Openings and closings with reconstruction criteria: a ... · Openings and closings with reconstruction criteria: a study of a class of lower and upper ... phological closings with

Journal of Electronic Imaging 14(1), 013006 (Jan–Mar 2005)

Openings and closings with reconstructioncriteria: a study of a class of lower and upper

levelingsIvan R. Terol-Villalobos

Centro de Investigacioń y Desarrollo Tecnolo´gico en Electroquı´micaParque Tecnolo´gico Quere´taro

Sanfandila-Pedro EscobedoCP.76700-APDO 064, Quere´taro, Mexico

E-mail: [email protected]

Damian Vargas-VazquezCentro Universitario

Facultad de Ingenierıá de la Universidad Autońoma de Quere´taroCerro de las Campanas S/N, APDO

76000, Qro. Me´xico

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Abstract. A study of a class of openings and closings is investi-gated using reconstruction criteria. The main goal in studying thesetransformations consists of eliminating some inconveniences of themorphological opening (closing) and the opening (closing) by recon-struction. The idea in building these new openings and closingscomes from the notions of filters by reconstruction and levelings. Inparticular, concerning the notion of levelings, a study of a class oflower and upper levelings is carried out. The original work of level-ings is due to Meyer, who proposes this notion and introduces somecriteria to build the levelings in the general case (extended levelingsand self-dual transformations). We see the criteria proposed byMeyer as reconstruction criteria during the reconstruction processfrom a marker image into the reference image. We show that newopenings and closings are obtained, enabling intermediate resultsbetween the traditional opening (closing) and the opening (closing)by reconstruction. Some applications are studied to validate thesetransformations. © 2005 SPIE and IS&T. [DOI: 10.1117/1.1866149]

1 Introduction

Morphological filtering is one of the most interesting sujects of research in mathematical morphology~MM !. Mor-phological filters are nonlinear transformations that locamodify geometric features of images. The basic morplogical filters are the morphological openings and the mphological closings with given structuring elements. In geeral, this element is a set that describes a simple shapeprobes the image and provides information about it.using basic filters, we can build others with different filteing characteristics~e.g., alternating filters, sequential altenating filters, morphological center, etc!. These filterspresent several inconveniences. In general, if the undeable features are eliminated, the remaining structuresbe changed. Recently, filters by reconstruction~a class of

Paper 020076 received Oct. 12, 2000; revised manuscript received Jan. 24,accepted for publication May 14, 2004; published online Feb. 24, 2005.1017-9909/2005/$22.00 © 2005 SPIE and IS&T.

01300Journal of Electronic Imaging

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geodesic transformations! have become powerful tools thaenable us to eliminate undesirable features without affeing desirable ones. Intensive work has been done oncharacterization of these transformations~see Serra andSalembier,1 Crespoet al.,2 and Serra3 among others!. Thesetransformations by reconstruction, which form a classconnected filters, involve not only the homotopy modifiction but also, opening, closing, alternated filters, alternatsequential filters, and even new transformations calevelings.4 In particular, the morphological connected filtehave been studied and characterized in the binary case~seeSerra and Salembier1 and Crespo5!. However, as expresseby Serra,3 since connected filtering has proved its efficienin image segmentation, coding, and motion prediction, ia paradox that it uses an underlying axiomatic for conntivity that is strictly binary. Serra studied connectivitwithin the framework of lattices. Other interesting studiconcerning connectivity were made by Heijmans6 and byRonse.7 Heijmans introduces the notion of grain operatowhile Ronse gave another family of axioms to characterconnectivity in terms of separating pairs of sets and coparing them to those proposed by Serra.8 The works ofHeijmans and Ronse are also concerned with binstudies.

From a practical point of view, filters by reconstructioare built by means of a reference image~mask! and amarker image included in the reference. The marker imgrows by iterative geodesic operations, while stayingways inside the reference image. When building filtersreconstruction~by iterating geodesic dilations until stabiity!, this geodesic term is generally avoided and the notof a connected component is introduced to describe thfilters. For a complete study of reconstruction transformtions see the work of Vincent,9 where the author presents

2;

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formal definition of gray-scale reconstruction and proposeveral efficient algorithms.

One of the most interesting characteristics of filtersreconstruction is that they enable the complete extractiothe marked objects by preserving the edges.10 Not only arethe contours preserved, but also no new regional extreare created by these transformations.4 The merging and ex-tinction processes of extrema can be studied by meancritical functions.11 Due to these interesting filtering chaacteristics, another approach for characterizing connefilters was presented by Meyer.4,12 By defining monotoneplanings and flattenings and by combining both notiolevelings are introduced; these definitions enable us to han algebraic framework for studying connected filters ingray-level case. Serra13 extended these concepts by meaof a marker approach and the activity mappings notion

In this paper, the concept of levelings is used to bunew openings and closings that enable us to obtain inmediate results between the morphological opening~clos-ing! and the opening~closing! by reconstruction. The interest of obtaining intermediate results is to avoid tinconveniences of the morphological opening~closing! andthe opening~closing! by reconstruction. While the morphological opening modifies the remaining structures,opening by reconstruction sometimes reconstructs undable regions during the reconstruction process. This ismain reason to introduce the concept of reconstructionteria. All the work is presented in the discrete case amore specifically with real-valued images.

This paper is organized as follows. Section 2 presethe concepts of filters by reconstruction, connectivity claand lower and upper levelings. In Sec. 3, we modify socriteria for constructing levelings to propose openings aclosings. Since the construction of these transformatirequires the morphological opening and the opening byconstruction, some comments relating the behavior ofoperators studied here with those of the morphologopening and opening by reconstruction are presenteSec. 4. Finally, Sec. 5 presents two applications that illtrate the interest of these transformations.

2 Some Concepts of Morphological Filtering

We next describe some basic concepts of morphologfiltering that are useful in this paper.

