Page 1

Computing the q-index for Tsallis Nonextensive Image

Segmentation

Paulo S. Rodrigues

Artificial Intelligence Group

Centro Universit´ ario da FEI

S˜ ao Bernardo do Campo, S˜ ao Paulo, Brazil

psergio@fei.edu.br

Gilson. A. Giraldi

Computer Science Department

National Laboratory fir Scientific Computing

Petr´ opolis, Rio de Janeiro, Brazil

gilson@lncc.br

Abstract—The concept of entropy based on Shannon Theory of

Information has been applied in the field of image processing and analysis

since the work of T. Pun [1]. This concept is based on the traditional

Boltzaman-Gibbs entropy, proposed under the classical thermodynamic.

On the other hand, it is well known that this old formalism fails to explain

some physical system if they have complex behavior such as long rang

interactions and long time memories. Recently, studies in mechanical

statistics have proposed a new kind of entropy, called Tsallis entropy (or

non-extensive entropy), which has been considered with promising results

on several applications in order to explain such phenomena. The main

feature of Tsallis entropy is the q-index parameter, which is close related

to the degree of system nonextensivity. In 2004 was proposed [2] the first

algorithm for image segmentation based on Tsallis entropy. However,

the computation of the q-index was already an open problem. On the

other hand, in the field of image segmentation it is not an easy task to

compare the quality of segmentation results. This is mainly due to the

lack of an image ground truth based on human reasoning. In this paper,

we propose the first methodology in the field of image segmentation for

q-index computation and compare it with other similar approaches using

a human based segmentation ground truth. The results suggest that our

approach is a forward step for image segmentation algorithms based on

Information Theory.

Index Terms—Image segmentation; q-entropy; Tsallis entropy

I. INTRODUCTION

Image segmentation plays an important role on the basis of

computational vision tasks, such as image analysis, recognition and

tracking, to name a few. This is a basic but important and complex

problem which has been intriguing the researches for decades. The

issue behind image segmentation is to decompose the image into

regions of coherent properties in an attempt to identify objects and

their parts.

Gray level image segmentation techniques can be classified into the

following categories: thresholding, methods based on feature space,

edge detection based methods, region based methods, fuzzy logic

techniques and neural networks. Besides, most of these methods can

be extended to color images by representing color information in an

appropriate color space. In addition, it is possible to combine more

than one approach to achieve better performance such as Zhu and

Yuille [3], Shi and Malik [4], Malik et al. [5] and Ma and Manjunath

[6].

Among these methods, those based on threshold are fundamental

for our work. Thresholding is a large class of segmentation techniques

that are based on the assumption that the objects can be distinguished

and extracted from the background by their gray levels. The output of

traditional thresholding operations is a binary image whose intensity

pattern distinguish the foreground (e.g. gray level 0) from the

background (e.g. gray level 255). Interesting surveys underlining the

thresholding segmentation can be found in references [7] and [8].

In general, threshold selection can be categorized into two classes:

local methods and global methods. By the way, a threshold can be

based on different criterions, such as Otsus method [9], minimum

error thresholding [10], and entropic methods [1], [11], to name a

few.

Applying the concept of entropy in order to segment a digital

image is a common practice since PUN’s work [1] shows how to

find out a threshold that maximizes the information measure, the

very celebrated Shannon entropy, of the resulting binary image. Other

works following the same philosophy were proposed, e.g, Kapur et

al [11] maximized un upper bound of the total a posteriori entropy

in order to obtain the threshold level. Abutaleb [12] extended the

method using two-dimensional entropies. Li and Lee [13] and Pal [14]

used the directed divergence of Kullback-Leibler for the selection of

the threshold, and Sahoo et al. [8] used the Reiny entropy model

for image thresholding. A recent review about entropy methods for

image segmentation can be read in [15].

The concept of Shannon entropy was proposed in the Theory

of Information based on Boltzaman-Gibbs entropy for the context

of classic thermodynamic. However, for several decades it is well

known that this concept fails to explain some phenomena which have

complex behaviors such as long range interactions and long-time

memories [16], [17]. Such systems are called “non-extensive sys-

tems”, and those following the BGS formalism are called “extensive

systems”.

