A Method of Pixel Unmixing by Classes based on the Possibilistic

Similarity

B. Alsahwa

1,2

, S. Almouahed

, D. Guériot

1,2

and B. Solaiman

1,2

Image & Information Processing Dept., Telecom Bretagne, Institut Mines-Télécom, Brest, France

Lab-STICC UMR CNRS 3192 - Laboratoire en sciences et technologies de l'information, de la communication et de la

connaissance (Institut Mines-Télécom-Télécom Bretagne-UEB), Brest, France

Keywords: Possibility Distribution, Possibilistic Similarity, Contextual Information, Pixel Unmixing, Classification.

Abstract: In this paper, an approach for pixel unmixing based on possibilistic similarity is proposed. This approach

uses possibility distributions to express both the expert’s semantic knowledge (a priori knowledge) and the

contextual information. Dubois-Prade’s probability-possibility transformation is used to construct these

possibility distributions starting from statistical information (learning areas delimitated by an expert for each

thematic class in the analyzed scene) which serve, first, for the estimation of the probability density

functions using the kernel density estimation. The pixel unmixing is then performed based on the

possibilistic similarity between a local possibility distribution estimated around the considered pixel and the

obtained possibility distributions representing the predefined thematic classes. The obtained similarity

values are used in order to obtain the abundances of different classes in the considered pixel. Accuracy

analysis of pixels unmixing demonstrates that the proposed approach represents an efficient estimator of

their abundances of the predefined thematic classes and, in turn, higher classification accuracy is achieved.

Synthetic images are used in order to evaluate the performances of the proposed approach.

1 INTRODUCTION

An important difficulty related to image

classification task stems from the existence of

“mixed” pixels (Tso and Mather, 2009). These

mixed pixels contain a mixture of more than one

class of different thematic classes contained in the

analyzed scene. They arise mainly due to spatial and

spectral resolving power limitations of the used

sensor. In the case of spatial resolving power

limitation, the mixed pixels extents cover more than

one class in the observed scene. The pixel response

is then a mixture of the covered underlying thematic

classes (e.g. remote sensing platforms flying at a

high altitude) or thematic classes are combined into

a homogeneous mixture (e.g. sand grains on a

beach), which can occur regardless of the spatial

resolution of the sensor. For the other case (spectral

resolving power limitation), the spectrum of each

mixed pixel is composed of a collection of

constituent spectra or “endmembers” (Van der Meer,

1997). It is important to notice that these two types

of resolution have an inextricable relationship to one

another (Tso and Mather, 2009). Indeed, high

spectral variability of local areas of the analyzed

scene becomes apparent as the spatial resolution

becomes finer. Therefore, using advanced sensors

with higher spatial resolving power may not

necessarily enable improved classifications when the

pixel-based images classification systems are used.

Hence, a method of pixel unmixing by classes

becomes very important in many applications of

image analyses where subpixel detail is valuable and

more accurate classification results are needed.

In an unmixing approach, given a mixed pixel,

the goal is to find the constituent thematic classes

and the set of corresponding fractions or

“abundances” that indicate the proportion of each

thematic class present in the pixel. Several

approaches to subpixel analysis have been

employed. Among them, three are widely used.

These are fuzzy maximum likelihood approaches

(Wang, 1990), fuzzy c-means approaches (Foody

and Cox, 1994), and linear mixture models, or

spectral mixture analysis (Plaza et al., 2010).

However, in all these approaches, the subpixel

analysis is usually conducted using multispectral

image or hyperspectral image. In this paper, the

220

Alsahwa B., Almouahed S., Guériot D. and Solaiman B..

A Method of Pixel Unmixing by Classes based on the Possibilistic Similarity.