2.1 Basic Notions of Morphological Filtering

Morphological filters are increasing and idempotent traformations. The basic morphological filters are the morplogical openinggmB and the morphological closingwmB

with a given structuring elementmB, where in this paper,Bis an elementary structuring element~333 pixels! that con-tains its origin. Note thatB is the transposed set (B5$2x:xPB%) and m is a homothetic parameter. In thpaper, the homothetic parameter takes only integer valThe morphological opening is an antiextensive filter athe morphological closing is an extensive filter. Thetransformations are expressed by means of the morphocal dilationdmB and morphological erosionemB . Thus,

gmB~ f !~x!5dmB@emB~ f !#~x! and


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wmB~ f !~x!5emB@dmB~ f !#~x!, ~1!

where the morphological erosion and dilation are resptively expressed by

emB@ f ~x!#5∧$ f ~y!;yPmBx% and

dmB@ f ~x!#5∨$ f ~y!;yPmBx%,

where∧ is the inf operator~∨ is the sup operator!.

Remark. The structuring element used in this papermBis the elementary structuring elementB given by a set com-posed of 333 pixels centered at the origin affected byhomothetic parameterm>0. Thus,mB is a set formed by~2m11!3~2m11! pixels centered at the origin. Whenm50,the structuring elementmB is a set made up of one poin~the origin!. Negative values ofm are forbidden. In thefollowing, we avoid the elementary structuring elementB.The expressionsgm and gmB are equivalent~i.e., gm

5gmB) and a similar situation exists for all transformtions. When the homothetic parameter ism51, the structur-ing elementB is also avoided~i.e., dmB5dB5d).

2.2 Filters by Reconstruction and ConnectedClass—Opening and Closing byReconstruction

2.2.1 Filters by reconstruction

The notion of reconstruction is a very useful concept pvided by MM. The reconstruction transformations are conected filters that enable us to modify minima or maximwithout considerably changing the structure of the remaing components. Geodesic transformations are used to bthe reconstruction transformations. In the binary case,geodesic dilation of size 1 of a setY inside the setX isdefined as

dX1~Y!5XùdB~Y!5Xùd~Y!.

To build a geodesic dilation of sizem, the geodesic dilationof size 1 is iteratedm times; that is,

dXm~Y!5dX

1dX1¯dX

1~Y!m times

.

Observe that geodesic dilations are extensive transfortions if the origin is contained in the structuring elemei.e., Y#dX

m(Y). In Fig. 1~a! the marker image in whitecolor and the reference image in gray color are illustratwhile Figs. 1~b! and 1~c! show geodesic dilations of size 6and 230, respectively. When filters by reconstructionbuilt, the geodesic transformations are iterated until idepotence is reached@see Fig. 1~d!#. Then, by iterating suc-cessive elementary geodesic dilations of a setY inside amaskX, the connected components ofX whose intersectionwith Y is nonempty are progressively flooded. The dutransformation ofdX

m(Y) is the geodesic erosioneXm(Y),

which is built by iteratingm times the geodesic erosion osize 1. The reconstruction transformation in the gray-lecase is a direct extension of the binary one. Consider

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Openings and closings . . .

functions f and g, with f >g ( f <g). The reconstructiontransformations by geodesic dilation and geodesic erosexpressed, respectively, byR( f ,g) andR* ( f ,g) of f from gare defined by

R~ f ,g!5 limn→`

d fn~g!5d f

1d f1¯d f

1~g!until stability

,

~2!

R* ~ f ,g!5 limn→`

e fn~g!5e f

1e f1¯e f

1~g!until stability

,

where e f1(g)5 f ∨eB(g) is the geodesic erosion, an

d f1(g)5 f ∧dB(g) is the geodesic dilation that are obtaine

by means of morphological dilation and erosion. Whenfunction g is equal to the erosionem( f ) or to the dilationdm( f ) of the original function by a given structuring element, we obtain the opening and the closing by reconstrtion:

gm~ f !5 limn→`

d fn@em~ f !#5d f

1d f1¯d f

1@em~ f !#Until stability

,

wm~ f !5 limn→`

e fn@dm~ f !#5e f

1e f1¯e f

1@dm~ f !#Until stability

. ~3!

Figures 2~b! and 2~c! illustrate the morphological opening gm and the opening by reconstructiongm of the imagein Fig. 2~a!. Observe the great difference between boopenings.

Fig. 1 (a) Mask image in gray color made up of two connectedcomponents and the marker in white color; (b) and (c) geodesicdilations of size 60 and 230, respectively, of the marker inside themask; and (d) geodesic dilations until stability of the mark inside themask (reconstruction transformation).


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2.2.2 Connected class—the pointwise opening gx

Consider the arc connection inZ2 ~or in R2). A setX,Z2

~or X,R2) is connected, if for all pairs of pointsx, y in X,there exists a path linkingx andy, that is included inX. InSerra,8 connectivity is generalized by the introduction ofconnectivity class given from the following definition:

Definition 1. ~Connectivity class!. A connectivity classCis defined on the subsets of a setE @`(E)# when

1. BPC and for allxPE, $x%PC.

2. For each family$Ci% in C. ùiCiÞB⇒ø

iCiPC.

This definition is equivalent to the definition of a familof connected pointwise openings$gx ,xPE% associated toeach point ofE.

Theorem 1. ~Connectivity characterized by openings!.The definition of a connectivity classC is equivalent to thedefinition of a family of openings$gx ,xPE% such that

1. ;xPE, gx($x%)5$x%.

2. ;x, yPE and A#E, gx(A) and gy(A) are eitherequal or disjoint, i.e.,gx(A)5gy(A) or gx(A)ùgy(A)5B.

3. ;xPE andA#E, x¹A implies gx(A)5B.

When the operationgx is associated with the usual connectivity in Z2, the openinggx(A) can be defined as theunion of all paths that containx and that are included inA.Thus, when a spaceE is equipped with the openinggx ,connectivity issues inE can be expressed usinggx . A setA,Z2 is connected if and only ifgx(A)5A for all x in A.General relationships for the opening and closing by recstruction using some increasing criteria can be expressemeans of the notion of a connectivity class.1,5

2.3 Lower and Upper Levelings

Another characterization of the connectivity on discregray-level images was made by Meyer.4,12 The family ofconnected filters, called filters by reconstruction, has imptant features: the filters of this family never create regionextrema~minima or maxima!, and contours are preservedThese important characteristics, originated the proposit

Fig. 2 (a) Original image and (b) and (c) morphological opening gm

and opening by reconstruction gm , with m56.

