Research ArticleTrajectory-Based Morphological Operators A Model forEfficient Image Processing
Antonio Jimeno-Morenilla1 Francisco A Pujol1 Rafael Molina-Carmona2
Joseacute L Saacutenchez-Romero1 and Mar Pujol2
1 Departamento de Tecnologıa Informatica y Computacion Universidad de Alicante PO Box 99 E-03080 Alicante Spain2Departamento de Ciencia de la Computacion e Inteligencia Artificial Universidad de Alicante PO Box 99 E-03080 Alicante Spain
Correspondence should be addressed to Francisco A Pujol fpujoldticuaes
Received 6 November 2013 Accepted 9 March 2014 Published 14 April 2014
Academic Editors Z Chen J Shu and F Yu
Copyright copy 2014 Antonio Jimeno-Morenilla et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Mathematical morphology has been an area of intensive research over the last few years Although many remarkable advanceshave been achieved throughout these years there is still a great interest in accelerating morphological operations in order forthem to be implemented in real-time systems In this work we present a new model for computing mathematical morphologyoperations the so-called morphological trajectory model (MTM) in which a morphological filter will be divided into a sequenceof basic operations Then a trajectory-based morphological operation (such as dilation and erosion) is defined as the set of pointsresulting from the ordered application of the instant basic operationsTheMTMapproach allowsworkingwith different structuringelements such as disks and from the experiments it can be extracted that our method is independent of the structuring elementsize and can be easily applied to industrial systems and high-resolution images
1 Introduction
During the last 10ndash15 years many different improvementshave been proposed in order to implement morphologicaloperations in a more efficient way than the original Serrarsquosalgorithm [1] The methods for increasing the speed ofmorphological operators can be divided into two maingroups (i) algorithms that reduce the computation time ofthe morphological filters through the decomposition of thestructuring element into smaller sets and (ii) algorithms thateliminate the inherent redundancies in the calculation of themorphological operations achieving as a consequence anincrease in the processing speed of such operations
To a large extent the most popular morphological meth-ods to achieve this speed enhancement are within the firstgroup One of the most important ones is the van Herk andGil-Wermanrsquos (HGW) algorithm [2 3] which is among thefastest methods in order to implement erosions and dilationsin gray-tone images having a computational complexityindependent of the size of the structuring element In general
terms this method is designed for 1D elements and workswith linear structuring elements composed of horizontalandor vertical segments Its main strength comes fromthe fact that only 3 comparisons are needed to obtain theoutput pixel In [4] an improvement of this algorithm wasmade achieving only 15 comparisons per output pixel butincreasing significantly the computational complexity of themethod since it implies the generation of ordered listsHGWrsquos method has had a great impact since it was firstproposed and up to now a great amount of works continue todevelop new improvements of this algorithm many of theminclude some kind of hardware optimization [5ndash7]
In [8] authors extended the HGWrsquos model to the case ofusing lines with arbitrary orientations In addition in theirwork from 2001 [9] they improved their model so that two-dimensional structuring elements are also taken into accountmaking use of recursion An interesting set of extensionswerealso presented in this paper among which they include amethod to approximate discrete disks by using cascades ofdilations with periodic lines
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 801587 9 pageshttpdxdoiorg1011552014801587
2 The Scientific World Journal
On the other hand an algorithm for calculating grayscalemorphological operations with flat arbitrary-shaped struc-turing elements was presented in [10] Their approach isindependent of both the image content and the numberof necessary gray levels The use of arbitrary elements isinteresting particularly in cases where a structuring elementcannot be decomposed into smaller ones Essentially thealgorithm decomposes a structuring element into a seriesof chords that can be understood as a series of pixels ofmaximum extent and considers each chord as a horizontalstructuring element It also uses a look-up table to store theminimum intensity values of the pixels belonging to eachchord The experiments were performed with a wide varietyof elements and showed that this method improves manyothers that were developed to decompose some specific typesof structuring elements It also allows working with floatingpoint data
Another improvement in the decomposition of arbitrary-shaped structuring elements has been proposed in [11] Thedecomposition method is recursive and is optimized byusing genetic algorithms thus improving the results of otherwell-known decomposition approaches based on geneticalgorithms such as the ones described in [12ndash14]