DOI: 10.5220/0004826202200226

In Proceedings of the 3rd International Conference on Pattern Recognition Applications and Methods (ICPRAM-2014), pages 220-226

ISBN: 978-989-758-018-5

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

subpixel analysis is conducted using one gray level

image. This approach is proposed under the closed

world assumption. For the analyzed image I, let

Ω={C

, C

, ..., C

} an exclusive and exhaustive set

of M predefined classes. Generally, Each pixel P

from this image I, can be represented by a vector X

= (x

, x

…, x

)

of “N” measurements. Each

measurement x

, n=1,…,N, is the output of the given

sensor resulting of one physical parameter related to

the imaged scene. In the proposed approach, X is

limited to one measurement N=1 (e.g. gray level or

pixel brightness) i.e. only one output of the given

sensor. Prior knowledge is also assumed to be given

as an initial set of learning areas extracted from the

considered image and characterizing the M

considered classes from the expert point of view.

The class representation is done by means of

possibility distribution in order to deal with the

ambiguity as well as the uncertainty in the expert

description (Rabah, 2011). This possibilistic

representation constitutes an efficient and a flexible

tool corresponding to the way the experts express

their own semantic knowledge. For this purpose, the

probability-possibility transformations are adopted.

The Kernel Density Estimation (KDE) approach

(Epanechnikov, 1969) is first used to estimate the M

probability density functions from the learning set.

Then they are transformed into M possibility

distributions using Dubois-Prade transformations

(Dubois and Prade, 1983).

Each pixel P

from the image I can be considered

as being of a “homogeneous sub-region”. In this

case, a local possibility distribution 

(x) can be

estimated which express the possibility degree to

observe the pixel P

in the considered sub-region.

These possibility distributions (The M possibility

distributions as well as the local ones) using the

possibilistic similarity concept will lead to identify

thematic class components present in the mixed

pixels which, in his turn, would improve the

classification results.

This paper is organized as follows. In the next

section, the basic concepts of possibility theory are

introduced. The notion of similarity measures is the

subject of the third section. In the fourth section, the

proposed approach will be detailed. Section 5

presents the experimental results obtained when the

proposed approach is applied using synthetic

images.

2 POSSIBILITY THEORY

Possibility theory is devoted to handle epistemic

uncertainty, i.e. uncertainty in the context where the

available knowledge is only expressed in an

ambiguous form. This theory was first introduced by

Zadeh in 1978 as an extension of fuzzy sets and

fuzzy logic theory to express the intrinsic fuzziness

of natural languages as well as uncertain information

(Zadeh, 1978). It is well established that

probabilistic reasoning, based on the use of a

probability measure, constitutes the optimal

approach dealing with uncertainty. In the case where

the available knowledge is ambiguous and encoded

as a membership function into a fuzzy set defined

over the decision set, the possibility theory

transforms each membership value into a

possibilistic interval of possibility and necessity

measures (Dubois and Prade, 1980). The use of

these two dual measures in possibility theory makes

the main difference from the probability theory.

Besides, possibility theory is not additive in terms of

beliefs combination and makes sense on ordinal

structures (Dubois and Prade, 1992). The basic

concepts of a possibility distribution, the dual

possibilistic measures (i.e. possibility and necessity

measures), and the probability-possibility

transformation are briefly presented in the following

subsections.

2.1 Possibility Distribution

Let us consider an exclusive and exhaustive universe

of discourse Ω = {C

, C

,..., C

} formed by M

elements C

, m = 1, ..., M (e.g., thematic classes,

hypothesis, elementary decisions, etc).

Exclusiveness means that one and only one element

may occur at time, whereas exhaustiveness refers to

the fact that the occurring element certainly belongs

to Ω. A key feature of possibility theory is the

concept of possibility distribution, denoted by ,

assigning to each element C

 a value from a

bounded set [0,1] (or a set of graded values). This

value (C

) encodes our state of knowledge or

belief, about the real world and represents the

possibility degree for C

to be the unique occurring

element.

2.2 Possibility and Necessity Measures

Based on the possibility distribution concept, two

dual set measures, the possibility Π and the necessity

Ν measures are derived. For every subset (or event)

A, these two measures are defined as follows:





() maxπ(C )





(1)

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221





N( ) 1 ( ) min 1 π(C )



  