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of an algebraic framework, made by Meyer,4,12 to charac-terize these filters. The following definitions are proposby Meyer.12

Definition 2. An imageg is a leveling of the imagef ifand only if ;(p,q) neighbors:

g~p!.g~q!⇒ f ~p!>g~p! and g~q!> f ~q!.

The set of neighbors of a pixel are considered withspect to the underlying grid of square or hexagonal typetwo dimensions. In this paper, we consider the lower aupper levelings, defined next.

Definition 3. A function g is a lower leveling of a func-tion f if and only if ;(p,q) neighbors:

g~p!.g~q!⇒g~q!> f ~q!.

Definition 4. A function g is an upper leveling of a function f if and only if ;(p,q) neighbors:

g~p!.g~q!⇒g~p!< f ~p!.

The criteria for building lower and upper levelings agiven by the following criteria:

Criterion 1. A functiong is a lower leveling of a functionf if and only if @g> f ∧d(g)#.

Criterion 2. A function g is an upper leveling of a function f if and only if @g< f ∨e(g)#.

In criteria 1 andd ande are the morphological dilationand erosion by an elementary structuring element, coposed by 333 pixels centered at the origin. Criteria 1 andenable the generation of algorithms for producing lowand upper levelings. Given two functionsg andf, we trans-form g into a lower leveling~resp. upper leveling! of fusing the following algorithm:

while g, f do g85 f ∧d~g!,

@resp. while g. f do g85 f ∨e~g!#,

until criterion 1 ~resp. criterion 2! is satisfied everywhereBy comparing the opening~resp. the closing! by recon-struction given by Eq.~3! with the criterion 1~resp. crite-rion 2!, it is possible to observe that the opening~resp.closing! by reconstruction is a lower leveling~resp. an up-per leveling!. For the opening by reconstructiong5em( f ), f we do g85 f ∧d(g) until criterion 1 is satisfied:g5 f ∧d(g). At each iteration, the outputg85 f ∧d(g) be-comes the new input functiong5g8. In other terms, usinggeodesic dilations we do, at stepk11, f ∧d@d f

k(g)# until ata given stepn.k, d f

n(g)5 f ∧d@d fn(g)#. In fact, Meyer12

showed that an opening by reconstruction is a lower leving and a closing by reconstruction is an upper levelingthe same work,12 extended levelings are defined using geeral connected notion. In particular, levelings are propoby Meyer using the following criteria:


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Criterion 3. A function g is a lower leveling~resp. anupper leveling! of a function f if and only if $g> f ∧@g∨gld(g)#% ~respectively$g< f ∨@g∧wle(g)#%).

Criterion 4. A function g is a lower-leveling~respectivelyan upper-leveling! of a function f if and only if: @g> f ∧(g∨dgl(g))# ~respectively@g< f ∨(g∧ewl(g))#).

Criterion 3~similar for criterion 4! enables the determination of algorithms for producing extended lower and uper levelings. According to criterion 3, given two functiong and f, we transformg into a extended lower leveling~resp. extended upper leveling! of f:

while g, f do g85 f ∧@g∨gld~g!#,

$resp. while g. f do g85 f ∨@g∧wle~g!#%,

until criterion 3 is satisfied everywhere.In this paper, we focus mainly in criteria 3 and 4. Crit

rion 3 was used in Meyer and Maragos,14 while criterion 4was proposed by Meyer.12

3 Openings and Closings with ReconstructionCriteria

The main drawbacks of the morphological opening~clos-ing! and the opening~closing! by reconstruction were already described. Remember that with the opening by recstruction the elimination of some structures cannotachieved~it reconstructs all connected regions during treconstruction process!. This inconvenience called leakagwas reported by Salembier and Oliveras.15 The notion ofpseudoconnectivity was introduced to avoid this probleThe goal in this section is to introduce an opening~and aclosing! that enables us to obtain intermediate resultstween the morphological opening~closing! and the opening~closing! by reconstruction. The process to build these ntransformations involves the use of a reference image amarker image, as in the reconstruction transformatiocase. Thus, a reconstruction process of the reference imby means of geodesic dilations~or geodesic erosions! isused, but a criterion is taken into account that restrictsreconstruction of some regions. We present only the casthe new openings, but a similar procedure can be madethe closings.

3.1 Openings (Closings) with ReconstructionCriteria

First, let us analyze criterion 3, which establishes thatg isan extended lower leveling of a functionf if and only if g> f ∧@g∨gld(g)#. To transform a functiong into a lowerleveling of a function f, we iterate the relationshipg85 f ∧@g∨gld(g)# until criterion 3 is satisfied. Howeverthis expression will be simplified by imposing a conditioto functiong. The interest of this simplification is to buildopenings~resp. closings! that enable us to overcome thinconveniences of the opening~resp. closing! by recon-struction. As already expressed, the marker grows by iating geodesic dilations inside the reference. Thus, inopening by reconstruction@by iteratingg85 f ∧d(g) untilstability#, the reconstruction of the referencef from themarker g is made without restrictions@see Figs. 1~b! to

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1~d!#. Therefore, the goal is to restrict this reconstructiaccording to a size criterion and to avoid the reconstrucof some regions. It is shown later that the algorithmproducing an extended lower leveling, by iteratingg85 f ∧@g∨gld(g)#, can be seen as a reconstruction procof f from g with restrictions.

Since the termgld(g) of the expressionf ∧@g∨gld(g)#plays the main role in the reconstruction process ofmarkerg @in a manner similar to the dilationd(g) of ex-pression f ∧d(g) for the opening by reconstruction#, wemainly analyze this term along the following lines. Letfandg be the reference and the marker images, respectivSincegl is anti-extensive, the following inequality can bestablished:

gld~g!<d~g!.

However, nothing can be expressed forgld(g) andg. Forl50, only the following relationship is verified:

g<gld~g!5d~g!.

However, forl>1, even if the inequalitygld(g)<d(g) issatisfied, the equationg<gld(g) is not necessarily trueMoreover, it would be interesting to know how the behaior of the successive iterations ofgld(g) is when com-pared to the successive iterations ofd(g). To better under-stand the behavior of thegld(g), let us consider someconditions for the marker imageg. The function given byg5gm( f ) is selected as the marker image and not the esion function as it is the case of the opening by reconstrtion. However, an opening by reconstruction can be aobtained using the morphological opening as the marimage. In fact, sincegm( f )< f , the same result can be obtained, by dilatingm timesem( f ) using elementary morphological dilationsd, or by dilatingm timesem( f ) using geo-desic dilations of size 1, i.e.;

gm~ f !5dd¯d@em~ f !#m times

5d f1d f

1¯d f

1@em~ f !#m times

.

