Adaptive Gaussian Mixture Estimation and Its

Application to Unsupervised Classification of

Remotely Sensed Images

Sumit Chakravarty 1, Qian Du

1, Hsuan Ren

2

1Department of Electrical Engineering and Computer Science

Texas A&M University-Kingsville, Texas 78363

2 National Research Council, Washington DC 20001

Abstract This paper addresses unsupervised statistical

classification to remotely sensed images based on mixture

estimation. The application of the well-known technique,

Expectation Maximization (EM) algorithm to multi-

dimensional image data is to be investigated, where Gaussian

mixture is assumed. The number of classes can be estimated

via Neyman-Pearson detection theory-based eigen-thresholding

approach, which is used as a reference value in the learning

process. Since most remotely sensed images are nonstationary,

adaptive EM (AEM) algorithm will also be explored by

localizing the estimation process. Remote sensing data is used

in the experiments for performance analysis. In particular,

comparative study will be conducted to quantify the

improvement from the adaptive EM algorithm.

IndexTerms: Remote sensing imagery, Classification, EM

algorithm, Adaptive EM alogirthm.

I. INTRODUCTION

Due to the recent advance in remote sensing instruments,

the spectral resolution of remotely sensed image is

significantly improved as well as the image quality. Such

improvement provides the possibility of the precise object

identification. Since the spatial resolution or the area covered

by a single pixel is very large (typically several square

meters for images acquired by an airborne sensor, and

several hundred square meters for images acquired by a

spaceborne sensor), many materials are embedded in this

area. The radiance of a pixel is usually considered as the

mixture from all these materials. So image analysis in remote

sensing actually deals with mixed pixel processing instead of

pure pixel processing in standard digital images.

In most cases of remote sensing, the prior information

about the image scene is unavailable, and unsupervised

classification has to be implemented [1]. An intuitive way to

deal with the involved mixing problem is to assume

Gaussian mixture and estimate mixing parameters via

maximum likelihood estimation process, which results in the

well-known EM algorithm [2]. Then the membership

assignment (i.e., hard classification) of a pixel can be

achieved by using the maximum likelihood criterion, or soft

classification is implemented by generating membership

probability. The EM algorithm can be applied to

multispectral and hyperspectral images by assuming

multivariate Gaussian distribution. Also, the algorithms can

be made adaptive in order to capture the local statistics and

fit in the nonstationary case, referred to as adaptive EM

(AEM) [3]. Since the number of classes is unknown in

unsupervised classification, the EM and AEM algorithms are

modified to automatically select the class number.

II. EXPECTATION MAXIMIZATION (EM) ALGORITHM

The EM algorithm is an iterative technique for

maximum-likelihood estimation, which is widely used in

signal processing area. Assume that a

multispectral/hyperspectral remote sensing image of size

N1N2 with L spectral bands contains K classes with prior

probability being denoted as k , Kk 1 . The

probability density function (pdf) of the i-th pixel vector ix

is given by

=

=

K

k

ikki pp1

)()( xx (1)

where 21,,1 NNi = . If each class is Gaussian distributed,

( ) ( ) ( )

=

kikT

kiLikp xx

x

1

2/12/ 2

1exp

2

1)(

(2)

0 - 7 8 0 3 - 7 9 3 0 - 6 / $ 1 7 . 0 0 ( C ) 2 0 0 3 I E E E

0-7803-7929-2/03/$17.00 (C) 2003 IEEE 1796

where k and k are the mean vector and covariance matrix of the k-th class, respectively. The prior probability

k is non-negative and satisfies the following relationship

11

==

K

k

k . (3)

The whole image can be well approximated by an

independent and identically distributed random field X, and

the corresponding joint pdf is

= =

=

21

1 1

)()(NN

i

K

k

ikk pp xX . (4)

The task is to estimate the k , k and k such that )(Xp can be maximized, Kk 1 . The resulting

iterative algorithm is summarized as follows.

1) Initialization: initialize the algorithm with )0(

k ,)0(

k and )0(

k , Kk 1 .

2) Estimation step: compute the membership probability by

using

=

=K

k

ikm

k

ikm

kmik

p

pz

1

)(

)()(

)(

)(

x

x

(5)

for Kk 1 , where m is the iteration index.

3) Maximization step: update the parameter estimates by

using

=

+=

21

1

)(

21

)1( 1NN

i

m

ij

m

k zNN

(6)

=

++

=

21

1

)(

)1(21

)1( 1NN

i

im

ijmk

mk z

NNx

(7)

Tmki

mk

NN

i

im

ijmk

mk z

NN))((

1 )1()1(

1

)(

)1(21

)1(21

++

=

++

= xx

(8)

4) If the difference between the parameters in the m-th and

( )1+m -th iterations is less than a prescribed threshold , the algorithm is terminated. Otherwise, set

1+= mm and go to Step 2.

The number of classes K is initialized using a relatively

large number. If the prior probability of the k-th class k is less than a threshold , this class will be removed, and

1= KK . The estimation result using the Neyman-Pearson detection theory-based eigen-thresholding approach

in [4] can be adopted as the initial value of K.