(2)

where, A

denotes the complement of the event

2.3 Possibility Distributions Estimation

based on Pr- Transformation

Many methods are proposed in the literature in order

to estimate the possibility distributions from a

limited prior knowledge in order to represent the

existing thematic classes. These methods can by

divided into two categories: the first category

reproduces fuzzy set theory concepts by using the

standard and predefined membership functions and

then applying Zadeh’s postulate for which

possibility values numerically duplicate the

membership ones, but have a different semantic

significance (Medasani et al., 1998). In fact, Zadeh’s

postulate transforms membership degrees (to a fuzzy

set describing an ambiguous concept) into

possibility degrees (describing the uncertainty

concept). This estimation category is well adapted to

the case where the available expert’s knowledge is

expressed using an ambiguous description over the

set of thematic classes that can be modeled by the

standard membership forms. The second category is

based on the use of statistical data like methods of

probability-possibility transformations, histogram

based methods [4], and learning based methods

(FCM, nearest neighbour techniques, neural

networks, etc) (Medasani et al., 1998).

As we consider that the available expert’s

knowledge is expressed through the definition of

learning areas representing different thematic

classes, i.e. statistical data, we will focus on the

second category. Several Pr- transformations are

proposed in the literature. Dubois et al. (Dubois and

Prade, 1983) suggested that any Pr- transformation

of a probability distribution function Pr, into a

possibility distribution , should be guided by the

two following principles:

 The probability-possibility consistency

principle. This principle is expressed by

Zadeh (Zadeh, 1978) as: “what is probable

is possible”. Dubois and Prade formulated

this principle by indicating that the induced

possibility measure Π should encode upper

probabilities:

() Pr(), AAA 

(3)

 The preference preservation principle

ensuring that any Pr- transformation

should satisfy the relation:

Pr( ) Pr ( ) ( ) ( ), , AB ABAB



   

(4)

Verifying these two principles, a Pr-

transformation turning a probability distribution Pr

(defined by probability values Pr(

}), C

, m

= 1, 2,..., M) into a possibility distribution  (defined

by (C

), C

, m = 1, 2, ..., M) has been

suggested by Dubois et al. (Dubois and Prade,

1983). This transformation, called symmetric Pr-

transformation, is defined by:



mm j m

j=1

π(C )= ( C )= min Pr( C ), Pr( C )











(5)

In our study, this transformation is considered in

order to transform the probability distributions into

possibility distributions.

3 SIMILARITY MEASURES

In order to quantify the similarity between two

objects or two pieces of information (e.g. possibility

distributions) a similarity function is used. This

function has no single definition and depends on the

way these pieces of information are represented (e.g.

similarity function is proportional to the inverse of

distance metrics between the examined pieces of

information).

Considering the expert’s predefined set of M

thematic classes contained in the analyzed image,

={C

, C

..., C

}, a set of M possibility

distributions can be defined as follows:





π : 0,1

(P) π ((P))



where D refers to the definition domain of the

observed feature x(P) (e.g. gray level). For each

class C

, 

(x(P)) associates each pixel PI,

observed through a feature x(P)D, with a

possibility degree of belonging to the class C

, m =

1, ..., M.

Considering two classes C

and C

from the set

, different possibilistic similarity or distance

functions “Sim” can be defined between their two

possibility distributions π

and π

. The behaviour

of these functions can be studied in order to obtain a

better discrimination between classes C

and C

. To

do this, calculating a similarity matrix Sim(π

, π

)

informs us about such inter-classes behaviour and

ICPRAM2014-InternationalConferenceonPatternRecognitionApplicationsandMethods

222

will help in choosing the right measure in the given

context:











CC CC

mm mn

CC CC

nm nn

π ,ππ,π

Sim Sim

Sim=

Sim Sim







(6)

Evaluation of similarity between classes was studied

in our previous work (Alsahwa et al, 2013). Many

existing possibilistic similarity and distance

functions, which are the most frequently

encountered in the literature, are used for this

purpose.

The similarity measure Sim

∞

derived from the

∞

-norm called Maximum distance (equation 8), a

particular case of the Minkowski Lp-norm (equation

7), was the most suitable among the selected

functions to describe the similarity between the two

classes.



