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Signal Processing 87 (2007) 2125–2150
Morphological contrast measure and contrast enhancement:
One application to the segmentation of brain MRI
Jorge D. Mendiola-Santiban
˜ez
a,
, Iva
´n R. Terol-Villalobos
b
,
Gilberto Herrera-Ruiz
a
, Antonio Ferna
´ndez-Bouzas
c
a
Doctorado en Ingenierı
´
a, Universidad Auto
´noma de Quere
´taro, 76010 Quere
´taro, Me
´xico
b
CIDETEQ, S.C., Parque Tecnolo
´gico Quere
´taro S/N, San Fandila-Pedro Escobedo, 76700 Quere
´taro, Me
´xico
c
Instituto de Neurobiologı
´
a, UNAM Campus Juriquilla, 76001 Quere
´taro, Me
´xico
Received 19 March 2006; received in revised form 20 December 2006; accepted 20 February 2007
Available online 4 March 2007
Abstract
In this paper a morphological contrast measure is introduced. The quantification of the contrast is based on
the analysis of the edges, which are associated with substantial changes in luminance. Due to this, the contrast
measure is used to detect the image that presents a high visual contrast when a set of output images is analyzed.
The set of output images is obtained by application of morphological contrast mappings with size criteria.
These contrast transformations are defined under the notion of partitions generated by the set of flat zones of the
image; therefore, they are connected transformations. In addition, an application to the segmentation of white and grey
matter in brain magnetic resonance images (MRI) is provided. The detection of white matter is carried out by means of a
contrast mapping with specific control parameters; subsequently, white and grey matter are separated and their ratio is
calculated and compared with manual segmentations. Also, an example of segmentation of white and grey matter in MRI
corrupted by 5% noise is presented in order to observe the performance of the morphological transformations proposed in
this work.
r2007 Elsevier B.V. All rights reserved.
Keywords: Morphological contrast measure; Contrast mappings; Connected transformations; MRI segmentation
1. Introduction
In mathematical morphology (MM) contrast
enhancement is based on morphological contrast
mappings as described by Meyer and Serra [1,2].
The main idea of these transformations is to
compare each point of the original image with
two primitives; as a result, the nearest value with
respect to the original image is selected. The
reported primitives in the literature can be openings
and closings [1,2], erosions and dilations [3,4],or
white and black top-hats [3,5], in addition such
primitives can be connected or morphological.
In Fig. 1, two primitives, the original function and
the final result of certain contrast mapping are
illustrated.
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0165-1684/$ - see front matter r2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.sigpro.2007.02.008
Corresponding author. Tel./fax: +52 442 1921200x6023.
E-mail addresses: mendijor@uaq.mx
(J.D. Mendiola-Santiban
˜ez),famter@ciateq.net.mx
(I.R. Terol-Villalobos),gherrera@uaq.mx (G. Herrera-Ruiz),
fabouzas@servidor.unam.mx (A. Ferna
´ndez-Bouzas).
On the other hand, a special class of contrast
mappings denominated morphological slope filters
1
(MSFs) was introduced in [4,6,7]. The images
processed by MSFs have a well-defined contrast;
this is so because in each point of the output image,
the gradient value is greater than the filter para-
meter or has a zero value. The original idea of this
proposal from a practical point of view, is to modify
the gradient image by working on the original
image, without imposing markers, as is the case of
the watershed transformation [8].
Subsequently, a family of sequential MSFs was
introduced in [6,7]. The application of sequential
MSFs allows a better control on the image contrast;
however, the major drawback of using these filters
can be seen around contours, since new information
is generated in some configurations of blurred
edges. In Fig. 2, we observe this behavior ; here
‘‘p’’ is a high contrast point. In Fig. 2(b) the high
contrast contour is preserved; while a new edge
appears during the processing because the flat zone
(region with the same grey level) is broken.
Due to this situation, in [9] a class of connected
MSF was proposed. These contrast transformations
involve the connectivity concept [10–12], i.e., they
preserve or remove connected components (flat
zones) without generating new contours during the
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Nomenclature
Nomenclature
m;l;a;bscalars ( i.e. positive numbers)
Bstructuring element
A;E;X;Zeuclidean or digital space under study
Zneuclidean space
x;ypoints in Zn
}set of all subsets of Zn
;empty set
PðxÞelement of the partition containing x
fðxÞnumerical function of x
Cconnected set
FxðfÞflat zone of a function fat point x
ðf;PfÞðxÞgrey level value of the connected
component obtained by FxðfÞ
emðf;PfÞðxÞerosion on the partition of size m
induced by f
dmðf;PfÞðxÞdilation on the partition of size m
induced by f
gmðf;PfÞðxÞopening on the partition of size m
induced by f
jmðf;PfÞðxÞclosing on the partition of size m
induced by f
gradmmðf;PfÞðxÞmorphological gradient on the
partition of size minduced by f
Thwmðf;PfÞðxÞ;Thbmðf;PfÞðxÞwhite and black
top-hat on the partition of size minduced
by f
rðxÞproximity criterion
W3
l;m;b;aðf;PfÞðxÞmorphological three states con-
trast mappings with size criteria on the
partition induced by f
Zmðf;PfÞðxÞopening spectrum on the partition
induced by f
Thwbl;mðf;PfÞðxÞself-complementary top-hat
volðxm;lÞratio between the volume of the top-hat
spectrum and the total of the bright and
dark regions
C;Lcontrast and luminance
VCpvariation of contrast in function of
certain parameters p
emðfÞðxÞmorphological erosion of size m
dmðfÞðxÞmorphological dilation of size m
Btransposed of the structuring element B,
i.e.,
B¼fx:x2Bg
~
gtðfÞopening by reconstruction of size t
~
jtðfÞclosing by reconstruction of size t
Primitives
Original
function f
Contrast
mapping
Fig. 1. Example of contrast mapping.
1
The notion of these transformations is different to the slope
transform described in [13]; since MSFs were proposed as a class
of nonincreasing filters based on morphological gradient criteria
for contrasting and segmenting images; while the slope transform
is a re-representation of morphology onto the morphological
eigenfunctions.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–21502126
processing. Here, the flat zone and partition
concepts are fundamental to introduce the con-
nectivity notion, as well as the basic morphological
transformations at partition level in the numerical
case [9].
Once the morphological transformations at parti-
tion level were defined, the morphological contrast
mappings with size criteria were introduced as
connected transformations in [14,15]. Here, the sizes
of the primitives (opening and closing on the
partition of size m) were considered variable para-
meters. This originates a problem, since adequate
values for the sizes of the primitives and the
parameters involved in the proximity criterion [1]
must be calculated. In particular, in this class of
contrast mappings, the sizes of the primitives are a
very important point, because the proximity criter-
ion is obtained as a function of the primitives
(opening and closing on the partition of size m). In
addition, the proximity criterion compares a func-
tion fwith the primitives of variable size; in a way
similar to that proposed in [1].
In this paper, the problem of obtaining adequate
sizes for the primitives associated in the morpholo-
gical contrast mappings is solved. In order to
compute such parameters, a morphological contrast
measure, as well as a method that works similarly to
the granulometric density [16–19] is employed.
On the other hand, in the literature, several
methodologies have been provided to quantify the
contrast in the spatial domain. For example, in
Morrow et al. [20], the contrast in an image is
measured by using the mean grey values in two
rectangular windows centered on a given pixel. In
[21], Peli briefly describes other examples of contrast
measures, among them, the root mean square (rms)
contrast, which is used to compare the contrast of
two different images employing a statistical method.
From the point of view of visual contrast, the
Weber–Fechner law is a psychophysical model
widely used to quantify the contrast in accordance
to human visual perception [21–26].