In relation to this a mathematical morphology algorithmfor spatially variant square structuring elements was devel-oped in [15] achieving very low temporal cost and memoryrequirements In addition they proposed an efficient hard-ware implementation of morphological operations based onthis type of structuring elements this hardware architectureis presented as a hardware accelerator for the dilation anderosion operations in embedded systems
Recently in [16] a method for the development of mor-phological filters that runs in linear time with respect tothe image size and is constant in time with respect to thesize of the structuring element was proposed It is basedon the decomposition of a rectangular structuring elementinto one-dimensional segments then it eliminates redundantvalues and finally the result is encoded by calculating thedistance between every change of the valueThe authors claimthat it is possible for this method to achieve an efficientreal-time implementation having also very small memoryrequirements and supporting floating-point data
Regarding the algorithms to eliminate redundanciesthe one proposed by [17] is particularly interesting whichintroduced the concept of anchor defined as the positionin which a signal 119891 is not affected by the application of acertain operator Ψ Their approach is based on the searchfor anchors for the erosion and the closing and it allows amorphological operation to run with one-dimensional struc-turing elements about 30 faster than previous methodsHowever the main drawback is that this method requiresthe use of histograms so that its extension to two and threedimensions is not straightforward and the improvement incomputation speed is therefore minimized In addition thealgorithmdepends greatly on the image content Another sig-nificant work is described in [18] which is focused on imagebinarization using morphological operators To do this theso-called quick-closing and quick-opening are defined theseoperations have reduced computational cost and remove
the redundant comparisons in the neighborhood of everypixel The method works efficiently but only for square-shaped structuring elements
To sum up from this revision two conclusions emergefirst of all there is still a great interest from many researchgroups in order to accelerate morphological operations bothfor improving the basic morphological algorithms and fortheir hardware implementation on the other hand most ofthese investigations are based on optimizing morphologicalfilters for those cases where there is a dependence on theshape of the structuring element with the squarerectangularelements being the ones that get better results in terms ofoptimization While there are several studies to work witharbitrary-shaped elements there is still much work to do inthis field and some interesting structuring elements suchas disks or ellipses are difficult to decompose or to beapproximated by polygons
As a consequence in this work we show a new mathe-matical morphology approach the morphological trajectorymodel (MTM) which takes into account the trajectory inwhich a morphological operation is applied As it will beshown our method is independent from the structuring ele-ment size and can be easily applied to industrial systems andhigh-resolution images To complete our task in Section 2we shall define the so-called trajectory-based morpholog-ical operators Then in Section 3 the computation of thetrajectory-based filters is shown and afterwards Section 4considers some of the experiments completed to verifythat our system behaves properly Finally some importantremarks to our work as well as some future research tasksare summarized in Section 5
2 The Morphological Trajectory Model
The classical morphological model has a nondeterministicnature as it is defined over elements of a set without orderrestrictions in the access to these elements In our newapproach the morphologic operations will be restricted tosupport an order The order of the morphology operation isimportant because it will represent the structuring elementtrajectory As a result in this section an order relationbetween the elements in a set will be included so that asequence of operations could be established and therefore adeterministic component will be added to the morphologicalparadigm Let us define some terms first
21 Preliminary Definitions Let 119864 be the domain where thesets to be treated are defined Let us assume that in general119864 equiv 119877119899 Let 119883 sube 119864 be a subset of 119864 Thus in thecase of two-dimensional objects 119864 equiv 1198772 and for three-dimensional objects 119864 equiv 1198773 and consequently 119883 would bea two-dimensional or three-dimensional object respectivelyNotice that the domain is defined in a real space so themethod is suitable for any continuous domain Images can beconsidered a particular case where the domain is discretized
Let In(119883) be a function that obtains the inner part ofa set (ie an object) that is its result is object 119883 withoutits borders This function is defined as the set of positions of
The Scientific World Journal 3
(x minus cx)2 + (y minus cy)
2 = r2
(x minus cx)2
a2+
(y minus cy)2
b2+
(z minus cz)2
d2= 1
Figure 1 Examples of several possible representations of structuring elements The leftmost figures show the analytical expressions of SEson the right the corresponding classical SEs are shown
BX
crarr
(a)
XB
c
Γ(k)
rarr
(b)
XB
pc
rarr
(c)