This means:

gm~ f !5d f1d f

1¯d f

1

Until stability$d f

1d f1¯d f

1@em~ f !#m times

%

5d f1d f

1¯d f

1@gm~ f !#Until stability

,

gm~ f !5 limn→`

d fn@em~ f !#5 lim

n→`

d fn@gm~ f !#.

Then, a similar result can be obtained when the morplogical erosion or the morphological opening by a structing elementmB is used as a marker image. On the othhand, the morphological opening and the morphologicallation satisfy the following property~Serra8!:

Property 1. For all pair of parametersl1 andl2, withl1<l2, dl2( f )5gl1@dl2(f) #.


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By applying this property togld(g) usingg5gm( f ) asthe marker, we get;l<m11 ~negative values of the homothetic parameters are forbidden!:

gld@gm~ f !#5glddmem~ f !5gldm11em~ f !,

then

gld@gm~ f !#5gldm11em~ f !5dm11em~ f !5d@gm~ f !#,

and

gm~ f !,gld@gm~ f !#5d@gm~ f !#.

Specifically, the equation gld@gm( f )#5d@gm( f )#means that when the marker image is given byg5gm( f )for ;l<m11, the output images of the successive itetions of expressiongld(g) are similar to those of the successive iterations ofd(g), and we haveg,gld(g)5d(g). Thus, the termg∨gld(g) of f ∧@g∨gld(g)# incriterion 3 is simplified by

g∨gld~g!5gm~ f !∨gldgm~ f !5gm~ f !∨dgm~ f !5dgm~ f !,

since;l, l<m11, gm( f ),gld@gm( f )#5d@gm( f )#.Then, we work withvl, f

1 (g)5 f ∧gld(g) with the con-dition g5gm( f ). This operatorvl, f

1 is iterated until stabil-ity is reached, as it is the case of the reconstruction traformations, i.e.,

limn→`

vl, fn ~g!5vl, f

1 vl, f1

¯vl, f1 ~g!

Until stability

.

We illustrate later that the function values of the successoutput images of the transformationgld(g) in vl, f

1 (g)5 f ∧gld(g) increase or stop only wheng5gm( f );i.e., gld@vl, f

k (g)#>vl, fk (g). Then the sup operation

@g∨gld(g)# in f ∧@g∨gld(g)# does not affect the outpuimages under successive applications, thus, the simplexpressionvl, f

1 (g)5 f ∧gld(g) is used. A similar study en-ables the simplification off ∨@g∧wle(g)# in criterion ~3!by al, f

1 (g)5 f ∨wle(g) with the conditiong5wm( f ).We have shown that the expressiongld(g) and d(g)

with g5gm( f ) for ;l<m11, have a similar behavior under the successive iterations; however, when the refereimage is used@ f ∧gld(g)#, the reconstruction proceschanges and the openinggl plays the role to restrict thereconstruction to some regions of the image. Then, wvl, f

1 (g) is iterated until stability, an opening and a closinof sizem are obtained.

Proposition 1. Let gl and wl be the morphologicalopening and closing of parameterl, respectively. Thetransformations given by the following relations:

gl,m~ f !5 limn→`

vl, fn @gm~ f !#5vl, f

1 vl, f1

¯vl, f1 ~gm~ f !!

Until stability

and

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wl,m~ f !5 limn→`

al, fn @wm~ f !#5al, f

1 al, f1

¯al, f1 @wm~ f !#

Until stability

are an opening and a closing of sizem, respectively,where vl, f

1 @gm( f )#5 f ∧gld@gm( f )# and al, f1 @wm( f )#

5 f ∨wle@wm( f )#;l<m11.Now, let us compare the expressionf ∧@g∨dgl(g)#

used in criterion 4 with the one used by criterionf ∧@g∨gld(g)#. It is possible to show that the output images of the successive iterations ofgld(g) are similar tothose ofd(g) usingg5gm( f ) as the marker.

Since glgm( f )5gmgl( f )5gmax(m,l)(f), wheremax~m,l! is the maximum value ofm andl,

dgl~g!5dgl@gm~ f !#5dgm~ f ! ;l<m.

Also, the expressionf ∧@g∨gld(g)# can be simplified byf ∧dgl(g) with g5gm( f ). Thus, the output images of thoperationsdgl(g) andgld(g) are similar tod(g) with theconditions ;l<m and ;l<m11, respectively. Howeverby iterating both expressions,f ∧dgl(g) and f ∧gld(g)until stability, different output images are obtained. Thuanother opening~also a closing!, as expressed by propostion 1, can be established usingf ∧dgl(g).

Proposition 2. Let gl and wl be the morphologicaopening and closing of parameterl, respectively. Thetransformations given by the following relations:

g&l,m~ f !5 limn→`

vl, fn @gm~ f !#5vl, f

1 vl, f1

¯vl, f1 @gm~ f !#

Until stability

,

w& l,m~ f !5 limn→`

al, fn @gm~ f !#5al, f

1 al, f1

¯al, f1 @wm~ f !#

Until stability

,

are an opening and a closing of sizem, respectively,where vl, f

1 @gm( f )#5 f ∧dgl@gm( f )# and al, f1 @wm( f )#

5 f ∨ewl@wm( f )#;l<m.Henceforth, the notationvl, f

1 @gm( f )# is invariably beused forf ∧gld@gm( f )# and f ∧dgl@gm( f )#, and similarlyfor the closingal, f

1 @wm( f )#. Figures 3~a! to 3~d! illustratethe openingsgl,m , g&l,m , gm , andgm , respectively, of theoriginal image in Fig. 2~a!, while Figs. 3~e! to 3~h! showthe tophat transformations computed for the three opengl,m , g&l,m , gm , and gm . The binary images are obtaineby a threshold of the arithmetical difference betweenoriginal image and each opening.