III. ADAPTIVE EXPECTATION MAXIMIZATION (AEM)

Since most remotely sensed images are nonstationary,

adaptive EM algorithm (AEM) is also be explored by

localizing the estimation process. The basic idea is to use a

small window w moving around the image. The prior kestimation in Eq. (6) only depends on pixels covered by the

window, w. This makes all unsupervised procedures valid in

the nonstationary case [3]. The small image cube (for

multispectral / hyperspectral image) overlapped by the small

window, w is treated as a multi-dimensional image and the estimation step (E step) of the EM algorithm is applied. This

results in the calculation of the probabilistic membership

)( ikz of each pixel in the window. Using the obtained

probabilistic membership the prior probability ( k ) for individual class is obtained. This prior probability is assigned

as local probability )~( k to the center pixel of the window. The window is then moved to the next position in the image

and the process repeated. The prior value obtained for each

pixel of the image is used in the global EM algorithm, to

iteratively estimate the k and k such that )(Xp can be maximized.

IV. EXPERIMENTS

The image used in experiments is about a view of

Michigan Technological University (MTU) campus area

acquired with its Visible Fourier Transforms Hyperspectral

Imager. This instrument records hundreds of narrow spectral

bands in the visible to near infrared part of the spectrum. The

false color composite in Fig. 1 displays trees as red,

indicating their high reflectance in the near infrared region.

The light blue area of the image corresponds to urban

presence. The darker blue area of the image is attributed to

water bodies.

When applying the EM classification the three major

groups in the image could be easily segmented out. If the K

was set to any value higher than three, after several

iterations the value of K was reduced to three. The output of

the maximum likelihood classifier showed the three major

groups in Fig. 2. The first group in Fig. 2(a) represents the

segmented vegetation region as white while the rest of the

0-7803-7930-6/$17.00 (C) 2003 IEEE

0-7803-7929-2/03/$17.00 (C) 2003 IEEE 1797

image is left dark. The second group in Fig. 2(b) is the

representation of the urban region. The third group in Fig.

2(c) depicts the water bodies including the lake and rivers.

When the adaptive EM algorithm was applied, more land

cover patterns could be classified. In addition to the urban

area and water bodies classified as in Fig. 2, more patterns

could be distinguished. For instance, if the K was initialized

as four, then four classes were resulted as in Fig. 3. Now the

vegetation in Fig. 2(a) was further classified into two classes

in Fig. 3(a) and Fig. 3(c), denoted as vegetation 1 and

vegetation 2. If we compare them with Fig. 1, we find that

Fig. 3(c) corresponds to the area with a shade of red distinct

from the rest brown-red vegetation, and represents another

vegetation pattern. Since this area is small and represents a

local phenomenon, the global EM was unable to capture it

and just considered it as part of the vegetation. Comparing

the water bodies of the global EM in Fig. 2(c) and the

adaptive EM in Fig. 3(b), we can see that Fig. 2(c) has a lot

of tiny granules while in Fig. 3(b) the background is clearer.

This is due to finer classification achieved by the adaptive

EM than by the global EM. By comparison of the urban

classification in Fig. 2(b) and Fig. 3(d), the distinction is also

saliently observed in the adaptive EM result in Fig. 3(d)

where the contours of the periphery distinctly resemble those

blue area in the original image in Fig.1. As observed, the

global EM was unable to provide similar degree of accuracy.

It is worth noting that such improvement in classification is

achieved at the sacrifice of computational time.

IV. CONCLUSIONS

The application of EM and adaptive EM algorithms to

remote sensing image is investigated. The number of

resulting classes in the EM and adaptive EM algorithms can

be automatically selected. Using the adaptive EM algorithm,

local statistics in an image scene can be captured and

modeled. As a result, small classes can be more accurately

detected and classified at the sacrifice of computational

time, comparing to the high potential of being merged into

large classes when using EM algorithm for global statistics.

REFERENCES

[1] R. A. Schowengerdt, Remote Sensing, Models and

Methods for Image Processing, Academic Press, 1997.

[2] A. P. Dempster, N. M. Laird and D. B. Rubin,

Maximum likelihood from incomplete data via the EM

algorithm, Journal Royal Statistics Society, Vol. 39,

No. 1, pp. 1-21, 1977.

[3] A. Peng and W. Pieczynski, Adaptive mixture

estimation and unsupervised local Baysian image

segmentation, Graphical Models and Image

Processing, Vol. 57, No. 5, pp. 389-399, 1995. [4] C.-I Chang and Q. Du, A noise subspace projection

approach to determination of intrinsic dimensionality for hyperspectral imagery, Proceedings of 1999 European Symp on Image and Signal Processing for Remote Sensing V, pp.34-44, Florence, Italy, September 1999.

Fig1. The original image

(a) vegetation (b) urban area

(c) water bodies

Fig 2. Classification result using EM algorithm.

(a) vegetation 1 (b) water bodies

(c) vegetation 2 (d) urban area

Fig 3. Classification result using AEM algorithm.

0-7803-7930-6/$17.00 (C) 2003 IEEE

0-7803-7929-2/03/$17.00 (C) 2003 IEEE 1798

Index:

CCC: 0-7803-5957-7/00/$10.00 2000 IEEE

ccc: 0-7803-5957-7/00/$10.00 2000 IEEE

cce: 0-7803-5957-7/00/$10.00 2000 IEEE

index:

INDEX:

ind:

footer1: 0-7803-8367-2/04/$20.00 2004 IEEE

01: 3

02: 4

03: 5

04: 6

05: 7

06: 8

07: 9

08: 10

09: 11

10: 47