Figure 2 Geometric description of an instant basic operation(a) Initial position (b) Transformation of object 119883 (c) Distancecomputing
the center of a solid 119899-ball of radius 120576 so that the ball is insidethe object
In (119883) = 119909 isin 119864
exist120576gt 0 119861 (119909 120576) sub 119860 (1)
where 119861(119909 120576) is a solid 119899-ball of center 119909 and radius 120576On the other hand let Fr(119883) be a function relating a set
to its border so that all the points belonging to the objectcontour are obtained
Fr (119883) = 119883 minus In (119883) (2)
As mentioned before structuring elements are an essen-tial tool to develop morphological operators For methodsthat use classical mathematical morphology the structuringelement (SE) can be seen as a group of pixels Howeverin our trajectory-based approach the structuring elementwill be defined on the basis of the geometric definition ofits frontier so that any representation of the SE that allowsthe extraction of its frontier is valid for our method Anespecially interesting case is the use of analytical expressionsto define the SE because this continuous representationgives as a result an adaptive precision and a more efficientcomputation than classical SE definitions as it will be shownin the following sections Figure 1 illustrates this concept withtwo cases of analytical SEs (left) and classical SEs (right)For instance the first row shows a circular SE which canbe analytically expressed as (119909 minus 119888
119909)2 + (119910 minus 119888
119910)2 = 1199032
corresponding to a circumference centered on point (119888119909 119888119910)
of radius 119903 instead of the classical neighborhood of pixelsdefined by the area of the circle as shown on the right Thisfact can be also extended to three-dimensional SEs as shownon the second row in Figure 1
This equation-based definition for the structuring ele-ment is used here for simplicity but notice that our approachcan be extended to any other frontier-based definitionThough the use of analytical or classical SEs in ourmodel doesnot add any restriction for computing the trajectory-basedmorphological operators (as it will be shown the method isbased on a distance calculation) the analytical expression ispreferable for efficiency and precision reasons
22 Instant BasicOperations Amorphological operationwillbe divided into a sequence of unitary or basic operationsThis sequence will guarantee the resulting order of the wholeoperation Since every basic operation will correspond to aparticular position of a structuring element along a trajectorythat is performedduring a period of timewe call them instantbasic operations
Let us define the instant basic operator⊙Γ(119896)
for any giveninstant 119896 as follows
119883⊙Γ(119896)
119861 = 119901 isin 119864 119901 = distV (119861 119883 ∙ Γ (119896)) sdot V and 119861119901cap 119883 =Oslash
(3)
where 119883 is the target object 119861 the structuring element 119861119901
are copies of the structuring element centered at every point119901when it touches the boundary of119883119864 equiv 119877119899 Γ(119896) represents anhomogeneous transformationmatrix in 119877119899+1 times119877119899+1 obtainedfor a particular real value 119896 and distV is the Euclidean distancebetween the structuring element and the transformation ofthe object 119883-obtained by postmultiplying every element ofthe set 119883 by the homogeneous transformation matrix Γ(119896)-computed in the direction addressed by vector V In otherwords this operation obtains the structuring element centerwhen it touches the boundary119883 following direction V
A graphical example of this operator is shown in Figure 2Thus an object 119883 is transformed applying a 2D rotationmatrix over its center 119888 For this case 119896 could representthe number of degrees in that transformation matrix so itsvalues are in the [0 2120587) range Once the object is transformed(Figure 2(c)) the distance between 119861 and 119883 in the directionV is applied to the center of 119861 in order to obtain the resultof the instant basic operation (ie point 119901) For differentand ordered real 119896 values (using the lt relation in 119877) we willobtain a new set of structuring element centers 119861
119901that touch
the boundary of 119883 These centers will also be ordered in thegeometric space due to the use of different rotation matrixes
4 The Scientific World Journal
The calculation of function distV is the most time-consuming operation in (2) Here the description of thestructuring element plays a crucial role We propose threemethods for obtaining the distance distV between thestructuring element and the target object in the directionaddressed by vector V
(i) In the case of having an analytical description for boththe SE and the target object an analytical expressionmay be obtained for calculating the distance as wellThis way the distance calculation is straightforwardand its computation time will be low
(ii) When no object can be described using an analyticalexpression (neither the SE nor the target object) thedistance is obtained using a discrete method bothobjects must be discretized to obtain the points intheir boundaries and the distance is obtained point-to-pointThis is theworst case and the computationalcost depends on the discretization precision