3.2 Marker and Criteria Option

To analyze the behavior of the termgld(g), a conditionwas imposed to the marker imageg ~Sec. 3.1!. The choiceof g5gm( f ) instead ofg5em( f ) enables us a better controof the output image. Let us study the behavior off ∧gld(g)by consideringg5em( f ) as the marker function. That is,

gld~g!5gld@em~ f !#5glgem21~ f !5glem21~ f !

5d~dl21el21!eem21~ f !5dgl21em~ f !.


s

According to the expressiongld(g)5dgl21em( f ), atthe first step of the transformationf ∧gld@em( f )#, an open-ing of sizel21 is applied to the markerg5em( f ). Then,the marker of the transformation given byem( f ) is changedby em1l21( f ), which means that some components of toriginal marker@em( f )# can be eliminated at the first iteration. Thus, the notion of markerg is lost @marker given bya size criterionem( f )] and the true marker is given by acombination between the markerg5em( f ) and the openinggl21 .

Now, let us analyze both openingsgl,m( f ) andg&l,m( f )to better understand the behavior from a geometrical poof view ~at the pixel level!. The following study is made inthe binary case. First, let us study the behavior of the oping gl,m( f ) using expressionXùgld@gm(X)#. We have atstepk before stability:

vl,Xk @gm~X!#5Xùgld$vl,X

k21@gm~X!#% ;l<m11.

In this case, the output imagevl,Xk @gm(X)# is composed of

the union of the setvl,Xk21@gm(X)# and the pointsx

PX $x¹vl,Xk21@gm(X)#% such that for eachx there exists a

point y with lBy,d$vl,Xk21@gm(X)#% andxPlBy .

Figure 4 shows a simple example that illustrates the dferent steps of the transformation. Figures 4~a! and 4~b!show the original input setX and the openinggm(X) of size

Fig. 3 (a) to (d) Openings gl,m , g&l,m , gm , and gm for m56 and l53and (e) to (h) tophat transformations for the openings gl,m , g&l,m ,gm , and gm .

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m52 in dark gray color. Figures 4~c! to 4~e! illustrate thereconstruction process steps 1, 2, and 3, respectively,l51. In Fig. 4~h! the structuring elementlBy5By ~l51!at pointy in dark gray color and its dilated setd(lBy) inlight gray color are illustrated. At step 1, the input svl,X

0 @gm(X)#5gm(X) is dilated, and then tested by the slB. The points in light gray color~three points! in Fig. 4~c!hit the structuring elementlBy,d@gm(X)# and they areadded togm(X) to obtain vl,X

1 @gm(X)#; while at step 2,using the same procedure, only two points@light graycolor in Fig. 4~d!# are added tovl,X

1 @gm(X)# to formvl,X

2 @gm(X)#. At step 3@Fig. 4~e!#, the stability is reachedby adding the pointsx1 , x2 , and x3 to vl,X

2 @gm(X)# toform vl,X

3 @gm(X)#.Now, consider the openingg&l,m( f ), in the binary case,

using expressionXùdgl@gm(X)#. At stepk before stabil-ity, we have

vl,Xk @gm~X!#5Xùdgl$vl,m

k21gm@~X!#% ;l<m.

The transformation Xùdgl$vl,Xk21@gm(X)#% interacts

with X in such a way that some regions ofX are selectedto form the output image. The output imagevl,X

k @gm(X)#is composed, according to the expressiXùdgl$vl,X

k21@gm(X)#%, by the union of the setvl,X

k21@gm(X)# and the pointsxPX$x¹vl,Xk21@gm(X)#% such

that there exists a pointy with lBy,vl,Xk21@gm(X)# andx

Pd(lBy).Figures 4~f! and 4~g! illustrate the reconstruction pro

cess, steps 1 and 2, respectively, withl51. At step 1, theinput set vl,X

0 @gm(X)#5gm(X) is tested bylB and thepoints in light gray color~three points! in Fig. 4~f! that hitthe dilated of the translated structuring element at poy @d(lBy)# are added togm(X) to obtain vl,X

1 @gm(X)#;while at step 2, using the same procedure, only two po@light gray color in Fig. 4~g!# are added tovl,X

1 @gm(X)# to

Fig. 4 (a) Original image X; (b) opening gm(X) for m52, (c), (d), and(e) output images vl,X

1 @gm(X)#, vl,X2 @gm(X)#, and vl,X

3 @gm(X)#, re-spectively, using Xùgld@gm(X)# with l51; (f) and (g) output imagesvl,X

1 @gm(X)# and vl,X2 @gm(X)#, respectively, using Xùdgl@gm(X)#

with l51; and (h) structuring elements lBy and d(lBy).


h

t

obtain vl,X2 @gm(X)#. At step 2, the stability is reached

because it is not possible to find a pointy withBy,vl,X

2 @gm(X)# such that at least one of the pointsx1 ,x2 , andx3 belongs tod(By).

Using this previous analysis, let us illustrate the diffeences between both openings. Figure 5~a! shows the origi-nal image, while Fig. 5~b! illustrates the binary image obtained by a threshold of the original image between 80 a255 gray levels. The morphological opening and the oping by reconstruction, withm57, are illustrated in Figs.5~c! and 5~d!. Finally, the openingsgl,m and g&l,m withparametersm57 andl53, were used to compute the outpuimages in Figs. 5~e! and 5~f!, respectively. Notice that whengl,m is used, the small holes do not considerably affethe reconstruction. Intuitively, this behavior is due to thconditions required to add a pointxPX to the output im-age. At stepk we have that a pointxPX, such thatx¹vl,X

k21@gm(X)#, belongs tovl,Xk @gm(X)# if there exists

lBy,d$vl,Xk21@gm(X)#% andxPlBy .

This means that at stepk, the output image at stepk21 is dilatedd$vl,X

k21@gm(X)#%, and some small holes arfilled before they are tested bylBy . Then, it seems that theopeninggl,m has a better behavior than the openingg&l,m ,but this depends on practical problems. For instance, inexample in Fig. 6, the typical problem reported by Salebier and Oliveras,15 of connected operators is illustratedUsing the morphological closing the word LETTER waeliminated but the remaining structures are modified@seeFig. 6~b!#; while using the closing by reconstruction thremaining structures are not changed but the word hasbeen eliminated@see Fig. 6~c!#. Now, by applying the clos-ings with reconstruction criteriawl,m andw& l,m the word iseliminated without considerably changing the remainistructures@see Figs. 6~d! and 6~e!#. However, the closingw& l,m better eliminates the word LETTER than the closinwl,m . Furthermore, the openingg&l,m and the closingw& l,mhave other properties that are described in the following

Fig. 5 (a) Original image, (b) threshold of the original image, (c) and(d) morphological opening gm and opening by reconstruction gm withm57, (e) opening gl,m with m57 and l53, and (f) opening g&l,m withm57 and l53.

