspecxj!c which is inversely proportional to the spectral
distance dxjmc :
pspecxj!c = d,1
xjmc
Pc0 d,1
xjmc0  ;
kX
c=1
pspecxj!c = 1 : 1
The algorithm iterates two alternating steps: a updating the membership
weights pspecxj!c, and b re-estimating the cluster centers mc =Px pspecxj!cx=Px pspecxj!c. In contrast to the hard k-means, all pixels
are used for the computation of each cluster center, weighted with their respective
fuzzy membership pspecxj!c. Convergence to a local minimum of
a global objective function has been shown Bezdek 1981.
Considering contextual image information,we use Markov random eld modelling,
and assume that the conditional spatial probability pspatxj!c of pixel
x depends only on the pixels x0 in its spatial neighborhood Nx Li 1995.
As the neighborhood Nx we here use a ll window around pixel x except
the pixel itself.
For the interaction between neighboring pixels, Besag 1986 has suggested
a neighborhood potential Ux = Px0
2Nx 1 , x;x0 , with x;x0 = 1
for equal classes !x = !x0, and 0 otherwise.
In this paper we use a re ned `fuzzy' neighborhood potential Uxj!c, based
on the current memberships P!cjx0 de ned in Eq. 4 of the neighboring
pixels x0. The potential Uxj!c and then the spatial membership pspatxj!c
are de ned as
Uxj!c =
X
x02Nx
1 ,P!cjx0 2
pspatxj!c = 1
Ze, Uxj!c
;
kX
c=1
pspatxj!c = 1 3
where 0 is a factor to weight the in uence of the spatial context. The
spatial membership pspatxj!c for class !c is large if the neighboring pixels x0
have large memberships P!cjx0 for the same class !c, and small if they tend
to belong to other classes !c0 . Computation of the normalization constant
Z is unnecessary here, as it cancels out in Eq. 4. The joint spectral-spatial
membership P!cjx of pixel x to belong to class !c then is de ned to be :
P!cjx = pspecxj!c 
 pspatxj!c
Pc0 pspecxj!c0  
 pspatxj!c0 
;
kX
c=1
P!cjx = 1 : 4
Using the additionalspatial features, we again can perform hard and fuzzy kmeans
clustering, depending on whether the cluster centers are re-estimated
from only those pixels currently assigned to each cluster, or from all pixels
using their current memberships as weights.
3 Results on Simulated and Remotely Sensed
Multispectral Imagery
In order to evaluate the e ect of the additional contextual memberships
Eq. 3, we have simulated a test image with n = 2 spectral bands and k = 2
spectral classes !1 and !2 Fig. 1. The spectral vectors x = x1;x2
t of each
class are scattered randomly around the two cluster centers m!1 and m!2,
forming uncorrelated Gaussian distributions. In the original spectral feature
space the two clusters Fig. 1, right, true cluster centers marked by crosses
are almost indistinguishable by eye appraisal due to the extreme scatter.
The root mean square scatter of both clusters is equal to the Euclidean
distance between the cluster centers.
A number of runs was performed with four di erent algorithms. The number
of clusters k = 2 and random initial centers seed coordinates for the
cluster centers were supplied. The resulting classi cation accuracies i.e.,
the relative number of correctly labeled pixels and the cluster center estimation
accuracy mean relative deviation of coordinates between true and
estimated class centers are given in Table 1. Typical classi cation results
of each method are shown in Fig. 2. We observe that the fuzzy k-means
performs slightly better in the estimation of the class center coordinates,
but not in the classi cation. Also, the hard k-means is improved by the
additional spatial features in classi cation, but not in cluster center accuracy.
For fuzzy k-means with additional contextual memberships, however,
the results indicate clearly that not only the classi cation results are improved,
but that also the coordinates of the cluster centers are estimated
with signi cantly improved accuracy.
Good convergence of the iterative algorithm can be achieved by starting the
iteration with = 0, i.e., without spatial in uence, and then increasing
gradually towards = 1. For each intermediate -value, convergence is
waited for before the spatial in uence is increased Fig. 4.
Another interesting observation is that the classi cation and cluster center
accuracy does not deteriorate when the number of classes k is over-estimated.
We performed another series of runs with k = 3 and k = 4 and random
cluster seeds provided. With a fuzzy k-means on purely spectral features this
decreases the classi cation accuracy as well as the cluster center estimation
Fig. 3. In contrast, with contextual fuzzy k-means the super uous classes
are basically not populated at all, and the centers of the actually existing
Gaussian distributions are found correctly. Classi cation and cluster center
accuracy with over-estimated k = 4 clusters is indeed the same as for the
correct k = 2 case Table 1, bottom line.