 

i=1

pC C C C

mn m n

L π ,π = π x-π x



(7)



 

iiCC C C

mn m n

i=1

maxL π ,π = π x-π x



(8)



π ,π 1

Sim



(9)

4 THE PROPOSED PIXEL

UNMIXING APPROACH

As previously detailed, samples initial sets are used

to estimate the probability density functions of every

thematic class. These functions are transformed into

possibility distributions through the application of

the Pr- Dubois-Prade’s transformation. A local

possibility distribution (P

) is constructed around

each pixel of the analyzed image I.

The similarity measure Sim

∞

is used to quantify

the similarity between this local possibility

distribution and each of the M estimated possibility

distributions. Figure 1 shows the estimated

possibility distributions in the case of synthetic

image composed of two classes generated by a

Gaussian distribution.

All the measured similarity values between

possibility distributions of classes C

, C

and the local

possibility distribution for every pixel in the image I

are

transformed into percentages as the following:

  

i0 m0

iCp Cp

aSimπ ,π /Simπ ,π







(10)

Figure 1: Synthetic image, possibility distributions of

classes C

, C

and the local possibility distribution in a

subzone around the pixel of interest P

where a

) is supposed to be the “abundance” of

the i

predefined thematic class in the considered

pixel P

. ∑Sim

∞

serves as a normalizing factor.

It is worthwhile to notice that high overlapping

case (high discrimination complexity) between the

predefined thematic classes is treated in the

proposed approach. In the case of low overlapping

(low discrimination complexity), the “abundance” of

a predefined thematic class in the considered pixel

is roughly inversely proportional to the distance

between the pixel vector and the mean of that class

(Wang, 1990).

The simplest and most widely used approach, the

linear mixture model (Adams et al., 1986), is used in

the proposed unmixing approach. This model is

based on the assumption that a linear combination

exists between the pixel brightness and the M

predefined thematic class. The spectral reflectance

of a pixel is the sum of the spectral reflectances from

the predefined thematic classes weighted by their

relative “abundances”:

BaB







(11)

where B is brightness value of the considered pixel

, B

is brightness value of the i

predefined

thematic class (i.e. mean of all brightness values of

the pure pixels contained in the i

class), and a

is it’s

0 50 100 150 200 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

gray level

possibility

Poss.dist.(C

)

Poss.dist.(C

)



)

Subzone around P

,σ

)

,σ

)

Learning zone

AMethodofPixelUnmixingbyClassesbasedonthePossibilisticSimilarity

223

abundance in the considered pixel P

. There are two

constraints on the abundances that should be

satisfied: the abundances must all be non-negative to

be meaningful in a physical sense (a

≥ 0) (Keshava,

2003), and must sum to one (∑a

=1).

A classification step is conducted at the end of

the proposed approach. This step consist in the

process of assigning a class to the considered pixel

by determining the nearest class via the similarity

function Sim

∞

used to measure the similarity

between this pixel’s local possibility distribution and

possibility distributions of each of the M classes.

5 EXPERIMENTAL RESULTS

In many applications, collecting mixed pixels and

determining their exact abundances of the

predefined thematic classes is very difficult.

Therefore, a 550×250 pixel synthetic image, given in

figure 3, is generated. This image is composed of

eleven sectors. The first and second sector is

assumed to contain two “pure” thematic classes

generated by two Gaussian distributions G(m1=100,

σ1=15) and G(m2=150, σ2=15). Pixels of sectors

from three to eleven are generated as a linear

mixture of the first and second sector pixels. The

abundances of class C1 and class C2 in these mixed

pixels is varying incrementally by 10%. For

instance, the abundance of class C1 in the third

sector is 10% (resp. abundance of class C2 is 90%)

and in the forth sector 20% (resp. abundance of class

C2 is 80%), etc. 7×7 pixel learning zones positioned

by the expert (as being representative areas of the

considered thematic classes) are also illustrated on

the generated image.

5.1 Estimation of Classes’ Abundances

in the Mixed Pixels

Using the learning zones, the initial estimation of the

class probability density functions are established

based on the KDE (Kernel Density Estimation)

approach. The application of the Pr- Dubois-

Prade’s transformation allows obtaining the

possibility distributions for each class in the

analyzed image.

A 3x3 pixel window centered on each pixel is

considered as the local spatial possibilistic context

and then local probability density functions are

established based on the KDE approach. The

application of the Pr- Dubois-Prade’s

transformation allows obtaining the local possibility

distributions.