In 1988, Tsallis proposed a new formalism for the generalization

of BGS entropy, which is called q-entropy or Tsallis entropy. This

new entropy has reached relative success in explanation complex

phenomena for several applications. The main feature of Tsallis

entropy is the introduction of a q parameter, called extensiveness

parameter. It has been proved in the literature that each physical

system is close related to a specific value for q, and to achieve the

optimal q value for a specific physical system is a challenge and has

been issue of great debates between current researchers. A complete

list of non-extensive systems is vast and can be fully find in [17].

In 2004, Albuquerque et. al [2] applied the concept of non-

extensive entropy for mamographic gray scale images. They assume a

probability distribution of gray scale luminance, one for background

and other for foreground class of pixels. Then, they take the threshold

that maximizes the separation between these two classes. The work of

Albuquerque et al was an advance of the method based on Shannon

entropy for image segmentation.

The main drawback of Tsallis entropy, as well as the algorithm

proposed by [2] is the choose of the q-index. Since it is not an

intuitive idea, several applications should to randomly choose its

value. Up to our knowledge, there is no automatic method or theory

proposed for its automatic computation.

Another important questions remain regarding the image segmen-

tation methods. How to measure the quality of output segmentation.

This is due to the lack of a database for ground truth and due to the

difficult to build a function to compare the similarity between two

Page 2

segmentations.

The work described in this paper present three contributions. firstly,

we introduce an automatic method to the computation of q parameter;

then we present a database for ground truth, which was segmented by

human subjects. Finally, we present a new similarity measure which

compute the quality of image segmentation in x and y euclidian

dimensions and z luminance dimension.

This paper is organized as follows. In Section II we introduce the

q-entropy under the context of non-extensive systems and explains

the original non-recursive method. In Section III we show how to

compute the q parameter. In Section IV we explain our database. Is

Section V we present our proposed measurement method. In section

VI we explain how we carried out our experiments and in final

Section VII we discuss our main conclusions.

II. THEORETICAL BACKGROUND

The traditional equation for entropy, over a probability density

function p(x), also called Boltzmann-Gibbs-Shannon entropy (BGS),

is defined as:

S = −

i

?

piln(pi)

(1)

Generically speaking, systems which can be described by Equation

(1) are called extensive systems and have the following additive

property: Let A and B be two random variables, with probability

densities functions A = (a1,...,an) and B = (b1,...,bn),

respectively, and S be the entropy associated with A or B. If A and

B are independent, under the context of the Probability Theory, the

entropy of the composed distribution1verify the so called additivity

rule:

S(A ∗ B) = S(A) + S(B)

This rule was used by several researchers of Computational Vi-

sion Systems to achieve an optimal threshold aiming to separate

foreground from background of intensity images [18], [19]. The

general idea, historically presented by T. Pun [1], considers the gray

level histogram with L bins a symbol source, with all the symbols

statistically independent.

This traditional form of entropy is well known and for years

has achieved relative success to explain several phenomenon if

both the effective microscopic interactions and the effective spatial

microscopic memory are short-ranged. Roughly speaking, when the

system does not has such behavior, the standard formalism became

only an approximation, and some kind of extension appears to became

necessary. A complete review about this theory can be see [16], [17],

[20].

Recent developments based on the concept of non-extensive en-

tropy, also called Tsallis entropy, have generated a new interest in

the study of Shannon entropy for Information Theory [21]. Tsallis

entropy (or q-entropy) is a new proposal for the generalization of

BGS traditional entropy applied to non-extensive physical systems.

The non-extensive characteristics of Tsallis entropy have been

applied through the inclusion of a parameter q, which generates

several mathematical properties which the general equation is the

following:

Sq(p1,...pk) =1 −?k

(2)

i=1pq

i

q − 1

(3)

1we define the composed distribution, also called direct product of A =

(a1,...,an) and B = (b1,...,bn), as A∗B = {aibj}i,j, with 1 ≤ i ≤ n

and 1 ≤ j ≤ n

where k is the total number of possibilities of the whole system and

the real number q is the entropic index that characterizes the degree

of non-extensiveness.