In our case, the morphological contrast measure
is obtained from a local analysis of the enhanced
image trying to detect important changes in
luminance in a way that is psychophysically valid,
i.e., representative of the apparent or perceived
contrast. In this way, luminance is associated with
the contours of the image, since changes in the
contours produce modifications in the contrast.
Moreover, the morphological contrast measure
introduced in this work will be useful to select
adequate parameters involved in morphological
contrast mappings with size criteria. About this
proposal, there are two main aspects to mention.
First, notice that in MM several transformations
have been proposed in order to enhance the
contrast, for example, morphological contrast
mappings [1–3,5,7], morphological slope filters
[4,6], morphological center [3,27], and morphologi-
cal top-hat [28], among others. However, there is
not a defined morphological contrast measure
capable of quantifying contrast in the processed
images.
Second, although the study is directed to obtain
some parameters involved in morphological con-
trast mappings with size criteria, an important
contribution of this work is the introduction
and formalization of the method to quantify
contrast from the point of view of mathematical
morphology.
On the other hand, the proposals given in this
paper will be useful to perform the segmentation of
white and grey matter in a frontal lobule area from
slices of brain magnetic resonance images (MRI). In
order to better appreciate the segmentation of white
and grey matter, some MRI slices belonging to a
simulated brain database are segmented; these
analyzed slices have the characteristic of being
corrupted by the presence of 5% noise [29].
Finally, this paper is organized as follows. In
Section 2, a background of different transforma-
tions and concepts related to mathematical
morphology is presented. The connectivity notion,
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pp
ab
Fig. 2. (a) Original function, (b) MSF output.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–2150 2127
the flat zone concept and the basic transformations
on the partition are provided in Section 2.1. In
Section 2.3, morphological contrast mappings are
defined at partition level. A method to obtain
adequate sizes for the closing involved in contrast
mappings on the partition is presented in Section 3.
A method to quantify the contrast in the processed
images is introduced in Section 4. A practical
example is given in Section 5.1.3, in which the grey
and white matter are segmented in a frontal lobe
area from brain MRI. While in Section 5.2, a
segmentation of white and grey matter is carried out
on slices of brain MRI in the presence of noise. In
Section 6, a brief explanation about the computing
complexity of the transformations on the partition
is discussed.
2. Background on morphological connected
transformations
2.1. Connectivity and connected transformations
Serra [27] established connectivity by means of
the connected class concept.
Definition 1 (Connected class). A connected class C
in }ðEÞis a subset of }ðEÞsuch that: (i) ;2C; (ii)
8x2E;fxg2C; (iii) for each family fCigin C,
TCia;)SCi2C,
where }ðEÞrepresents the set of all subsets of E.
An element of Cis called a connected set. An
equivalent definition to the connected class notion is
the opening family expressed by the next theorem
[27].
Theorem 1 (Connectivity characterized by open-
ings). The definition of a connectivity class Cis
equivalent to the definition of a family of openings
fgx;x2Egsuch that: (a) 8x2E;gxðxÞ¼fxg; (b)
8x;y2E and A E;gxðAÞ¼gyðAÞor gxðAÞ\gy
ðAÞ¼;; (c) 8AE and 8x2E;8xeA)
gxðAÞ¼;.
When the transformation gxis associated with the
usual connectivity (arcwise) in Z2(Zis the set of all
integers), the opening gxðAÞcan be defined as the
union of all paths containing xthat are included in
A. When a space is equipped with gx, the
connectivity can be expressed using this operator.
A set AZ2is connected if and only if gxðAÞ¼A.
In Fig. 3, the behavior of this opening is illustrated.
The connected component of the input image X
where point xbelongs (Fig. 3(a)) is the output of the
opening gxðXÞ, while the other components are
eliminated (Fig. 3(b)).
In order to introduce the morphological contrast
mappings with size criteria in the following subsec-
tion, the next definitions are presented [9,15,30]:
Definition 2. Given a space E, a function P:E!
}ðEÞis called a partition of Eif: (a) x2PðxÞ;
x2E; (b) PðxÞ¼PðyÞor PðxÞ\PðyÞ¼; with
x;y2E.
PðxÞis an element of the partition containing x.If
there is a connectivity defined in Eand, 8xthe
component PðxÞbelongs to this connectivity, then
the partition is connected.
Definition 3. The flat zones of a function f:Z2!Z
are defined as the connected components (largest) of
points with the same value of the function.
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Fig. 3. Extraction of connected components. (a) Binary image X, (b) the opening gxðXÞextracts the connected component in Xwhere
point xbelongs.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–21502128
The operator FxðfÞwill represent the flat zone of
a function fat point x.
Definition 4. An operator is connected if and only if
it extends the flat zones of the input image.
The term ‘‘extends’’ in latter definition means that
the flat zones of the image are enlarged during the
processing by merging contiguous flat zones.
Definition 5. Let xbe a point of Z2equipped with
gx. Two flat zones FxðfÞand FyðfÞin Z2are
adjacent if FxðfÞ\FyðfÞ¼gxðFxðfÞ[FyðfÞÞ.
Definition 6. Let xbe a point in Z2equipped with
gx. The set of flat zones Axadjacent to the
component extracted by Fxis given by AxðfÞ¼
fFx0ðfÞ:x02Z2;FxðfÞ[Fx0ðfÞ¼gxðFxðfÞ[Fx0
ðfÞÞg.
In Definition 6, the element extracted by FxðfÞ
belongs to the set of adjacent flat zones, since AxðfÞ
fulfills the criterion of being reflexive.
The notion of adjacent flat zones is illustrated in
Fig. 4. The original image is located in Fig. 4(a). In
Figs. 4(b) and 4(c) two adjacent flat zones are
shown, while in Fig. 4(d), the adjacency is exempli-
fied according to the expression FxðfÞ[FyðfÞ¼
gxðFxðfÞ[FyðfÞÞ.
The introduction of the morphological transfor-
mation using the flat zone notion for the numerical
case is provided in [9].In[9] a new representation
for the grey level image was provided. Such
representation is given by the original function f
and the partition Pfinduced by fusing the notion
of flat zone. This implies that the operators are
going to act on the pair ðf;PfÞ, where the element
ðf;PfÞðxÞis taken as the grey level value of the
connected component obtained by FxðfÞ. Thus, the
morphological dilation and erosion on the partition
induced by fare given by [9]:
dðf;PfÞðxÞ¼maxfðf;PfÞðyÞ:Fy2Axg, (1)
eðf;PfÞðxÞ¼minfðf;PfÞðyÞ:Fy2Axg. (2)
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Fig. 4. Adjacent flat zone concept. (a) Image fwith 14 flat zones; (b) flat zone in point x,FxðfÞ; (c) flat zone in point y,FyðfÞ; and (d) two
adjacent flat zones, i.e., FxðfÞ[FyðfÞ¼gxðFxðfÞ[FyðfÞÞ.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–2150 2129
The dilation dmand erosion emof size mare obtained
iterating mtimes the elemental dilation and erosion
given in Eqs. (1) and (2):
dmðf;PfÞðxÞ¼dd dðf;PfÞðxÞ
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
mtimes
, (3)
emðf;PfÞðxÞ¼ee eðf;PfÞðxÞ
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
mtimes
. (4)
The opening and closing on the partition of size m
induced by fare:
gmðf;PfÞðxÞ¼dmðemðf;PfÞ;PfÞðxÞ, (5)
jmðf;PfÞðxÞ¼emðdmðf;PfÞ;PfÞðxÞ. (6)
The morphological gradient transformations of size
mon the partition are presented as follows:
gradmmðf;PfÞðxÞ¼dmðf;PfÞðxÞemðf;PfÞðxÞ. (7)
On the other hand, the top-hat transformation was
proposed by Meyer [28]. This operator allows the
detection of peaks (respectively, valleys) of certain
height (respectively, depth) and certain thickness,
having also granulometric (anti-granulometric)
characteristics. This approach allows the classifica-
tion of regions of the image by size and height. The
top-hat transformation is divided in white and black
top-hat. In the case of dealing with clear regions
the white top-hat is employed, and it is obtained as
the subtraction between the original image and the
opening transformation. Whereas for the dark
regions the black top-hat is used, this transforma-
tion is determined by the subtraction between the
closing transformation and the original image. As
follows, top-hat expressions build with the opening
and closing on the partition are presented:
Thwmðf;PfÞðxÞ¼ðf;PfÞðxÞgmðf;PfÞðxÞ, (8)
Thbmðf;PfÞðxÞ¼jmðf;PfÞðxÞðf;PfÞðxÞ. (9)
Generally, transformations (8) and (9) are followed
by a thresholding operation in order to obtain a
binary image containing certain bright structures
with a given size and contrast. Indeed, the top-hat
approach leads to a size distribution involving
contrast of the image. This contrast operator is
widely used in image segmentation [8,28].