(iii) Finally a mixed method is proposed when only oneobject (the SE or the target object) can be describedusing an analytical expression In this case the dis-tance is obtained between the points in the surface ofthe discretized object and the other object as a wholeusing its analytical description The computationalcost is much lower than for the purely discretemethod
23 Trajectory-Based Morphological Operators In this sec-tion the instant basic operator ⊙
Γ(119896)is applied in order to
achieve a whole morphologic operation We are particularlyinterested in defining the two fundamental operations in themorphologic paradigm the dilation and the erosion Dueto the fact that frontiers of objects and structuring elementsare only taken into account to compute the instant basicoperations the goal is to obtain only the boundary of dilationand erosion
231 Dilation In general terms this operation is classicallydefined as the place of the center positions of the structuringelement 119861 when it touches a set119883 [18]
Derived from (3) and (5) we define the instant basicdilation which specifies a center position touching theboundary of a set119883 but from the outside
119883oplusΓ(119896)
119861 = 119901 isin 119864
119901 = distV (119861119883 ∙ Γ (119896)) sdot V and 119861119901cap 119883 =Oslash and 119861
119901cap In (119883) =Oslash
(6)
Using the instant basic dilation we define the trajectory-based dilation 119883oplus
Γ119861 as the set of points resulting from
the repeated and ordered application of the instant basicdilation for the normalized 119896 range [0 1] The trajectorydefined by Γ(119896) must cover all the surface of the objectin the normalized range Note that only boundary pointsare computed and that the frontier of the trajectory-baseddilation is expressed in (6)
119883oplusΓ119861 = ⋃119896isin[01]
(119883oplusΓ(119896)
119861)
= 119909 isin 119864 119861119909cap 119883 =Oslash and 119861
119909cap In (119883) = Oslash
(7)
Trajectory-based dilation can orientate the structuringelement in any position on the object boundary by means ofhomogeneous transformations which are a combination oftranslations and rotations This feature is not supported byclassical dilations In addition partial dilations of objects arenow also possible when a subrange of 119896 is chosen
232 Erosion Classically this operationmdashwhich is com-monly used for image filteringmdashis defined as the place of thecenter positions of the structuring element119861when it is forcedto be inside a set119883
119883Θ119861 = 119909 isin 119864 119861119909sube In (119883) (8)
In our context we are interested only in the erosionboundary which is the place of the center positions that touchthe frontier of set119883 from the inside
Fr (119883Θ119861) = 119909 isin 119864 119861119909sube In (119883) and 119861
119909cap Fr (119883) =Oslash (9)
Derived from (3) and (9) we define the instant basic ero-sion which specifies a center position touching the boundaryof a set119883 but from the inside
119883ΘΓ(119896)
119861 = 119901 isin 119864
119901=distV (119861119883 ∙ Γ (119896)) sdotV and 119861119901sube In (119883) and 119861
119901cap Fr (119883) =Oslash
(10)
Consequently we define the trajectory-based erosion119883ΘΓ(119896)
119861 as the set of points resulting from the repeatedand ordered application of the instant basic erosion for thenormalized 119896 range [0 1] As in the case of dilation thetrajectory defined by Γ(119896) must cover all the surface of theobject in the normalized rangeThe frontier of the trajectory-based erosion is
119883ΘΓ119861 = ⋃119896isin[01]
(119883ΘΓ(119896)
119861)
= 119909 isin 119864 119861119909sube In (119883) and 119861
119909cap Fr (119883) = Oslash
(11)
Figure 3 shows an example of dilation and erosion appliedto a 2D image The black part corresponds to the classicaloperation result The frontier is computed by means of theassociated trajectory-based operator
The Scientific World Journal 5
Morphological operation
Original object boundaries Dilation
Erosion
SE
Figure 3 Classical morphological operations on 2D images On the left a morphological dilation On the right a morphological erosionThe structuring element (SE)mdasha circle of 20 pixels in radiusmdashis shown at the top left corner
Table 1 Characteristics of the algorithms based on classical mathematical morphology versus the morphological trajectory model
Classical morphology Morphological trajectory model
Application space Finite group of pixels as a discretization ofEuclidean space
2D Euclidean space (extensible to 119899-D Euclideanspace)
Objects Based on a complete image Based on the boundary of each object(discretecontinuous)
SEs Group of pixels Geometric representation of the frontier
Method On each pixel it operates in a neighborhoodenvironment defined by the SE
The minimum distance of the SE center is calculatedin a direction V on a trajectory defined by Γ(119896)
Result Erosiondilation as group of pixels Frontier points of the erosiondilation operation
Figure 4 Partial morphological erosion as a subset of the completeerosion (over the subrange defined by the dotted line)
As with dilation trajectory-based erosion can orientatethe structuring element in any position on the object bound-ary and define partial erosion of objects when a subrange of119896 is chosen (see Figure 4)