In the limit q → 1, Equation (3) meets the traditional BGS entropy

defined by Equation (1). These characteristics give to q-entropy more

flexibility to explain several physical systems, which can not be

properly explained by traditional BGS formalism. Then, this new kind

of entropy does not fail to explain the traditional physical systems

since it is a generalization.

Furthermore, a generalization of some theory may suppose the

violation of one of its postulates. In the case of the generalized

entropy proposed by Tsallis, the additive property described by

Equation (2) is violated in the form of Equation (4), which apply if

the system has a non-extensive characteristic. In this case, the Tsallis

statistics is useful and the q-additivity describes better the composed

system. In our case, the experimental results (Section VI) show that

it is better to consider our systems as having non-extensive behavior.

Sq(A ∗ B) = Sq(A) + Sq(B) + (1 − q)Sq(A)Sq(B)

In this equation, the term (1 − q) stands for the degree of non-

extensiveness. Note that, as we said before, when q → 1, this

equation meets the traditional Equation (2).

Recently, Albuquerque et al. [2] proposes an algorithm using the

concept of q-entropy to segment general images. Their idea is quite

the same as that proposed by T. Pun, however, under the concept of

Tsallis entropy, having the followinf formalism. Suppose an image

with L gray-levels. Let the probability distribution of these levels

be P = {pi = p1;p2;...;pL}. Then, we consider two probability

distribution from P, one for the foreground (PA) and another for

the background (PB). We can make a partition at luminance level t

between the pixels from P into A and B. In order to maintain the

constraints 0 ≤ PA ≤ 1 and 0 ≤ PB ≤ 1 we must re-normalize

both distribution as:

PA :p1

(4)

pA,p2

pA,...,pt

pA

and

PB :pt+1

pB

,pt+2

pB

,...,pL

pB

where pA =?t

entropy for each distribution as SA =

1−?k

q−1

Equation (4), for two statistically independent systems, we can

compute the pseudo-additive property of systems A and B as:

1 −?t

q − 1

+(1 − q)1 −?t

q − 1

To accomplish the segmentation task, in the work of M. Al-

buquerque et al. [2] the information measure between the two

classes (foreground and background) is maximized. In this case, the

luminance level t is considered to be the optimum threshold value

(topt), which can be achieved with a cheap computational effort of

i=1pi and pB =?L

i=t+1pi.

Now, following the Equation (3), we calculate the a priori Tsallis

1−?t

i=1(pi

q−1

pA)q

and SB =

i=t+1(pi

pB)q

. Allowing the pseudo-additive property given by

SA∗B(t) =

i=1(pi

pA)q

+

1 −?L

1 −?L

i=t+1(pi

q − 1

i=t+1(pi

q − 1

pB)q

i=1(pi

pA)q

pB)q

(5)

topt = argmax[SA(t) + SB(t) + (1 − q)SA(t)SB(t)]

Note that the value t which maximizes Equation (6) depends on

mainly the entropic parameter q. Up to now in the literature the

value of q which generates topt is not explicitly calculated and must

(6)

Page 3

Fig. 1.

to q = 0.46, is the optimal q used for initial segmentation (Fig. 2-middle).

Sq/Smax as a function of q range. The lower value, corresponding

be defined empirically. In this paper, we propose an algorithm to

compute the optimal q-value, which is justified and described in the

next section.

III. COMPUTATION OF THE q INDEX

Considering the background and foreground of an image as inde-

pendent physical (sub)systems, the very celebrated strategy proposed

by T. Pun [1] for image segmentation was to use the additive

property (Equation (2)) of the extensive systems to achieve the

optimal threshold between both (sub)systems. This idea comes from

the fact that the maximum possible information is transferred when

the maximal global entropy is achieved through the sum of the

both systems. The same argument works for nonextensive systems.