2.2. Transformations by reconstruction
Into MM there are defined two connected
morphological filters called opening and closing by
reconstruction. These morphological filters have
next characteristics [31,32]: (i) the generation of
new features is avoided; and (ii) several details are
eliminated without considerably modifying the
remainder structures of the image.
When filters by reconstruction are built, the basic
geodesic transformations, the geodesic erosion and
the geodesic dilation of size 1 are iterated until
idempotence is reached [32]. The geodesic dilation
d1
fðgÞand the geodesic erosion e1
fðgÞof size one are
given by d1
fðgÞ¼f^dðgÞwith gpfand e1
fðgÞ¼
f_eðgÞwith gXf, respectively. When the function g
is equal to the erosion or the dilation of the original
function, we obtain the opening and the closing by
reconstruction [31,32], i.e.:
~
gtðfÞ¼ lim
n!1 dn
fðetðfÞÞ and ~
jtðfÞ¼ lim
n!1 en
fðdtðfÞÞ,
(10)
where the morphological erosion etBand dilation
dtBare expressed by etBðfÞðxÞ¼^ffðyÞ:y2t
Bxg
and dtBðfÞðxÞ¼_ffðyÞ:y2t
Bxg.^is the inf opera-
tor and _is the sup operator. In this work, Bis an
elementary structuring element (3 3 pixels) that
contains its origin.
Bis the transposed set ð
B¼
fx:x2BgÞ and tis an homothetic parameter. An
example to illustrate how these connected transfor-
mations work is presented in Fig. 5. Notice how the
maxima or minima of the image are flattening, while
the remain of the image is preserved; also note in
Figs. 5(b) and 5(d) that the creation of new contours
is avoided.
2.3. Morphological contrast mappings
The main idea of contrast mappings consists in
selecting at each point of the analyzed image the
grey level value between different patterns (primi-
tives) in accordance with some proximity criterion
(as a particular example, see Fig. 1). In the
literature, several examples of contrast mappings
have been reported. For example, see the Kramer
and Bruckner transformation [33]. This transforma-
tion modifies the contrast by means of two
primitives equivalent to the morphological erosion
and dilation defined by order-statistical filters
[34,35]. Here the proximity criterion is defined as a
comparison metric between the morphological
internal and external gradients. Another example
of contrast mappings are the MSFs. These filters
were introduced in [4,6,7]. In these contrast map-
pings, the morphological erosion and dilation are
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J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–21502130
used in separate ways, i.e., two contrast mappings
are built. One of them uses the original function and
the morphological erosion, while the other employs
the original function and the morphological dila-
tion. Due to this, the degradation in the output
image is less marked, than if it were processed by the
Kramer and Bruckner transformation. In morpho-
logical slope filters, the proximity criterion corre-
sponds to a criterion given in function of the
internal and external morphological gradients.
However, when the erosion or dilation is used
as primitive to build contrast mappings, a risk
exists of degrading the processed image. This
problem was considered by Serra [2], who solved it
by using idempotent transformations as primitives.
But, in the case of working at pixel level, if the size
of the structuring element is large, the use of
idempotent transformations does not ensure that
the degradation in the processed image will be
avoided, unless the primitives are connected trans-
formations. This situation is illustrated in [5]. Here
the opening and closing by reconstruction [10] are
utilized as primitives, but instead of using a
proximity criterion for selecting the primitives, a
contrast criterion given by the top-hat transforma-
tion is used.
In this work, contrast mappings have the
characteristic of being connected, since the trans-
formations employed are defined on the partition
generated from the set of flat zones of the image. As
follows, three-state contrast mappings with size
criteria on the partition induced by fare considered.
These transformations are composed of three
primitives, opening on the partition, closing on the
partition and the original image (see Eqs. (5) and
(6). The proximity criterion [1] presented in Eq. (11)
considers the bright and dark regions of the image:
rðxÞ¼ jlðf;PfÞðxÞðf;PfÞðxÞ
jlðf;PfÞðxÞgmðf;PfÞðxÞ. (11)
Eq. (12) establishes a three-state contrast mapping
with size criteria on the partition
W3
l;m;b;aðf;PfÞðxÞ¼
jlðf;PfÞðxÞ;0prðxÞob;
ðf;PfÞðxÞ;bprðxÞoa;
gmðf;PfÞðxÞ;aprðxÞp1:
8
>
<
>
:(12)
Images in Fig. 6 illustrate the performance of some
contrast mappings mentioned above. The original
image is displayed in Fig. 6(a), while the output
image in Fig. 6(b) corresponds to the Kramer and
ARTICLE IN PRESS
Fig. 5. (a) Original function and marker g; (b) closing by reconstruction; (c) original function and marker g; and (d) opening by
reconstruction.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–2150 2131
Bruckner transformation after 20 iterations at pixel
level. The image in Fig. 6(c) is obtained after
applying MSF with slope f¼40 at pixel level; the
image processed with the morphological contrast
mapping on the partition is presented in Fig. 6(d).
In this case the parameters are m¼5, l¼7,
a¼0:388, and b¼0:686. Note that, the images in
Figs. 6(b) and 6(c) are degraded, which does not
occur in the case of the image in Fig. 6(d). Here, the
image has been enhanced and the modification of
contours is avoided; this situation can be observed
in Fig. 6(h), where the contours are preserved, while
in Figs. 6(f) and 6(g), the contours are modified. In
order to get a better appreciation of the images in
Figs. 6(e)–(h), a thresholding operation is applied
between sections 95–255; the resulting output
images are presented in Figs. 6(i)–(l).
The main advantage of working at partition level
is that the generation of new contours is avoided.
The former situation occurs because the employed
transformations are connected. On the other hand,
notice from Eqs. (11) and (12) that there are four
parameters to be determined: m,l,aand b. In this
work, the parameters mand lare obtained by a
graphic method described in Section 3, whereas
suitable values for the parameters aand bare
obtained from a quantifying contrast method
introduced in Section 4.