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4 Some Comments About the Openings withReconstruction Criteria

4.1 Inclusion Relations between the Openings gm ,g&l,m , gl,m , and gm

The extreme values forl in the openingg&l,m ~l50 andl5m! are well-defined. Observe that forl50, the expres-sion vl, f

1 @gm( f )#5 f ∧dgl@gm( f )# becomesv0,f1 @gm( f )#

5 f ∧d@gm( f )#, which is the geodesic dilation of size 1Thus,

g&0,m~ f !5 limn→`

v0,fn @gm~ f !#5 lim

n→`

d fn@gm~ f !#5gm~ f !.

Now, for l5m, at the first iteration we havevm, f1 @gm( f )#

5 f ∧dgm@gm( f )#5 f ∧d@gm( f )#. Then, gm( f )<vm, f

1 @gm( f )#5 f ∧d@gm( f )#5d f1@gm( f )#< f . At the sec-

ond step,vm, f2 @gm( f )#5 f ∧dgm@d f

1gm( f )# and, sincegm isa strong filter we obtain vm, f

2 @gm( f )#5 f ∧dgm( f )5d f

1gm( f ). We remember that a morphological filterC issaid to be strong if it satisfies the following robustness codition:

; f ,g C~ f !∧ f <g<C~ f !∨ f⇒C~ f !5C~g!.

Thus, for l5m, g&l,m5d f1@gm( f )# and an ordering can be

established between the openings

; f and ;l<m gm~ f !<g&l,m~ f !<gm~ f !.

In this manner, the openingg&l,m enables us to obtainresults intermediate betweengm and gm . This is a mostinteresting property, because the inconvenience of the csical openinggm and that of the opening by reconstructiogm can be overcome. Now, by analyzing the openinggl,m ,we observe that the extreme values are not well-defin

Fig. 6 (a) Original image, (b) morphological closing wm with m57,(c) closing by reconstruction wm with m57, (d) and (e) closings withreconstruction criteria wl,m , w& l,m , respectively, with m57 and l53.


-

.

For l50, the expressionvl, f1 @gm( f )#5 f ∧gld@gm( f )# be-

comesv0,f1 @gm( f )#5 f ∧d@gm( f )#, which is the geodesic

dilation of size 1. Thus,

g0,m~ f !5 limn→`

v0,fn @gm~ f !#5 lim

n→`

d fn@gm~ f !#5gm~ f !.

However, also forl51, the same output image is obtaineFor l51, the output image at stepk is given byvl, f

k @gm( f )#5 f ∧gd$v1,fk21@gm( f )#%. Using property 1,

vl, fk @gm( f )#5 f ∧d$v1,f

k21@gm( f )#%, which is the geodesicdilation of size 1. Finally, observe that this opening is dfined for l<m11, which means thatl can take a valuegrater thanm. Nevertheless, if the extreme values ofl arenot well-defined, the following relation betweengm , gl,m ,and gm can be established as for the openingg&l,m , i.e.,

; f and ;l<m11 gm~ f !<gl,m~ f !<gm~ f !.

4.2 Segmentation and Connectivity

The goal in binary image segmentation is to split the conected components into a set of elementary shapes. Inlembier and Oliveras,15 the segmentation by openings usinthe connectivity opening proposed by Ronse7 has been re-ported.

4.2.1 Segmentation by openings

Given a family of connected pointwise openingsgx , andan openinggm , a new family of connected pointwise openings sx can be created by the following rule:

a. sx~X!5gx@gm~X!# if xPgm~X!,

b. sx~X!5$x% if xPX/gm~X!,

c. sx~X!5B if x¹X.

The opening gm must verify the property gmgxgm

5gxgm . In other words, every connected component ofinvariant ofgm is itself and the invariant ofgm . The mainproblem of this approach is that in practice this waysegmenting the connected components, using the morlogical opening, leads to a loss of the shape informationthe remaining structures. This drawback can be attenuusing the openings with reconstruction criteria as illustraby Fig. 6, in particular, using the opening of propositionsince the opening given by proposition 1 does not satthe condition to create this connected pointwise openingfact, during the reconstruction process for obtaininggl,m ,the input set is dilated at each iteration before it is testedlB, and the reconstruction process of two or more conected components can interact to obtain the output imaThis is not the case of the openingg&l,m since at each stepthe input set is tested bylB before the structuring elemenis dilated. Let us illustrate this behavior by analyzing toperatorvl,X

1 when it is iterated to buildg&l,m .A set X is arcwise connected if any pair of pointsx, y

PX is linked by a path, entirely included inX. Where apath between two pixels of cardinalm is an m-tuple ofpixelsx0 , x1 ,...,xm such thatx5x0 andy5xm with xk and

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xk11 neighbors for allk. Similarly, a path to characterizethe connected components ofg&l,m can be defined undesome conditions. It is interesting to observe that the tralates oflB, insidevl,X

k21@gm(X)#, that hit the boundaries othe setvl,X

k21@gm(X)# enable us to decide which points aadded to the setvl,X

k21@gm(X)# to form the setvl,X

k @gm(X)#. Therefore, a pointyPX with y¹gm(X) isachieved at stepm by the reconstruction process cominfrom gm(X), if there exits a chain of pointsx0 , x1 ,...,xm ,with xk and xk11 neighbors for all k, such that ;k,lBxk,vl,X