Figure 2: Synthetic image composed of two classes and

their learning zones.

Abundances of the predefined thematic classes

in each sector, from three to eleven, can be estimated

from the possibilistic similarity values. In each of

these sectors, the proposed approach, using the

possibilistic similarity measure Sim

∞

, is applied on

all its pixels and their possibilistic similarity values

of each thematic class are calculated. The mean and

standard deviation of these possibilistic similarity

values for each class is given in the Table 1.

A close analysis of the obtained results shows

that the abundances of the predefined thematic

classes in the mixed pixels can be estimated from the

possibilistic similarity values. This estimation

conforms well to the values used in synthetic image

generation. For instance, it can be estimated that the

fifth sector contains about 28% of class C

and 72%

of class C

while the used values in synthetic image

generation are 30% of class C

and 70% of class C

The small values of standard deviation are another

indication that this estimation is quite consistent

with the values used in synthetic image generation.

5.2 Evaluation of the Improvement in

Overall Classification Accuracy

The above synthetic image (figure 2) is classified

using the proposed approach and the conventional

Bayesian approach, respectively. The classification

recognition rate is then calculated in order to

compare the classification results of the two

approaches (Table 2).

Analysis of the obtained results shows an overall

Sector 1 (class C

)

Sectors 3-11

(Linear mixture

of C

and C

)

Sector 2 (class C

)

Learning zone

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224

Table 1: Abundances of the predefined thematic classes in each sector.

(10%)

(90%)

(20%)

(80%)

(30%)

(70%)

(40%)

(60%)

(50%)

(60%)

(40%)

(70%)

(30%)

(80%)

(20%)

(90%)

(10%)

Mean (C

)

0.14 0.20 0.28 0.39 0,50 0.61 0.72 0.79 0.87

Std (C

) 0.07 0.08 0.09 0.10 0,10 0.09 0.08 0.08 0.06

Mean (C

)

0.86 0.80 0.72 0.61 0,50 0.39 0.28 0.21 0.13

Std (C

) 0.06 0.07 0.09 0.10 0,10 0.10 0.09 0.07 0.06

Table 2: Classification recognition rate of the predefined thematic classes in each sector calculating first by the proposed

approach and second by the Bayesian approach

Recognition rate %

(10%)

(90%)

(20%)

(80%)

(30%)

(70%)

(40%)

(60%)

(50%)

(60%)

(40%)

(70%)

(30%)

(80%)

(20%)

(90%)

(10%)

Proposed approach (C1)

0 1 1 11 49 93 99 100 100

Proposed approach (C2)

100 99 99 89 51 7 1 0 0

Bayesian approach (C1)

2 4 12 27 51 28 88 95 99

Bayesian approach (C2)

98 96 88 73 49 72 12 5 1

improvement in classification accuracy using the

proposed approach. This improvement has been

achieved 17% in some cases (e.g. C

(40%) and

(60%)). In addition to this improvement in

classification accuracy, the estimation of the classes’

abundances in the mixed pixels (section 5.1) enable

the assessing of the classification accuracy which, in

his turn, may integrate in the interpretation of the

analyzed scene. For instance, the classification of the

third sector is 100% class C

with a small deviation

of the assignment to its pixels (about 14% of class

) while the classification result of the forth sector

is also about 100% class C

but with a bigger

deviation of the assignment to its pixels (about 20%

of class C

). It is important to note that this

assessment of accuracy cannot be done using the

conventional pixel-based images classification

systems

6 CONCLUSIONS

In this study, a pixel unmixing approach was

developed based on the possibility theory. At the

first time, the spatial context is exploited to construct

a local possibility distribution around each

considered pixel. Secondly, the notion of

possibilistic similarity is used in order to assess the

similarity between the locale possibility distribution

and each of the class possibility distributions. The

first results on a synthetic image (compared to the

results obtained using a Bayesian approach) seem

promising. Information about pixel’s content of the

predefined thematic classes becomes available and

more classification accuracy is achieved. Hence, this

may lead to better interpretation of the analyzed

scene. A future research work will be to validate

these early results on various types of images with

more than two classes.

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