3. Opening and closing size determination
In the case of working at partition level, the
notion of the structuring element disappears (see
Eqs. (1)–(12)). In particular, it is necessary to
propose a method that enables to know the sizes
of the homotetic parameters mand linvolved in the
proximity criterion and the contrast mappings as
expressed in Eqs. (11) and (12). In these equations,
ARTICLE IN PRESS
Fig. 6. Examples of images processed by different contrast mappings. (a) Original image, (b) Kramer and Bruckner transformation after
20 iterations at pixel level, (c) MSF with f¼40 at pixel level, (d) morphological contrast mapping with size criteria, where m¼5, l¼7,
a¼0:388, and b¼0:686 at the partition level; (e) image contours of (a); (f) image contours of (b); (g) image contours of (c); (h) image
contours of (d); (i),(j),(k) and (l) thresholding operation between 95–255 sections of image in Figs. 6(e–h).
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–21502132
the sizes of opening and closing on the partition
must be calculated for every particular case.
In this work, a graphic method is employed in
order to find adequate values for the homotetic
parameters. The graphic method consists of drawing
information from the top-hat spectrum which is
obtained from the opening spectrum concept. The
opening spectrum is the image sequence created by
computing the difference between successive openings
fulfilling the granulometry definition [28,36–38],thus
the opening spectrum on the partition is:
Zmðf;PfÞðxÞ¼gmðf;PfÞðxÞgmþ1ðf;PfÞðxÞ
for m¼1;...;N1. ð13Þ
Classically, the opening spectrum given in Eq. (13),
enables to obtain information of the changes of size
distribution of the different structures detected in
the processed image. Notice that expression (13) can
be rewritten in terms of the white top-hat notion
(see Eq. (8)), i.e.:
Zmðf;PfÞðxÞ¼Thwmþ1ðf;PfÞðxÞThwmðf;PfÞðxÞ
for m¼1;...;N1. ð14Þ
Eq. (14) is known as the top-hat spectrum [38].
Notice that this expression yields information about
the contrast of the image, since the top-hat spectrum
gives information concerning changes in the white
regions as the size of the opening increases. On the
other hand, in the literature, the normalized open-
ing spectrum [18,39] is calculated from the ratio
between the area of the opening spectrum and the
area of the original image. In this work, the ratio
between the top-hat spectrum and the total of black
and white regions detected by top-hat transforma-
tions is denoted by xm;l. The total of black and white
regions is obtained from the sum of black and white
top-hats. The final result is also a top-hat known as
self-complementary top-hat [3], which is expressed
as follows:
Thwbl;mðf;PfÞðxÞ¼Thblðf;PfÞðxÞþThwmðf;PfÞðxÞ.
(15)
If volðfÞrepresents the volume detected in the
function f, the ratio between the volume of the top-
hat spectrum and that of the sum of bright and dark
regions in the processed image is written as follows:
volðxm;lÞ
¼volðThwmþ1ðf;PfÞðxÞÞ volðThwmðf;PfÞðxÞÞ
volðThwbl;mðf;PfÞðxÞÞ þ 1.
ð16Þ
Notice that the unit has been introduced in
the denominator of Eq. (16) to avoid any
indetermination.
On the other hand, parameters mand l, involved
in the morphological three-state contrast mapping
(See Eq. (12)), will be detected using a graphic
method obtained from Eq. (16). This expression
works similar to the granulometric density, and
allows the obtention of adequate sizes for the
opening or closing on the partition from a graph,
when one of these parameters is fixed. In this work
two examples are provided; in the first, the closing
size is maintained without change, while the open-
ing size varies within a certain interval. In the
second example, the closing size varies within a
certain interval, while the opening size is maintained
unchanged.
The objective of plotting volðzm;lÞvs mor volðzm;lÞ
vs lis to determine the interval in terms of mor l
sizes in which the main structure of clear or black
regions of the processed image is situated.
3.1. First example: closing size is maintained without
change, while the opening size varies within a certain
interval
The graphic method is illustrated in Fig. 7.
The original image is displayed in Fig. 7(a), while
in Fig. 7(b) a graph obtained from Eq. (16) is
presented. In this graph, the closing size is fixed at
l¼8, while the opening size mchanges within the
interval [40,41]. It is important to mention that,
experimentally, a large size for the closing and
opening on the partition (greater than 5) allows
adequate segmentations on MRI of the brain; this is
the reason for selecting l¼8. Nevertheless, if an
arbitrary value of lis selected, an adequate size m
for the opening on the partition may be obtained
from the graph drawn from Eq. (16). In the present
example, the main structure of clear regions in Fig.
7(b) is detected for mvalues comprised in the
interval 1 to 7. Therefore, an adequate value for the
size of the opening on the partition may be m¼7
[5,15,42].
In Fig. 7(c), the morphological three-state con-
trast mapping with parameters l¼8, m¼7,
a¼0:196, b¼0:392 is obtained. In this example,
parameters aand bwere selected without following
a particular method; however, an estimation of
these parameters will be done by means of
the morphological contrast method proposed in
Section 4.
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3.2. Second example: closing size varies within a
certain interval, while the opening size is maintained
unchange
This example is illustrated in Fig. 8. Here, the
opening size is maintained with m¼5, and l
changes within the interval [1,12]. The opening
size is selected empirically, as well as the vari-
able interval. One must bear in mind that assi-
gning large sizes for the primitives allows a
better detection of white and grey matter.
Note that different values of mand lare used
in the examples provided in the paper to illustrate
the performance of the proposals given in the
article.
On the other hand, if an arbitrary size mfor
the opening on the partition is selected, the closing
size lwill be detected from the graph derived from
Eq. (16).
In Fig. 8(b), the graph of Eq. (16) is obtained with
the established parameters. Note that there are two
intervals in which important structures of dark
regions are detected: ½1;7and ð7;12[5,15,42].
Therefore, two adequate values for the size of the
closing on the partition may be l¼7 and 12. Some
output images are presented in Figs. 8(c)–8(e).
These were obtained by fixing the same values for
aand bas in the first example and applying a
morphological three-state contrast mapping (see
Eq. (12)). Whereas the image in Fig. 8(f) was
obtained for different values of aand b,as
mentioned above, there are two values for the l
parameter. Note that if l¼7 is selected, the
morphological three-state contrast mapping is not
capable of modifying all ‘‘black’’ components, since
some of these can only be modified for values
of lambda greater than 7. However, if l¼12,
practically all ‘‘black’’ components are modified.
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0
5
10
15
20
25
30
123456789101112
Vol (μ∈ [1, 12], = 8)
μ
Opening size
ab
c
Fig. 7. Graphical method to detect the suitable size for the opening on the partition when the closing size is fixed from Eq. (16): (a) original
image, (b) graph of xm2½1;12;l¼8, and (c) morphological connected three-state contrast mapping with m¼7, l¼8, a¼0:196, and b¼0:392.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–21502134
In Figs. 8(e) and 8(f) this effect can be appreciated.
In the three-state morphological contrast mappings,
the proximity criterion determines which primitive
acts base on aand bparameters, as expressed in
Eq. (12). In Fig. 8(e) the behavior of the black
regions is observed. Here, black regions are merged
due to the closing on the partition as the bvalue
decreases, while in Fig. 8(f) the opening on the
partition and the original function predominate as
the avalue increases.
4. Contrast measure
The application of contrast transformations is a
common practice for enhancing characteristics of
interest in the processed images; however, it is not
common to have a quantification of the contrast to
select the enhanced image presenting the best visual
contrast. This situation occurs because the improve-
ment of images after being processed by some
contrast transformation is often quite difficult to
measure. In the literature, some definitions of
contrast measure have been reported, see for
example [20,21,40,41,43]. These methodologies are
not based on techniques of mathematical morphol-
ogy; therefore, the introduction of a morphological
contrast measure is important, since the improve-
ment of contrast using morphological transforma-
tions is widely used.