k @gm(X)# $vl,X0 @gm(X)#5gm(X)% and lBxk

hits the boundaries ofvl,Xk @gm(X)#, andyPd(lBxm). Ob-

serve that not only the size of the structuring elemenlplays a fundamental role in the reconstruction process,also the markergm(X), as illustrated by the following ex-ample in Fig. 7. The goal is to extract the leaf~tomato plantleaf! in the original image in Fig. 7~a!. After a binarizationstep of the original image@Fig. 7~a!#, we obtain a setXformed by the leaf and other regions that are arcwise cnected to the leaf@Fig. 7~b!#. The morphological openingsgm1(X) andgm2(X) with m1582 andm25155 are shownin gray color in Figs. 7~c! and 7~d!, respectively. Using theopeningsg&l,m1 andg&l,m2 with l518 the same output im-age @g&l,m1(X)5g&l,m2(X)# is obtained@Fig. 7~e!#. This isnot the case of the output image in Fig. 7~f! computed byg&l,m usingm581 andl518. In this example~l518!, thesame output image is obtained byg&l,m(X) for m betweenm1582 andm25155, where the valuem25155 is the criti-cal element of the morphological opening, i.e.,gm2(X)ÞB andgm211(X)5B. Thus, the well-known property othe opening by reconstruction, expressing thatgxgm1(X)ÞB and gxgm2(X)ÞB⇒gxgm1(X)5gxgm2(X), isverified in this example byg&l,m for m betweenm1582and m25155 @gxgl,m1(X)ÞB and gxgl,m2(X)ÞB⇒gxg&l,m1(X)5gxg&l,m2(X)].

In the general case, we require some conditions impoto the input set as expressed by the following property.

Fig. 7 (a) Original image, (b) binary image, (c) and (d) morphologi-cal openings gm1(X) and gm2(X) in gray color with m1582 and m15155, (e) output image of the openings g&l,m1(X) and g&l,m2(X) withl518, and (f) output image g&l,m(X) with m581 and l518.


-

t

-

d

Property 2. Let A be an arcwise connected componesuch that;x, yPA there exists a pathx0 , x1 ,...,xm withxk andxk11 neighbors for allk, entirely included inA, withlBxk,A for all k and such thatxPd(lBx0) and yPd(lBxm). Then, ;m1 , m2 with l<m1<m2 such thatg&l,m1(A)ÞB, g&l,m2(A)ÞB⇒g&l,m1(A)5g&l,m2(A)5A.

5 Some Applications

5.1 Magnetic Resonance Imaging

To illustrate the interest of the opening by reconstructwith propagation criteria, we applied it when performinthe segmentation of magnetic resonance imaging~MRI! ofthe brain. MRI is characterized for its high soft tissue cotrast and high spatial resolution. These two properties mMRI one of the most important and useful imaging modaties in diagnosis of brain related pathologies. An automa3-dimensional~3-D! segmentation of magnetic resonan~MR! brain tissue using filters by reconstruction was invetigated elsewhere.16 In contrast, we illustrate the importance of the opening by reconstruction with propagatcriteria by comparing it with the opening by reconstructioThese transformations are applied in a 2-dimensional~2-D!case. The purpose of this procedure was to segmenaccurately as possible, the skull and the brain as well asbackground. Figures 8~a! to 8~i! illustrate the procedure. Toeliminate the skull from the image, an opening by recostruction of size 12 was applied@Fig. 8~b!#. In this stage,the skull gray level is attenuated and a simple threshenables us to obtain the brain shown in Fig. 8~c!. Now, byusing the same procedure, but this time applying the oping g&l,m with m512 andl54 to the image in Fig. 8~a!, asimilar result was obtained in Fig. 8~e! by thresholding theimage in Fig. 8~d!. However, because in some 2-D imagethin connections exist between skull and brain, in socases, it is impossible to separate both regions, as iltrated in Fig. 8~g!. Using classical connectivity, the operator processes the skull and the brain on the image in F8~f! as a single object and the connected operator recstructs ‘‘too much.’’ By applyingg&l,m to the image in Fig.8~f! this drawback is eliminated. Figure 8~h! illustrates thetransformationg&l,m with m512 andl54. In this case, it ispossible to separate the brain from the skull without cosiderably affecting the structure of the brain@Fig. 8~i!#. Thetransformationg&l,m was successfully tested with 15 MR2-D images. These experiments especially showed a robehavior to eliminate the skull. A slight difficulty was observed in a few cases when the size of the brain~2-D im-ages! is smaller than the skull. A 3-D approach will ovecome this problem.

5.2 Image Segmentation of the Illness Caused byPhytophthora infestans

A second example concerns the study of thePhytophthorainfestansillness evolution in the tomato plant. This studybased on the color space HLS. The three components oHLS space are respectively hue, luminance, and satura@Figs. 9~a! to 9~c!#. The luminance~L! component was usedfor detecting the leaves and the H and S componentsdetecting thePhytophthora infestansillness. To use the tra-ditional morphological image segmentation technique

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MM, the so-called watershed-plus-marker approach,17 a setof markers that signal each region of the image, mustdetermined. This step requires a complete study of imagA simpler algorithm can be used in this case. After a binrization step of the leaf shown in Fig. 9~d!, we not onlyobtain a setX composed by the leaf to be studied, but alother components that are arcwise connected to the@see Fig. 9~e!#. Because the leaf to be analyzed is tgreater component on the image, the critical element ofmorphological opening can be used as the marker. Thawe select them value such thatgm(X)ÞB and gm11(X)5B, as illustrated in Sec. 4.2. However, in practice, tnotion of distance function enables us to obtain an efficialgorithm for selecting a marker. Remember that the dtance function associates with each pixelp of X its distanceto the background. Then, a distance function of the binimageX in Fig. 9~e! was computed@see Fig. 9~f!#. Since theleaf under study is the greater region on the image,global maximum of the distance function must be insithis region. Thus, ifG is the global maximum value of thedistance function (G5144 in this example! andY is the setrepresenting the region of the global maximum, the marwill be given by M5dm(Y) with m5G21. In Fig. 9~g!,the marker in gray color is illustrated; while Fig. 9~h!shows the reconstruction ofX from M computed by

Fig. 8 (a) Original image; (b) opening by reconstruction gm withm512 of the image (a); (c) brain detection by threshold from theimage (b); (d) opening g&l,m , for m512 and l54 of the image in (a);(e) brain detection by threshold from image (d); (f) original image;(g) opening by reconstruction gm with m512 of the image in (f); (h)opening g&l,m , for m512 and l54 of the image in (f); and (i) braindetection by threshold from image (h).