However, the formalism used in this case turns according to Equation

(5), where t is the optimal threshold which maximizes the self-

information.

As posed in Section II, the Tsallis formalism are a generalization of

the Shannon entropy, meeting the traditional system when q → 1.0

only. Thus, we can conclude that the q-entropy (as also has been

called this formalism) can capture both the nonextensivity and exten-

sivity behaviors. So, it is reasonable to investigate the segmentation

entropic approaches under both contexts. Later we will show that

for our image database we achieve better segmentation performance

(regarding the human reasoning) under nonextensive formalism.

Of course, the usage of a new parameter has an extra computational

price to pay, and despite of its class, each image or region may

demand for a different q value (including q = 1.0) in order to achieve

information maximization. Then, it is interesting to evaluate the value

of the computed entropy for each image in several ranges for q; e.g,

regarding sub-extensive systems (q < 1.0), extensive ones (q = 1.0)

and super-extensive ones (q > 1.0).

From the point of view of Theory of Information, as smaller the

maximum entropy Sq produced by a q value related to the theoretical

maximum entropy Smaxof a physical system (in this case, an image),

larger is the self-information contained in this system. This is a

well known principle of Theory of Information and yields to the

idea of that the optimal q-value can be reached by minimizing the

Sq/Smax ratio. Then, before applying the proposed formalism stated

by Equation (8), we compute the optimal q value underlining the

image. This is accomplished as the following. For each q value in

the range [0.01,0.02,...,2.0] we get the optimal q as that which

minimizes the Sq/Smax ratio. In this paper we work the hypothesis

that not only each natural image may behave as a singular non-

extensive system – and as such demanding for a different q value

for segmentation – but also its internal regions also may be singular

non-extensive ones – also demanding for different q values as well.

Later, experiments will show that this is a promise hypothesis.

In order to apply different q values to segment different image

regions, and to achieve most of the image’s main regions, we carried

out two levels of segmentation. Initially, we compute the q value

minimizing Sq/Smaxand apply the Equation (6) to get a first optimal

topt threshold, obtaining a first segmentation, separating background

(RB) from foreground (RF). Then, for each achieved region (RB

and RF) we compute new q values, treating RB and RF as different

physical systems, and apply the algorithm again, obtaining two new

topts as well. Thus, we can achieve at most four intensity separations

and several regions in the image. Fig. 2 shows an example. Fig. 2-left

is the original image, and the Fig. 2-middle is its first segmentation in

two regions (RBand RF), achieved with the optimal q = 0.46, which

corresponds to the minimal value of the curve of Fig. 1 (Sq/Smax).

Following the same idea for RB and RF regions, we compute new

q values by minimizing new Sq/Smax curves and achieve two new

optimal thresholds topt. The result can be seen in the Fig. 2-right.

In this case we found q = 0.15 for RB and q = 0.73 for RF,

suggesting sub-extensive system behavior for all regions.

Fig. 2.

with q = 0.46 achieving RBand RF; (right image) the final segmentation

with q = 0.15 and q = 0.73 for the previous RAand RF, respectively.

(left image) a natural image; (middle image) the first segmentation

IV. DATABASE

As discussed in the Section I, the task of automatic image seg-

mentation into its individual cognitive regions is already an open

problem. We can state at least two main reasons to not consider this

as an easy task: (i) a good segmentation does depend on the human

subjectiveness as well as its point of view and cognitive visual target;

and (ii) it is rare in the Computer Science and correlated research

areas finding a database for formal result comparisons. Typically,

researchers show their results on a few images and point out why the

results ‘look good’. It is not clear from these results if the technique

will work for other images from the same class. At the and of the

papers, the same question remains: “What is a correct segmentation”.

An alternative is to carried out a segmentation only in the context of

a system task, such as object recognition, as did Borra and Sarkar

[22].