In this work, a morphological contrast mea-
sure is proposed and treated from a point of
view of visual contrast. In psychovisual studies,
the contrast Cof an object with luminance Lagainst
its surrounding luminance Lsis defined as
follows [20]:
C¼LLs
Ls
. (17)
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0
0.05
0.1
0.15
0.2
0.25
0.3
123456789101112
Closing size
Vol (ξλ∈ [1, 12], μ = 5)
a
cd
ef
b
Fig. 8. Graphical method to detect the suitable size for the opening on the partition when the closing size is fixed in Eq. (16). (a) original
image, (b) graph of xl2½1;12;m¼5. Morphological connected three-state contrast mapping with: (c) m¼5, l¼7, a¼0:196, and b¼0:392; (d)
m¼5, l¼10, a¼0:196, and b¼0:392; (e) m¼5, l¼12, a¼0:196, and b¼0:392; and (f) m¼5, l¼12, a¼0:596, and b¼0:827.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–2150 2135
A perceptive contrast measure is a complex task,
since several conditions must be considered, for
example, state of adaptation of the observer, nature
of existent contours between adjacent areas, relation
between adjacent areas, size of the internal struc-
tures of the image, spatial frequency of the stimuli,
among others. In order to model the way the eye
perceives luminance changes, several contrast mod-
els have been introduced, for example, Weber law,
power law, Michelson law, just to mention a few
[23]. However, there is no universal measure which
can specify both the objective and subjective validity
of the enhancement method [41]. This situation is
illustrated with some attempts where a local
contrast measurement in the spatial domain is
obtained [40]. For example, the local contrast
proposed by Gordon and Rangayan [44] is defined
by the obtention of the average of intensity values
detected in two rectangular windows centered on a
current pixel. In order to improve the method
proposed by Gordon and Rangayan [44], Beghdadi
and Negrate proposed a local contrast measure
based on the local edge information of the image
[43]. In Agaian et al. [40], a quantifying contrast
method was proposed, in which the maximum
intensity and minimum intensity inside the block
were analyzed to calculate the measure of the
enhancement. Another example of contrast measure
can be found in Morrow et al. [20]; this approach
uses statistical quantifications based on the contrast
histogram.
Notice that each one of the contrast measures
mentioned above attempts to quantify contrast
enhancement in different ways, confirming the
absence of a universal method to measure contrast
in processed images.
In this paper, the morphological contrast measure
is oriented toward the analysis of contours of the
image. The modification of the contours in a
processed image will produce changes in the
contrast. For this reason, important changes in
luminance are associated with the contours of the
image. From the point of view of visual perception,
local visual information is combined to form a
global representation. In our case, each edge of the
image is analyzed as local information, in such a
way that a global representation yielded by all
contours of the processed image provides informa-
tion about contrast changes in the processed image.
In the next section, a morphological contrast
measure is proposed. The morphological contrast
measure will be useful to determine the best
parameters associated with certain morphological
contrast transformation. The parameters of interest
are obtained from a graphical method which
involves a set of output images. In particular, these
output images are generated by the application of
the morphological three-state contrast mappings
(see Section 2.3).
4.1. Morphological contrast measure based on image
edges
Edges are defined as significant local intensity
changes in the image; usually they are considered
step discontinuities. These important transitions of
local intensity are associated with significant varia-
tions of luminance. Due to this, edges are important
features to be analyzed since they produce changes
in the contrast. On the other hand, in psycho visual
studies, the finding of contours is the first step in
vision processing, since edges often coincide with
important boundaries in the visual scene [17]; this
process is called primal sketch. In order to detect the
edges of the image, there are for example first
derivative techniques, second derivative techniques,
among others [23]. In general, the implementation
of these approaches includes the convolution of the
signal with some form of linear filter. On the other
hand, in mathematical morphology, the extraction
of contours is carried out by the implementation of
morphological gradients [45].
In this work, the morphological gradient pre-
sented in Eq. (7) is used. The analysis of image
contours consists in quantifying in local mode the
variations of the maximum and minimum intensities
of the detected contours of the image into a window
Bcontaining its origin. Formally, this is expressed
as follows.
For the sake of simplicity, let us consider
max gradmðxÞ¼maxfgradmðf;PfÞðxþbÞ:b2Bgand
min gradmðxÞ¼minfgradmðf;PfÞðxþbÞ:b2Bg,where
xbelongs to the domain of definition of f, denoted
by Df. Expression (18) is proposed in order to have
an indirect measure of the variations of the contrast.
This quantification is denoted as VCp, where p
represents the parameters involved in the contrast
enhancement of the processed image,
VCp¼X
x2Df
½max gradmðxÞmin gradmðxÞ. (18)
max gradmðxÞand min gradmðxÞrepresent the
maximum and minimum intensity values of the
morphological gradient on the partition around
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point x. These values are taken from one set of
pixels contained in a window Bof elemental size
(3 3 elements) that contains its origin (notice that
the window Bcorresponds to the structuring
element). From Eq. (18), and in accordance with
the order-statistical filters [34,35], the maxima and
minima intensity values of the image in certain
neighborhood are the morphological dilation
dmBðfÞðxÞ¼_ffðyÞ:y2m
Bgand the morphological
erosion emBðfÞðxÞ¼^ffðyÞ:y2m
Bg(where ^is the
inf operator, _is the sup operator, and
Bthe
transposed of the structuring element B, i.e.,
B¼fx:x2Bg), in such a way that Eq. (18) is
rewritten as follows:
VCp¼X
x2Df
½dmBðgradmðf;PfÞðxÞÞðxÞ
emBðgradmðf;PfÞðxÞÞðxÞ.ð19Þ
However, the morphological gradient at pixel level
is expressed as follows:
gradmmBðfÞðxÞ¼dmBðfÞðxÞemBðfÞðxÞ. (20)
From Eq. (20), Eq. (19) is written as:
VCp¼X
x2Df
gradmmBðgradmðf;PfðxÞÞðxÞ. (21)
Notice from Eq. (21) that the edges of the processed
image are obtained by the application of the
morphological gradient on the partition, whereas
important variation of the detected edges are
obtained through the morphological gradient at
pixel level. Therefore, the morphological contrast
measure introduced in this work is a measure of the
intensity variations of the detected contours. Notice
that important differences between the maximum
and minimum intensities of the morphological
gradient in certain window Bconcern with sub-
stantial changes in the luminance of the image.
On the other hand, the main purpose of the
morphological quantitative contrast method in this
work consists in detecting the output image that
presents a good visual contrast from a set of output
images generated by some parametric contrast
transformation. For each one of the output
enhanced images, the VCpvalues are calculated
and plotted. The analysis of the obtained graph is
focused on its global maxima, since they provide
information about important changes in the ana-
lyzed edges. The following steps are employed for
selecting the maximum producing a good visual
contrast:
Step 1: Calculate and draw the graph VCpvs
parameters for a set of output enhanced images.
Step 2: A higher visual contrast will correspond to
VCpvalue associated with the global maximum in
the graph VCpvs parameters.
Notice that in the graph VCpvs parameters it is
possible to find more than one maximum, though the
largest change in luminance will be detected by the
global maximum (the maximum with highest alti-
tude), hence only the global maximum is employed.
Also it is important to mention that images with
high VCpvalues not necessarily mean good visual
contrast; for example, if an input image is processed
with a contrast enhancement transformation produ-
cing a large degradation in the output image, then
high VCpvalues can produce an output image
without good visual contrast. In consequence, the
election of the parametric values directly controlling
contrast transformations must be done with care in
order to avoid such problems. The performance of
this contrast measure is illustrated in the next section.
