Computing the qindex for Tsallis Nonextensive Image Segmentation.
ABSTRACT 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 BoltzamanGibbs 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 nonextensive 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 qindex 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 qindex 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 qindex 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; qentropy; Tsallis entropy

Article: Multiq Analysis of Image Patterns
[show abstract] [hide abstract]
ABSTRACT: This paper studies the use of the Tsallis Entropy versus the classic BoltzmannGibbsShannon entropy for classifying image patterns. Given a database of 40 pattern classes, the goal is to determine the class of a given image sample. Our experiments show that the Tsallis entropy encoded in a feature vector for different $q$ indices has great advantage over the BoltzmannGibbsShannon entropy for pattern classification, boosting recognition rates by a factor of 3. We discuss the reasons behind this success, shedding light on the usefulness of the Tsallis entropy.12/2011;  [show abstract] [hide abstract]
ABSTRACT: PURPOSE: Architectural distortion is an important sign of early breast cancer. We present methods for computeraided detection of architectural distortion in mammograms acquired prior to the diagnosis of breast cancer in the interval between scheduled screening sessions. METHODS: Potential sites of architectural distortion were detected using node maps obtained through the application of a bank of Gabor filters and linear phase portrait modeling. A total of 4,224 regions of interest (ROIs) were automatically obtained from 106 prior mammograms of 56 intervalcancer cases, including 301 truepositive ROIs, and from 52 mammograms of 13 normal cases. Each ROI was represented by three types of entropy measures of angular histograms composed with the Gabor magnitude response, angle, coherence, orientation strength, and the angular spread of power in the Fourier spectrum, including Shannon's entropy, Tsallis entropy for nonextensive systems, and Rényi entropy for extensive systems. RESULTS: Using the entropy measures with stepwise logistic regression and the leaveonepatientout method for feature selection and crossvalidation, an artificial neural network resulted in an area under the receiver operating characteristic curve of 0.75. Freeresponse receiver operating characteristics indicated a sensitivity of 0.80 at 5.2 false positives (FPs) per patient. CONCLUSION: The proposed methods can detect architectural distortion in prior mammograms taken 15 months (on the average) before clinical diagnosis of breast cancer, with a high sensitivity and a moderate number of FPs per patient. The results are promising and may be improved with additional features to characterize subtle abnormalities and larger databases including prior mammograms.International Journal of Computer Assisted Radiology and Surgery 03/2012; · 1.36 Impact Factor
Page 1
Computing the qindex 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
BoltzamanGibbs 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
nonextensive 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 qindex 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 qindex 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
qindex 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; qentropy; 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 twodimensional entropies. Li and Lee [13] and Pal [14]
used the directed divergence of KullbackLeibler 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 BoltzamanGibbs 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 longtime
memories [16], [17]. Such systems are called “nonextensive 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 qentropy 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 nonextensive 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 qindex. 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
qentropy under the context of nonextensive systems and explains
the original nonrecursive 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 BoltzmannGibbsShannon 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 shortranged. 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 nonextensive en
tropy, also called Tsallis entropy, have generated a new interest in
the study of Shannon entropy for Information Theory [21]. Tsallis
entropy (or qentropy) is a new proposal for the generalization of
BGS traditional entropy applied to nonextensive physical systems.
The nonextensive 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 nonextensiveness.
In the limit q → 1, Equation (3) meets the traditional BGS entropy
defined by Equation (1). These characteristics give to qentropy 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 nonextensive characteristic. In this case, the Tsallis
statistics is useful and the qadditivity describes better the composed
system. In our case, the experimental results (Section VI) show that
it is better to consider our systems as having nonextensive 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 qentropy 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 graylevels. 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 renormalize
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 pseudoadditive 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 pseudoadditive 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. 2middle).
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 qvalue, 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 qentropy (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 subextensive systems (q < 1.0), extensive ones (q = 1.0)
and superextensive 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 selfinformation contained in this system. This is a
well known principle of Theory of Information and yields to the
idea of that the optimal qvalue 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
nonextensive 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. 2left
is the original image, and the Fig. 2middle 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. 2right.
In this case we found q = 0.15 for RB and q = 0.73 for RF,
suggesting subextensive 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 edgemap corresponds to five segmentations superimposed
in order to observe the consistency between human subjects.
10 pairs of imagesegmentation from our 100 images used for
In the five edgemaps 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 edgemaps 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
nonextensive algorithm proposed in this paper, it is reasonable to try
to response the question about what qvalue most approaches machine
segmentation to human segmentation?; or, what we should to use?:
a random constant qvalue 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
nonextensive 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 interregions
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 edgemap, 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 edgemaps 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 edgemaps can be computed as the Euclidian
distance between the edgemap’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 edgemap’s histograms,
Mx and Hx, of the xdimension 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(MxHx) =
M
(Mx(i) − Hx(i))2.
(7)
Simy(MyHy) =
N
(My(i) − Hy(i))2.
(8)
Simz(MzHz) =
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 edgemaps of a machine and a human
segmentation:
Sim(MH) = 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 nonextensive entropy, but with a constant
manually chosen q value; and (NEA) our proposed method, which
is also based on generalized nonextensive 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 edgemap 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 edgemap
with the Canny operator; (iii) measure the Sim similarity, given by
Equation (10), between Ig edgemap and the output Canny edge
maps for BGS, NEC and NEA algorithms, respectivelly; (iv) Repeat
the steps iiii 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 nonincreasing 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)
edgemaps, 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 edgemap (e.g. given by NEC, NEA or BGS algorithm)
with a human edgemap (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(MH) = 0.0
and the worst match is reached when Sim(MH) = 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 rowcolumn
pair according to their similarity to human edgemap (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 nonextensive 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 qindex of Tsallis nonextensive 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 qindex, this choice is not an intuitive task as this value
is very related to physical nonextensive 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
nonextensive 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 nonextensive feature of
such natural images.
The use of nonextensive 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 nonextensive
methods as it opens new possibilities, not only for image analysis
area, but several others with nonextensive 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 301858/20071)
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.
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