The fuzzy clustering with purely spectral features on the one hand, and with
spectral-spatial features on the other hand, was applied to remotely sensed
multispectral scanner imagery  ight altitude 300 m with n = 10 spectral
bands. The airborne line scanner of DAEDALUS, Inc., is operated on
board of a Do 228 aircraft by the German Aerospace Research Establishment
DLR.
Some examplary classi cation results can be inspected in Fig. 5. The classi
cation results which utilize both spectral and spatial features appear
smoother and without grainy, pixel-size noise. Note that not only the pixelwise
classi cation but also the cluster centers di er when utilizing additional
spatial features. The such established clusters are clearly better suitable for
thematic image segmentation see e.g. the forest areas in Fig. 5, top.
test image 1. band true class image original spectral space
0 50 100 150 200 250
0
50
100
150
200
250
Figure 1: Simulated test image left with n = 2 spectral bands and k = 2 spectral
classes center with rms scatter equal to the distance between the cluster
centers in the spectral feature space right.
hard k-means fuzzy k-means fuzzy k-means
hard w context fuzzy w context fuzzy w context
Figure 2: Typical classi cation results of the various algorithms left and center.
On the right-hand side, estimated cluster centers are depicted in the spectral feature
space. The path of the migrating means from random seeds is indicated by
solid lines. For improved visualization, the original feature space was magni ed
and the scatter reduced.
classi cation accuracy cluster center accuracy
algorithm correctly classi ed pixels deviation from true centers
hard k-means 75 1 5 1.5
with context 91 1 5 1.5
fuzzy k-means 75 1 3.5 1
with context 99 1 0.3 0.1
Table 1: Results on simulated data. Accuracy of cluster center estimation and
classi cation for various algorithms. Error margins are estimated from several
runs with random cluster seeds.
spectral spectral-spatial Figure 3: Typical classication
results top, and
cluster centers found bottom,
with over-estimated
k = 4 instead of the correct
k = 2 for fuzzy kmeans
clustering,
on purely spectral left
and spatial-spectral features
right. On the
right-hand side, estimated
cluster centers are depicted
in the spectral feature
space. The path of
the migrating means from
random seeds is indicated
by solid lines. For improved
visualization, the
original feature space was
magni ed and the scatter
reduced.
0 20 40 60
0.00
0.05
0.10
0.15
0.20
iterations
0.2
β = 0.0
0.4
0.6
0.8
1.0
deviationfromtrueclustercenters
Accuracy vs. Rising Spatial Influence
Figure 4: Typical iteration process.
Convergence is achieved for each level
of spatial in uence . The spatial inuence
is raised gradually and yields increasing
accuracy of the cluster center
estimation.
image =980 nm spectral spectral-spatial
Figure 5: Multispectral aerial imagery of Nurnberg with n = 10 spectral bands
1 = 435nm, 10 = 2215nm, unsupervisedlyclassi ed by fuzzy k-means clustering
into k = 4 classes recorded in 1995, 300m altitude, atmospherically corrected,
various scales.
4 Conclusions
We have tested the e ects of using spatial context features in addition to
spectral features for unsupervised clustering and classi cation of multispectral
imagery. Our observations with simulated and remotely sensed multispectral
imagery can be summarized as follows:
 The additional use of spatial features can signi cantly improve the
classi cation labeling results of unsupervised clustering. The full
bene ts of additional spatial features are experienced when used in
conjunction with fuzzy clustering in contrast to hard clustering.
 Moreover, also the accuracy of the cluster center coordinates as estimated
by unsupervised clustering is signi cantly improved when using
additional spatial features in conjunction with fuzzy k-means.
 The additional spatial features avoid e ectively the deteriorating effect
of over-estimating the number of clusters k. The cluster centers
are found accurately by fuzzy k-means clustering even if the spectral
feature space in fact contains fewer than k clusters of Gaussian distri-
bution.
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BEZDEK, J.C. 1981: Pattern Recognition with Fuzzy Objective Function Algorithms,
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LI, S.Z. 1995: Markov Random Field Modeling in Computer Vision, Springer,
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spectral spectral-spatial
Figure 5b: Multispectral aerial imagery of Nurnberg with n = 10 spectral bands
1 = 435nm, 10 = 2215nm, classi ed by fuzzy k-means clustering into k = 4
classes recorded in 1995, 300 m altitude, atmospherically corrected.