5. One application to the segmentation of brain MRI
In this section, two examples of the proposals
given in this work are presented. The first example
deals with white and grey matter segmentation in a
frontal lobe area, while in the second example white
and grey matter are also segmented, but in this case
MRI are corrupted by introducing 5% noise.
As follows we describe the procedure used to
segment white and grey matter in a frontal lobe
region. This procedure is similar to that followed for
segmenting white and grey matter in MRI corrupted
by noise.
5.1. Segmentation of white and grey matter in frontal
lobe
The brain MRI-T1 presented in this example
belongs to the MRI-T1 bank of the Institute of
Neurobiology, UNAM Campus Juriquilla, Quere
´-
taro Me
´xico. The file processed and presented in
this article comprises 120 slices, from these 22
belong to a frontal lobule. The selection of the
different frontal lobule slices was carried out by a
specialist in the area of the same institute. The
segmentation of the skull for each brain slice is done
by means of the transformation proposed in [46];
our sole interest at this point is the segmentation of
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white and grey matter. The first three slices
of a frontal lobule without skull are presented in
Fig. 9(a).
The general idea is to apply contrast mappings on
the partition as a pre-processing step to enhance
clear regions, i.e., enhance the zones where white
matter is located. Note that during the enhancement
process, several clear regions will be merged, thus
white matter is obtained for certain grey levels.
The procedure followed to obtain white and grey
matter in frontal lobe is explained in the next
sections; however, in order to simplify the proce-
dure, the same contrast mapping on the partition
with specific parameters l,m,a, and bwill be
applied to all slices. This approximation is made for
two reasons: first, the intensities of white matter are
similar in all slices; second, to avoid a large and
inadequate process. Thus, the parameters l,m,a,
and bare obtained solely for the first slice of the
frontal lobe and applied to the remaining slices.
5.1.1. Determination of the opening size on MRI
The analysis carried out in this section corre-
sponds to the first slice of frontal lobe presented in
Fig. 9(a) as was mentioned previously.
The morphological contrast mappings on the
partition will be used to enhance the clear regions in
frontal lobe. This is achieved by attenuating the
dark regions, while the clear regions are maintained
or ‘‘hardly’’ modified. Subsequently, adequate
sizes for the opening and closing on the par-
tition involved in the contrast mappings must be
determined.
By means of Eq. (16), the size of the opening is
calculated. The size of the closing on the partition is
fixed at l¼15. This value is empirical, since,
experimentally a large size for the closing will give
adequate segmentations.
An adequate size for the opening on the partition
is calculated graphically from Eq. (16). The aim of
plotting the graph volðzm;lÞvs mis to determine an
interval given in terms of msizes, where the main
structures of clear regions of the processed image
are located. In Fig. 9(b) the graph volðzm;lÞvs mis
presented. Note that mtakes values within the
interval 1–12, while l¼15 is a fixed value. The
interval for mis considered within these values given
that the opening on the partition originates larger
modifications than the traditional morphological
opening [9,15].
The main structure of clear regions in Fig. 9(b) is
detected for mvalues in the interval 1–6. For this
reason an adequate value for the size of the opening
on the partition may be m¼6[5,15,42]. Hence, a
contrast mapping on the partition with parameters
l¼15 and m¼6 will be applied for all slices of the
analyzed frontal lobe area.
5.1.2. Determination of parameters aand b
The analysis presented in this section also
corresponds to the first slice of the frontal lobe
shown in Fig. 9(a). The determination of parameters
aand binvolved in the contrast mappings on the
partition is carried out by means of the morpholo-
gical contrast measure introduced in Section 4.1. In
other words, parameters aand bare associated with
the image presenting the ‘‘best’’ visual contrast.
The methodology consists in generating a set of
output images by means of a contrast mapping on
the partition with specific parameters land m, while
aand btake values within the interval ½0;1.
Subsequently, the contrast measure VCa;bis calcu-
lated for each image of the set by means of Eq. (21).
The image with the best visual contrast is obtained
from the graph VCa;bvs a;b; such image allows the
adequate determination of aand bvalues. In this
work, a set of 12 images is generated from a contrast
mapping on the partition with parameters l¼15
and m¼6, while aand btake values within the
interval ½0;1. The values for VCa;bare then
calculated for each output image; the graph VC a;b
vs a;bis presented in Fig. 9(c).
The image presenting the best visual contrast is
associated with the global maximum located in the
graph VCa;bvs a;b. In this example, the maximum
corresponds to the image with parameters a¼0:137
and b¼0:627. Hence, l¼15, m¼6, a¼0:137 and
b¼0:627 will be used as the specific parameters in a
contrast mapping for enhancing the clear regions in
all slices of the frontal lobe. This situation is
illustrated in Fig. 9(d).
5.1.3. Segmentation of white and grey matter in a
frontal lobe area
In order to illustrate the different steps of the
proposed algorithm to segment white and grey
matter in a frontal lobe region, Fig. 10 will be used.
The original image is located in Fig. 10(a1), while
the image processed by the contrast mapping (see
Eq. (12)) using the parameters calculated previously
is presented in Fig. 10(a2).
Algorithm to segment white and grey matter:
(i) Compute the threshold of the image in
Fig. 10(a2) between sections 90–255. The sections
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Opening Size
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
123456789101112
00.05 0.1 0.15 0.2
0.25 00.2
0.4 0.6 0.8
1
4
5
6
7
8
9
10
11
12
13
Contrast Variation
α
β
μ
×104
Vol (ξμ∈ [1,12] , λ =15)
VCα, β
a
b
c
d
Fig. 9. MRI segmentation. (a) First three slices of a frontal lobule; (b) graph of the volume calculated from Eq. (16), the opening size
varies within the interval ½1;12, while closing size is fixed to l¼15; (c) graph VCa;bvs a;b; and (d) contrast mapping applied to images in
Fig. 9(a) with l¼15, m¼6, a¼0:137 and b¼0:627.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–2150 2139
90–255 are obtained approximately from the nor-
malized histogram presented in Fig 10(a3); the
output image is presented in Fig. 10(a4).
(ii) Obtain from the original image (Fig. 10(a1))
the grey level values where the binary image in step
(i) takes the value of 1. At this point the white
matter is segmented (See Fig. 10(a5)).
(iii) Compute point by point the arithmetic
difference between the original image in Fig.
10(a1) and the image in step (ii). Here the grey
matter and other structures are detected. The output
image is presented in Fig. 10(a6).
(iv) Compute the threshold of the image ob-
tained in step (iii) between sections 70–255;
the output image is presented in Fig. 10(a7).
The sections 70–255 are obtained approxi-
mately from the normalized histogram prese-
nted in Fig. 10(a3). For values greater than
70 levels of intensity, the cerebrospinal fluid is
eliminated.
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0 100 200
0
2
4
6
8
x 10-3
white matter
grey matter
cerebrospinal
fluid
Histogram
a1 a2
a4 a5 a6 a7
a8 a9 a10 a11
a12 a13 a14
a3
Fig. 10. White and grey matter segmentation. (a1) Original image; (a2) enhanced image; (a3) normalized histogram of image in
Fig. 10(a2); (a4) thresholding between 90–255 sections of image in Fig. 10(a2); (a5) detection of white matter; (a6) difference between
image in Fig. 10(a1) and image in Fig. 10(a5); (a7) thresholding between 70–255 sections to detect grey matter of image in Fig. 10(a6); (a8)
detection of grey matter; (a9–a11) first three slices in which white matter is segmented; and (a12–a14) first three slices in which grey matter
is segmented.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–21502140
(v) Obtain from the original image the grey level
values where the binary image in step (iv) has the
value of 1. In this step the grey matter is segmented
(see Fig. 10(a8)).