Clearly, under the lack of a precise response to this question,

we need at least a “lighthouse” to follow as a relative point in

order to compare several techniques under the same database and

or parametrization. By the way, the Berkeley database, presented by

D. Martin and colleagues [23], can be considered a tentative in the

way to stand a point from which we can carried out measures.

Page 4

The Berkeley database consists of a public available ‘ground truth’

segmentation produced by humans for images of a variety of natural

scenes. This database has been continuously updated, and, at the

moment we were writing this paper, it had 1000 images with 481x321

RGB images from the Corel image database, which is also a large

usage database with 40,000 images widely used in Computer Vision

(e.g. [24], [25])

In our work we use a subset of 100 images from Berkeley database

with 5 segmentation by each image. Fig. 3 shows some examples

of images from this database and the 5 different segmentations

superimposed, where we can see the high degree of consistency

between different human subjects. Additional details of database

construction may be found in [23].

Fig. 3.

experiments. Each edge-map corresponds to five segmentations superimposed

in order to observe the consistency between human subjects.

10 pairs of image-segmentation from our 100 images used for

In the five edge-maps superimposed for the same image in Fig. 3

not all edges from each human subject meet each other. The effect

is that as more subjects choose the same line more this line is

highlighted. The contrary is also true. Then, in our work we use

100 edge-maps images as a base for comparison inter algorithms

(comparing their output segmentations under a same parametrization)

as well as intra algorithms (comparing their output segmentations

under different parameterizations).

Obviously, the Berkeley database may not be considered an abso-

lute ground truth. But, since it was generated by several independent

human subjects (having high degree of cognitive consistency) it is

reasonable to use this database as a relative point for segmentation

comparison. However, the divergence (in absolute value) of infor-

mation between some machine segmentation and the ground truth

(human segmentation) will not be taken as a segmentation quality

measurement. This database is only a base for relative comparison

between input algorithms or algorithm’s parameters. In the case of the

non-extensive algorithm proposed in this paper, it is reasonable to try

to response the question about what q-value most approaches machine

segmentation to human segmentation?; or, what we should to use?:

a random constant q-value or the proposed automatic calculation

for each image region? Besides, this is an open door for a posed

further question: which class of image is better segmented with a

non-extensive parameter q ?= 1.0?; or which images may be better

segmented with the traditional Shannon entropy?

V. SEGMENTATION MEASUREMENT

In order to measure the similarity between two segmentations

(in this case, between a human and a machine segmentation), we

need to define a similarity function. However, this is also a difficult

task and an open problem. Sezgin and Sankur [19], in their image

segmentation survey, proposed a set of five quantitative criteria in

order to measure the region luminance and shape uniformity of 20

classical methods for image segmentation. Since their criteria are not

based on ground truth data, it is an intrinsic quality judgment of the

segmented areas: e.g, an output segmentation with uniformity shape

regions may not approach to expected human segmentation.

On the other hand, measuring techniques based on ground truths

are also difficult to propose when the system demands for detecting

several image’s regions together, a common task in several computa-

tional vision applications. Also, the problem of match corresponding

boundaries carries the problem to detect their corresponding whole

regions, as well as their spatial localization. But in several Computer

Vision applications, this will be important to infer inter-regions

relationships.

Some algorithms can be useful as they tolerate any localization

error, approaching slightly to mislocalized boundaries. Then, simply

detecting coincident boundary pixels and consider all unmatched

pixels either false positives or misses would yields to severe low per-

formance. Clearly, as we can see from Fig. 3, the machine boundary

pixels assigned to ground truth boundaries must tolerate localizations

errors since even the ground truth data contains boundary localization

divergencies. Then, some slope correspondence in order to permit

small localization divergencies may be useful, as did the approach in

[23].

On the other hand, on a 2D edge-map, such as the Berkeley one,

we can find two kind of information: geometrical and luminance scat-

tering. The geometrical scattering measures the size and localization

of the region boundaries and the luminance scattering measures the

boundary intensities, which, for a human segmentation, it captures

the boundary cognitive consistency between all subjects.