.

f

e,

t-

r

vl,Xk (M ) ~until stability! using expression

Xùdgl@gm(M )# with l518. The valuel518 was calcu-lated by experimental tests using a set of images. Adetecting the tomato leaf in the luminance component@seeFig. 10~a!#, we reduce the problem of detecting thePhy-tophthora infestansillness by working within the tomatoleaf using HS information@see Fig. 10~b!#. Another proce-dure was used for this last step~illness detection!. Figure10~c! illustrates the detection of thePhytophthora infestansillness.

6 Conclusion

In this paper, a study of a class of transformations thatlower and upper levelings was presented. In the origiwork due to Meyer, the author puts forward some critefor constructing levelings. Here, some of these criteria wused to build new openings and closings. We showedthese openings and closings with reconstruction criteriaable us to obtain intermediate results between the morplogical openings~closings! and the openings~closings! byreconstruction. These transformations diminish someconveniences inherent to the morphological opening~orclosing! and the opening~or closing! by reconstruction. Infact, the opening and the closing with reconstruction criria do not reconstruct some regions linked by thin conntions, as in the case of opening and closing by reconstrtion. On the other hand, these transformations do

Fig. 9 (a) to (c) The three components of the original image hue,luminance, and saturation respectively; (d) luminance component;(e) binary image of image in (d); (f) distance function of the binaryimage; (g) binary image in white color and the marker M in graycolor; and (h) output image computed by vl,X

k (M) (until stability), forl518 of image in (e).

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considerably affect the remaining structures, as is the cin the morphological opening and closing. Finally, the iterest of these transformations was motivated by someproblems.

Acknowledgments

We would like to thank Marcela Sanchez Alvarez for hcareful revision of the English version. The author I. Terwould like to thank Diego Rodrigo and Darıó T. G. for theirgreat encouragement. This work was partially fundedthe government agency CONACyT~Mexico! under theGrants 25641-A and 41170.

References1. J. Serra and P. Salembier, ‘‘Connected operators and pyramids

Image Algebra Math. Morphology, Proc. SPIE2030, 65–76~1993!.2. J. Crespo, J. Serra, and R. Schafer, ‘‘Theoretical aspects of mor

logical filters by reconstruction,’’Signal Process.47, 201–225~1995!.3. J. Serra, ‘‘Connectivity on complete lattices,’’J. Math. Imaging Vision

9~3!, 231–251~1998!.4. F. Meyer, ‘‘From connected operators to levelings,’’ inMathematical

Morphology and Its Applications to Image and Signal Processing, H.Heijmans and J. Roerdink, Eds., pp. 191–198, Kluwer~1998!.

5. J. Crespo, ‘‘Morphological connected filters and intra-region smoo

Fig. 10 (a) Detection of tomato leaf in luminance component, (b)superposition of tomato leaf contour, and (c) Phytophthora infestansillness detection.


e

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ing for image segmentation,’’ PhD Thesis, Georgia Institute of Tenology ~1993!.

6. H. J. A. M. Heimans, ‘‘Connected morphological operators for binaimages,’’ Technical Report CWI No. PNA-R9708~Apr. 1997!.

7. C. Ronse, ‘‘Set-theoretical algebraic approaches to connectivitycontinuous or digital spaces,’’J. Math. Imaging Vision8, 41–58~1998!.

8. J. Serra, Ed.,Image Analysis and Mathematical Morphology, Vol. II,Academic, New York~1988!.

9. L. Vincent, ‘‘Morphological grayscale reconstruction in image anasis: applications and efficient algorithms,’’IEEE Trans. Image Pro-cess.2~2!, 176–201~1993!.

10. S. Beucher, ‘‘Segmentation d’Images et Morphologie Mathe´matique,’’PhD Thesis, Centre de Morphologie Mathematique, ENSMP, Ftainebleau, France~1990!.

11. I. R. Terol-Villalobos, F. Rodrı´guez-Garcıá, and C. Morales-Aguilloń,‘‘Some new results for morphological connected filters: image smentation and contrast enhancement,’’Recent Res. Devel. Opt. Eng.2,87–112~1999!.

12. F. Meyer, ‘‘Levelings,’’ inMathematical Morphology and Its Appli-cations to Image and Signal Processing, H. Heijmans and J. RoerdinkEds., pp. 199–206, Kluwer~1998!.

13. J. Serra, ‘‘Connections for sets and functions,’’Fund. Inform. 41,147–186~2000!.

14. F. Meyer and P. Maragos, ‘‘Nonlinear scale-space representationmorphological levelings,’’J. Visual Commun. Image Represent11~3!,245–265~2000!.

15. P. Salembier and A. Oliveras, ‘‘Practical extensions of connectederators,’’ inMathematical Morphology and Its Applications to Imagand Signal Processing, P. Maragos, R. W. Schafer, and M. K. ButEds., pp. 97–110, Kluwer~1996!.

16. J. Madrid and N. Ezquerra, ‘‘Automatic 3-dimensional segmentatof MR brain tissue using filters by reconstruction,’’ inMathematicalMorphology and Its Applications to Image and Signal Processing, P.Maragos, R. W. Schafer, and M. K. Butt, Eds., pp. 417–424, Kluw~1996!.

17. F. Meyer and S. Beucher, ‘‘Morphological segmentation,’’J. VisualCommun. Image Represent1, 21–46~1990!.

Ivan R. Terol-Villalobos received his BScdegree from Instituto Politecnico Nacional(I.P.N.), Mexico, his MSc degree in electri-cal engineering from Centro de Investiga-cion y Estudios Avanzados del I.P.N.,Mexico, his DEA degree in computer sci-ence from the University of Paris VI,France, and his PhD degree from the Cen-tre de Morphologie Mathematique, Ecoledes Mines de Paris, France. He is currentlya researcher with Centro de Investigacion

y Desarrollo Tecnologico en Electroquımica, Queretaro, Mexico. Hiscurrent research interests include morphological image processing,morphological probabilistic models, and computer vision.

Damian Vargas-Va zquez received his BSc degree from the Insti-tuto Tecnologico de Queretaro and his MSc degree in electrical en-gineering from the Autonomous University of Queretaro, Mexico. Hiscurrent research interests are algorithm techniques for image pro-cessing.

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Openings and closings with reconstruction criteria: a ... · Openings and closings with reconstruction criteria: a study of a class of lower and upper ... phological closings with