In Figs. 10(a9)–(a11), the white matter of images
in Fig. 9(a) are presented, as well as the grey matter
in Figs. 10(a12)–(a14).
On the other hand, the specialist in neuroanat-
omy in the Institute of Neurobiology, UNAM,
Campus Juriquilla, Quere
´taro Me
´xico, identifies
white and grey matter by intensity, and quantifies
white and grey matter by selecting different regions
of interest. In order to illustrate the difficulties
found to segment manually white and grey matter in
frontal lobe, Fig. 11 is presented. In Fig. 11(a)
the original images are shown (this images have
been used throughout the paper). In Fig. 11(b)
we have the output images after applying Eq. (12)
with parameters obtained in Sections 5.1.2 and
5.1.1, i.e., l¼15, m¼6, a¼0:137 and b¼0:627.
In Fig. 11(c), white matter has been detected by
means of the algorithm introduced previously.
Finally, in Fig. 11(d) a similar procedure followed
by the specialist in neuroanatomy is provided. In
this case a thresholding operation is carried out
between sections 90–255. The idea is to separate
clear regions. Subsequently, the thresholded image
is superposed to the original image to get a
reference. By comparing Fig. 11(c) and Fig. 11(d)
notice that in the latter, the thresholding operation
detects several clear regions that were not detected
in the images of Fig. 11(c). Nevertheless, many of
them do not correspond to white matter. This is the
main problem in manual segmentations, since the
specialist has to decide whether the regions corre-
spond to white or grey matter. Moreover, manual
segmentations are time-consuming procedures, and
many errors are introduced in the measurements,
because the selection of white or grey matter is
subjective. In the images of Fig. 11(c), white matter
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Fig. 11. Detection by intensity. (a) Original images; (b) enhanced images after applying the contrast mapping with the parameters
obtained in Section 5; (c) output images where white matter is segmented by using the algorithm introduced in Section 5.1.3; (d)
thresholding operation between 90–255 sections to the images in Fig 11(b).
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–2150 2141
is obtained as flat zones of regions that fulfill the
thresholding operation. The objective of applying
contrast mappings on the partition is to enhance
and merge clear regions; further on, white matter is
separated by a thresholding operation.
On the other hand, in the Institute of Neurobiol-
ogy, UNAM, Campus Juriquilla, Quere
´taro
Me
´xico, several specialists study the problem of
memory impairment related to aging. They com-
pare the index given by the ratio between white
and grey matter in the frontal lobe for different
brains. In this paper, the segmentation of the
frontal lobe called e2397 is presented; however,
the same procedure was applied for segmenting
four other frontal lobes. In Fig. 12(a) the set of
slices that form the frontal lobe of the brain e2397
is presented, whereas in Fig. 12(b) we have
the enhanced images after applying Eq. (12)
with parameters l¼15, m¼6, a¼0:137 and
b¼0:627. In Fig. 13(a), the segmentation of white
matter is presented. White matter in frontal lobe
was obtained by the application of the algorithm
given in this section. Finally, grey matter is
segmented following the algorithm provided also
in this section; output images can be observed in
Fig. 13(b).
Once that white and grey matter are segmented,
all the pixels different from zero in Figs. 13(a) and
(b) are counted. The volume of white matter
amounts to 24 731 pixels and that of grey matter
to 33 169 pixels; the ratio between grey and white
matter is equal to 1:341. The relation between white
and grey matter is compared with a manual
segmentation performed by an expert in the area;
the comparison gives a variation of þ5% with
respect to the manual procedure.
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Fig. 12. Set of images corresponding to frontal lobe of e2397 brain. (a) Original images; (b) enhanced images after applying the contrast
mappings with the parameters obtained in Section 5.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–21502142
Likewise, in our remaining segmentations of
frontal lobes, the ratios between white and grey
matter presented a variation of 5% with respect
to the manual method. The segmentation of white
and grey matter, as well as the ratios between
white and grey matter was validated by an expert of
the Institute of Neurobiology, UNAM, Campus
Juriquilla, Quere
´taro Me
´xico.
5.2. Segmentation of white and grey matter in the
presence of noise
In the last three sections a methodology to
segment white and grey matter in frontal lobe was
provided. However, in such images (see for example
Fig. 12) the noise level is less than 1%; therefore a
and bparameters can be obtained from the
morphological contrast measure without any pro-
blem. Nevertheless, if the processed image is
corrupted by noise, the methodology followed to
compute aand bparameters fails, because the
contrast measure involves the morphological gra-
dient, as expressed in Eq. (21).
Noise is an undesirable characteristic in MRI. It
reduces image quality and makes the segmentation
process troublesome. In our particular case, a
solution to this problem is a pre-filtering step on
the corrupted images. The segmentation quality is
conditioned by this step, and more precisely by
preserving the useful information. As follows, an
example where white and grey matter are separated
in some slices of a volume of MRI in presence of
noise is provided; here the reconstruction transfor-
mations (see Section 2.2) will be used as a pre-
processing step to the noisy images. The MRI data
volume was obtained from a simulated brain
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Fig. 13. White and grey matter segmentation in frontal lobe of e2397 brain. (a) Segmentation of white matter from the images in Fig.
12(b); (b) segmentation of grey matter from the images in Fig. 12(b).
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–2150 2143
database [29], which has the characteristic of being
corrupted by 5% noise.
2
In Fig. 14 eight slices
belonging to the file t1_icbm_normal_1mm_
pn5_rf 20½1:mnc are presented. While in Figs. 15
and 16, some output images illustrate the metho-
dology followed in this paper to segment white and
grey matter. In Fig. 15(a) we have the original image
corrupted by noise. This image corresponds to slice
107. In order to appreciate the noise in the original
image; a threshold was obtained between sections
150–255 (Fig. 15(b)). In Fig. 15(c), the normalized
log-histogram of the image in Fig. 15(a) is obtained.
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white
matter
grey matter
cerebrospinal
fluid
Log-histogr
Log-histogram
ab
c
de
f
Fig. 15. (a) Original image corrupted by 5% noise; (b) threshold of image in Fig. 15(a) between 150–255 sections; (c) normalized log-
histogram of image in Fig. 15(a); (d) filter ~
jt¼1ð~
gt¼1ðfÞÞ; (e) threshold of image in Fig. 15(d) between 124–177 sections; and (f) normalized
log-histogram of image in Fig. 15(d).
Fig. 14. (a) Slices of MRI corrupted by noise 5%. The images correspond to the slices 107–114 to the file
t1_icbm_normal_1mm_pn5_rf 20½1:mnc.
2
In accordance with the information given in http://
www.bic.mni.mcgill.ca/brainweb/, the noise in the simulated
images has Rayleigh statistics in the background and Rician
statistics in the signal regions. The ‘‘percent noise’’ number
represents the percent ratio of the standard deviation of the white
Gaussian noise versus the signal for a reference tissue.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–21502144
Notice the different regions where cerebrospinal
fluid, grey matter and white matter are detected.