The geometrical scattering between two edge-maps can be mea-

sured quantifying the divergence of information between both edge-

maps, for x and y dimensions, and the luminance scattering for

z dimension as well. The divergence of information on the x

dimension between two edge-maps can be computed as the Euclidian

distance between the edge-map’s (e.g. Mx histogram for machine

segmentation and Hx for the corresponding human segmentation)

of a M × N image. Then, in this paper, we propose to use the

following matching function between both edge-map’s histograms,

Mx and Hx, of the x-dimension in order to measure how far a

machine segmentation will be from a human segmentation in this

specific direction:

??

where Mx and Hx are the probability mass functions (luminance

histograms) of the boundary distribution along the x direction, and

M is the size of x distribution (image resolution in x direction).

Similarly, we propose the following matching functions for y and

z directions, respectively:

??

??

Simx(Mx|Hx) =

M

(Mx(i) − Hx(i))2.

(7)

Simy(My|Hy) =

N

(My(i) − Hy(i))2.

(8)

Simz(Mz|Hz) =

L

(Mz(i) − Hz(i))2.

(9)

Page 5

Fig. 4. Simulation Results under increasing gaussian noise.

where N and L are the size of y and z distribution, respectively.

Note that N is the image resolution in y dimension and L is the

total luminance levels (e.g. 256).

Thus, we propose the following matching function to measure

information between two edge-maps of a machine and a human

segmentation:

Sim(M|H) = Simx+ Simy+ Simz

VI. EXPERIMENTAL RESULTS

We have three types of segmentation to analyze: (BGS) that

proposed by T. Pun [1], which is based on traditional BGS entropy;

(NEC) that proposed by M. Albuquerque and colleagues [2], which

uses the generalized non-extensive entropy, but with a constant

manually chosen q value; and (NEA) our proposed method, which

is also based on generalized non-extensive entropy but with an

automatic calculated q value, according to Section III.

Firstly, it is interesting to observe how the three algorithms behave

under noise situation. Then, in the first experiment, we randomly

choose an image Im and its corresponding Ig edge-map from the

Berkeley database and apply the following four steps: (i) add to Im

gaussian noise with zero mean; (ii) apply the algorithms BGS, NEC

and NEA over the noisiness Im image and achieve an edge-map

with the Canny operator; (iii) measure the Sim similarity, given by

Equation (10), between Ig edge-map and the output Canny edge-

maps for BGS, NEC and NEA algorithms, respectivelly; (iv) Repeat

the steps i-iii 50 times (taking the average) for 10 increasing standard

deviations: from σ2= 0.005 to σ = 0.5, given a total of 500

segmentations for each algorithm. This approach is enough to curve

convergence.

In the graphic of Fig. 4, we clearly see the behavior of the

three algorithms under non-increasing SNR. According to Fig. 4

all three algorithms decrease their performance in approaching to

human segmentation of Berkeley image. The NEC algorithm slightly

overcomes the BGS algorithm for all values of gaussian noise. On

the other hand, our proposed algorithm NEA clearly overcomes both

as well. All three algorithms meet for σ2→ 0.05, since under this

value the noise is to high and there is few information to get.

It is quite subjective to get some conclusion about segmentation

algorithms based on visual inspection over their output regions or

(10)

Fig. 5.

is the NEC performance; (c) is the BGS performance.

Simulation of general performance. (a) is the NEA performance; (b)

edge-maps, since it is not an easy task to compare their size, spatial

position and amount of output regions. Also, it is not easy to compare

all results together instead of inspecting an unique image individually.

Thus, our experiments use the Equation (10) in order to match a

machine edge-map (e.g. given by NEC, NEA or BGS algorithm)

with a human edge-map (given by Berkeley database). Also, we

segment all 100 images from the Berkeley database with the three

algorithms. For the NEC algorithm we use a constant q = 0.5 value.

We normalize between [0,1] the result of Equation(10) in order to

measure which algorithm most approaches the Berkeley’s human

segmentation. A perfect match is reached when Sim(M|H) = 0.0

and the worst match is reached when Sim(M|H) = 1.0.

In the Fig. 5 we can see an overview of all three segmentations

for the whole database. Each row corresponds to an image from the

database. There is three main columns: the left most column corre-

sponds to NEA segmentation; the middle one corresponds to NEC

segmentation and the right most corresponds to BGS segmentation. In

order to clarify the visualization, we add a gray color to row-column

pair according to their similarity to human edge-map (Equation 10),

where white color corresponds to ‘high similarity (near to 1.0)’ and

black color corresponds to ‘low similarity (near to 0.0)’.

According to Fig. 5 it is clear that most of all black colors were

added to the right most column, indicating that most of all images

are better segmented using NEA or NEC non-extensive algorithms.

Besides, our proposed method has the advantage to automatically

compute the important q parameter. Numerically, for all 100 images,

only 22 were better segmented with BGS entropy, 42 with our

NEA and 36 with NEC. However, when we use only our NEA

approach against BGS, the result is 74 NEA segmentations against

26 segmentations of BGS traditional algorithm.

VII. MAIN CONCLUSIONS

In this paper we proposed an automatic methodology for comput-

ing the q-index of Tsallis non-extensive gray scale image segmenta-

tion. The experimental results suggest that this is a promise technique

in working with natural images even under noise influence. Besides,

the results indicate better performance against the traditional similar

algorithms when we compute q at running time.

The previous results have two folds. Firstly, it is possible to claim

that the automatic q computing methodology is that which most

Page 6

approximate the obtained regions to Berkeley human segmentation.

Even it is always possible to find manually a value to fit the

optimal q-index, this choice is not an intuitive task as this value

is very related to physical non-extensive image features. On the

other hand, the automatic method proposed here can became, in the

future, a start point of automatic methods for classifying images as

non-extensive versus extensive systems, whose implications are now

broadly investigated. The fact that most of our image database (about

74%) was better segmented (in terms of similarity to Berkeley human

segmentation) is a strong indication of the non-extensive feature of

such natural images.

The use of non-extensive methods for image segmentation is an

advance of the celebrated BGS entropic methods, and the alternative

for automatic q computing is also an advance for non-extensive

methods as it opens new possibilities, not only for image analysis

area, but several others with non-extensive physical behavior.

The next step is to test our proposed method to other characteristics

such as color, texture and spatial information such as gradient or

region shapes.

ACKNOWLEDGMENT

The authors would like to thank the CNPq (Project 301858/2007-1)

and CAPES (Project 094/2007), the Brazilian agencies for Scientific

Financing, as well as to FEI (Fundao Educacional Inaciana) a

Brazilian Jesuit Faculty of Science Computing and Engineering, for

the support of this work.

REFERENCES

[1] T. Pun, “Entropic thresholding: A new approach,” Comput. Graphics

Image Process, vol. 16, pp. 210–239, 1981.

[2] M. P. Albuquerque, M. P. Albuquerque, I. Esquef, and A. Mello, “Image

thresholding using tsallis entropy,” Pattern Recognition Letters, vol. 25,

pp. 1059–1065, 2004.

[3] S. C. Zhu and A. L. Yuille, “Region competition: Unifying snakes,

region growing, and bayes/mdl for multiband image segmentation,” IEEE

Transactions on Pattern Analysis and Machine Intelligence, vol. 18,

no. 9, pp. 884–900, 1996.

[4] J. Shi and J. Malik, “Normalized cuts and image segmentation,” IEEE

Transactions on Pattern Analysis and Machine Intelligence, vol. 22,

no. 8, pp. 885–905, 2000.

[5] J. Malik, S. Belongie, T. Leung, and J. Shi, “Contour and texture analysis

for image segmentation,” International Journal of Computer Vision,

vol. 43, no. 1, pp. 7–27, 2001.

[6] W.-Y. Ma and B. Manjunath, “Edgeflow: a technique for boundary detec-

tion and image segmentation,” IEEE Transactions on Image Processing,

vol. 9, no. 8, pp. 1375–1388, 2001.