The intensity levels where these regions are located
are relevant, given that the thresholding operations
in the algorithm proposed to segment white and
grey matter use this information. On the other hand,
the pre-processing step using the filter ~
jt¼1ð~
gt¼1ðfÞÞ
is presented in Fig. 15(d). Here the opening and
closing by reconstruction are applied with a size
t¼1; the intention is to suppress noise components
without affecting the remaining structures of the
image. This characteristic is very important, since
the image has been simplified without the introduc-
tion of new contours. In Fig. 15(e) a thresholding
operation between sections 124–177 is shown
to appreciate the noise in the image. While in
Fig. 15(f), the normalized log-histogram of the
image in Fig. 15(d) is displayed. By comparing
histograms in Figs. 15(c) and (f), it is evident that in
the latter some dark and clear regions were elimi-
nated. Several intensity levels were stretched as
well, resulting in enhancement of the image. In
Figs. 16(a) we obtain the opening size when the
closing size is fixed at l¼15. In this example, a high
value for the closing is selected, because we want to
suppress or merge the noise with the grey levels of
the image when it is processed by the contrast
mapping. As mentioned previously, the flat zones
that form the image will be extended or merged
when a transformation on the partition is applied
(see Fig. 18). From the graph in Fig. 16(a), notice
that an important structure of white regions can be
found in the interval m2½1;11, thus mis selected
with a value of m¼11. On the other hand,
parameters aand bare associated with the image
presenting the best visual contrast. Parameters a
and bwere deduced from the global maximum
located in the graphic presented in Fig. 16(b), in
this case, a¼0:078 and b¼0:313. Furthermore,
Fig. 16(c) shows the output image after applying
Eq. (12) with parameters, m¼11, l¼15, a¼0:078
and b¼0:313. While in Fig. 16(d) its normalized
log-histogram is presented. In this histogram notice
that white matter is formed by regions with intensity
levels around 150. The value of 150 will be useful in
the algorithm as the inferior threshold level to
separate white matter. The segmentation of white
and grey matter is then obtained by following the
algorithm given previously in Section 5.1.3; the
output images are presented in Figs. 16(e) and (f).
Although the algorithm to segment white and
grey matter works properly, some noise components
appear in the borders, as presented in Figs. 16(e)
and (f). Hence, a better segmentation will be
obtained if an efficient transformation to suppress
noise is utilized. Finally, in Fig. 17 a set of images
corrupted with 5% noise is processed following the
same procedure. Each row displays the original
image, output image following the pre-processing
step, enhanced image, white and grey matter.
6. Implementation of the transformations on the
partition
Though the bidimensional array (pixel matrix) is
a common way to represent an image, it does not
allow to deal effectively with the concept of
partition based on the notion of flat zone, since
there is not an easy access to the regions and to its
vicinity relations. The structure of data better
adapted to this problem is the graph made up of
nodes and arcs, where nodes represent flat zones
and arcs the adjacency between flat zones. This
structure is useful for the description and imple-
mentation of connected transformations on images
[47]; it is known as region adjacency graph.
Using this representation, connected transforma-
tions can be obtained in terms of this graph though
it is necessary to change the grey level of the
merging nodes. A graph structure was implemented
for the transformation introduced in this work.
Here the nodes represent the partition components
of the original image and the arcs describe the
adjacency between components. Fig. 18(a) illus-
trates the adjacency graph of the image presented in
Fig. 18(b), while Fig. 18(c) shows the image
obtained from the erosion on the graph.
The values to the interior of the nodes correspond
the grey level of the flat zones of the original image
(see Fig. 18(a)) and the grey levels of the trans-
formed image by the erosion (see Fig. 18(c)). The
value on the top and in the interior correspond to
the grey levels of the original image, while the
emphasized value at the bottom is the grey value of
the eroded image. The representation of the
adjacency graph allows the implementation of a
sequence of transformations in an efficient way. The
algorithm for the creation of the graph is relatively
fast. An image of the kind presented in Fig. 7(a)
with size 256 256 (65 536 pixels) containing 7353
flat zones (nodes) requires approximately 6 s to have
its adjacency graph on a computer Pentium III at
600 MHz. After creating the graph, a contrast
mapping (see Eq. (12)) of size l¼m¼15 is carried
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J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–2150 2145
out in 5 s approximately, i.e., 60 basic operations
(erosion and dilation of size 1) are undertaken
during this time.
7. Discussion and conclusions
In this paper we present a morphological contrast
mapping defined at partition level and a morpho-
logical contrast measure based on edge analysis.
The contrast measure is useful to find adequate
values for the parameters aand binvolved in the
contrast mapping. Moreover, a graphic method is
proposed to obtain the size of one of the primitives,
the opening or closing on the partition, when the
size of the closing or the opening on the partition is
fixed. The purpose of detecting suitable values for
the sizes of the primitives and the parameters aand
bthat appear in the contrast mapping is to get an
adequate contrast enhancement in the processed
image. Contrast enhancement not only serves to
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α
×105
μ
Log-histogram
Vol (μ∈ [1, 12], = 15)
VCα,β
Opening size
a
b
cd
ef
Fig. 16. (a) Graph to obtain the opening size; (b) graph to obtain the aand bparameters; (c) enhanced of image in Fig. 15(d) by means of
the contrast mapping with parameters m¼11, l¼15, a¼0:078 and b¼0:313; (d) normalized log-histogram of image in Fig. 16(c); (e)
detection of white matter; and (f) detection of grey matter.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–21502146
improve the image, but it is also useful in segment-
ing the image. This approach was mainly used to
detect white matter in frontal lobe sections as is
illustrated in Figs. 12 and 13.. The segmentation of
white matter consists in the separation of all clear
regions that are enhanced by the contrast mapping
with specific parameters. In this case, a thresholding
operation was used to separate white matter.
However, in Section 5.1.3, we mention that manual
segmentation is based basically on a thresholding
operation (see Fig. 11). The difference between the
two methods consists in the following: we separate
merged and enhanced flat regions, while in manual
segmentations that is not possible. Nevertheless,
ARTICLE IN PRESS
Fig. 17. Set of images. In each row is displayed the original image, image after pre-processing step, enhanced image, white and grey matter
images.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–2150 2147
when an image is processed at the partition level, the
main disadvantage is that it is strongly modified and
adjacent flat zones are merged as the size of the
primitives increases (see Fig. 18). Hence, the
practical problem solved in this paper, grey matter
regions having similar intensities than white matter
areas will be merged. To separate similar inten-
sities of grey and white matter is also a hindrance
in manual segmentations. That is the reason of
5% variations when our method and manual
segmentations are compared. On the other hand,
the segmentation of white and grey matter takes
20 min per lobe (more or less 17 slices are ana-
lyzed); this includes the calculation of all the
parameters involved in the contrast mappings. The
time spent by the neuroanatomist is dramatically
reduced, for she/he may employ almost three days
per lobe. However, the obtainment of opening and
closing sizes, as well as alpha and beta parameters
that control the contrast mapping are time consum-
ing, mainly alpha and beta parameters, since they
are derived from a set of images which must be
analyzed to select the image presenting the best
visual contrast related to the contrast measure. In
this case, the time employed for detecting such
parameters is approximately 15 min. This is the
main drawback when white and grey matter are
segmented.
On the other side, when noisy MRI is processed,
our method fails. This is because the quantifying
contrast method is defined as a function of the
morphological gradient (see Eq. (21)). However, by
introducing a pre-processing step to suppress noise
components, followed by the procedure discussed in
this paper to segment white and grey matter,
acceptable segmentations were obtained. Never-
theless, some noise borders can be appreciated in
the output images. Better segmentations will be
obtained if an improved pre-processing step to
suppress noise is applied.
Acknowledgments
The author Jorge D. Mendiola Santiban
˜ez thanks
CONACyT Me
´xico for financial support. The
author I. Terol would like to thank Diego Rodrigo
ARTICLE IN PRESS
Fig. 18. (a) Original image containing 9 flat zones; (b) region adjacency graph; and (c) output image obtained by an erosion size 1 on the
partition. Notice that the flat zones are extended.
J.D. Mendiola-Santiban˜ez et al. / Signal Processing 87 (2007) 2125–21502148
and Darı
´o T.G. for their great encouragement. This
work was partially funded by the government
agency CONACyT (Mexico) under the grant 41170.
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