[7] S. K. Fu and J. K. Mu, “A survey on image segmentation,” Pattern

Recognition, vol. 41, pp. 3–16, 1981.

[8] P. K. Sahoo, S. Soltani, and A. K. C. Wong, “A survey of thresholding

techiniques,” Comput. Vis. Graphics Image Process, vol. 41, pp. 233–

260, 1988.

[9] N. Otsu, “A threshold selection method from gray level histogram,” IEEE

Transaction on Systens, Man and Cybernetics, vol. SMC-8, pp. 62–66,

1978.

[10] J. Kittle and J. Illingworth, “Minimum error thresholding,” Pattern

Recognition, vol. 19, pp. 41–47, 1986.

[11] J. N. Kapur, P. K. Sahoo, and A. K. C. Wong, “A new method for gray-

level picture thresholding using the entropy of the histogram,” Comput.

Graphics Image Process, vol. 29, pp. 273–285, 1985.

[12] A. S. Abutaleb, “A new method for gray-level picture thresholding using

the entropy of the histogram,” Comput. Graphics Image Process, vol. 47,

pp. 22–32, 1989.

[13] C. H. Li and C. K. Lee, “Minimum cross entropy thresholding,” Pattern

Recognition, vol. 26, pp. 617–625, 1993.

[14] N. R. Pal, “On minimum cross entropy thresholding,” Pattern Recogni-

tion, vol. 26, pp. 575–580, 1996.

[15] C.-I. Chang, Y. Du, J. Wang, S.-M. Guo, and P. Thouin, “Survey

and compartive analysis of entropy and relative entropy thresholding

techniques,” IEE Proceedings, Vision, Image and Signal Processing, vol.

153, no. 6, pp. 837–850, Dec. 2006.

[16] C. Tsallis, “Nonextensive statistics: Theoretical, experimental and com-

putational evidences and connections,” Brazilian Journal of Physics,

vol. 29, no. 1, March 1999.

[17] ——, Nonextensive Statistical Mechanics and its Applications, ser.

Lecture Notes in Physics, S. Abe and Y. O. (Eds.), Eds. Berlin: Springer,

2001.

[18] C.-I. Chang, Y. Du, J. Wang, S.-M. Guo, and P. D. Thouin, “Survey

and comparative analysis of entropy and relative entropy thresholding

techniques,” in IEEE Proceedings, Vision, Image and Signal Processing,

vol. 153, no. 6, December 2006, pp. 837–850.

[19] M. Sezgin and B. Sankur, “Survey over image thresholding techniques

and quantitative performance evaluation,” Journal of Eletronic Imagin,

vol. 13, no. 1, pp. 146–165, January 2004.

[20] C. Tsallis, Introduction to Nonextensive Statistical Mechanics: approach-

ing a complex world.Springer, 2009.

[21] C. Shannon and W. Weaver, The Mathematical Theory of Communica-

tion.Urbana: University of Illinois Press, 1948.

[22] S. Borra and S. Sarkar, “A framework for performance characterization

of intermediate-level grouping modules,” IEEE Transaction on Pattern

Analysis and Machine Inteligence (PAMI), vol. 19, no. 11, pp. 1306–

1312, Nov. 1997.

[23] D. Martin, C. Fowlkes, D. Tal, and J. Malik, “A database of human

segmented natural images and its application to evaluating segmentation

algorithms and measuring ecological statistics,” in Proc. 8th Int’l Conf.

Computer Vision, vol. 2, July 2001, pp. 416–423.

[24] C. carson, M. Thomas, S. Belongie, J. M. Hellerstein, and J. Malik.,

“Blobworld: A system for region-based image indexing and retrieval,”

in Third International Conference on Visual Information Systems, Jun.

1999.

[25] O. Chapelle, P. Haffner, and V. N. Vapinik, “Support vector machines

for histogram-based image classification,” IEEE Transactions on neural

networks, vol. 10, no. 5, pp. 1055–1064, Sep. 1999.