Table 1 summarizes the main differences between themethods described in this section In classical mathematicalmorphology the calculation is made on a complete imagethat is the morphological operation does not distinguishwhether the pixels belong to a specific object or not it simplyapplies a calculation operation of supremum or infimum in aneighborhood environment In the morphological trajectorymodel (MTM) it is necessary to differentiate between theobjects given in the space since each object has a differentgeometric representation Furthermore this representationdefines the frontier of the object and not its interior Anotherimportant difference is the representation of the structuringelement whereas in traditional morphology it is treated asa subgroup of points (which is discretized for the case ofworking with images) the MTM considers the geometricfunction of the points that make up its frontier withoutbeing necessary to carry out a discretization of the structuringelement
3 Morphological Trajectory Computation
In this section the computation of the erosion for the MTMis presented The algorithm becomes straightforward if themorphological erosion concept defined in [1] is applied Firstof all the boundary curve 119862 of object 119883 is represented asa set of ordered points 119901 organized in collinear segments 119904For every point we compute the structuring element centerposition (1199011015840) that touches each point in the direction of acertain vector V
119901 which must be perpendicular from inside
object 119883 The center 1199011015840 will be valid only if the structuringelement placed at 1199011015840 is inside the shape (ie it will not collidewith the curve 119862) Note that for the dilation operation theprocedure will be the same but in this case the elementwill touch the boundary from the outside A pseudo codealgorithm for the erosion is presented in Algorithm 1
If a point 119901 presents a discontinuity in the first derivativewe generate a set of new vectors in order to cover the gap (seeFigure 5) From that new set we also compute new possiblestructuring element centers
Let us analyze now the computational cost of the MTMalgorithm in terms of the problem size The operator used is119874 to determine an upper limit of the computation cost Asshown in Algorithm 1 the algorithm essentially consists ofan external loop which is used to have access to every pointof the shape and two main function calls Letrsquos call 119899 to thenumber of points that represent the shape 119862 once it has beendiscretized If we use a constant step factor 119904 and the totallength of 119862 is 119871 then 119899 will be 119871119904 points
The function ObtainSECenter computes the center of thestructuring element when it touches a point119901 in the directionaddressed by vector V
119901 So this function depends on the SE
6 The Scientific World Journal
(1) For every 119901119894isin 119862 do
(2) 1199011015840119894= ObtainSECenter(119901
119894V119901119894)
(3) If not CollideSE(1199011015840119894 119862) then AddTrajectory(1199011015840
119894)
(4) Endfor
Algorithm 1 Basic pseudo-code algorithm for the morphological trajectory erosion
Table 2 Equations to calculate the intersection between a circle and a 2D segment
Segment function Circle function Segment-Circle intersection equation on 119905119909 = 119909
Dark-grey SE positions are discarded due to shape collision Notethat discontinuity at 119901
119889is solved by a vector swept generation
geometry For simple SEs such as circles rectangles andtriangles the function can be evaluated in a constant time ctEquation (12) shows this function for a circular SE of radius119877
ObtainSECenter (119901 V119901) = 119901 + 119877 sdot V
119901 (12)
The next function called CollideSE(1199011015840 119862) is true if theSE centered at point 1199011015840 is not completely inside shape 119862 andreturns false otherwise In order to evaluate this conditionthis function computes the intersection of the SE geometryand shape 119862 The cost of this function depends on therepresentation of 119862 For the experiments we have organizedthe shape into a set of contiguous segments that representsthe shape Then every segment is tested (at a constant time)and if a segment produces two or more intersections in theSE geometry then the function returns a true value Notethat in this case the discretization of shape 119862 will not be thesame as the one we used to determine the center positionsin the shape For shapes with a high degree of colinearity thenumber of segments will be reduced slightly Let us call119898 thatnumber of segments
The expressions in Table 2 show the quadratic equationused to determine the intersection between a circle cen-tered at the origin and a 2D segment defined between points
(1199091 1199101) and (119909
2 1199102) for a normalized range 119905 [0 1] As
a conclusion a double solution for variable 119905 in the range[0 1] will cause a true return in the function CollideSEotherwise the next segment will be analyzed
Finally the third function called AddTrajectory adds thenew center 1199011015840 to the list of successful centers at a constanttime so it is not considered for evaluating the cost
As a conclusion let us analyze the whole algorithm inorder to obtain an upper limit for the computational costThenext expression evaluates this cost
We must remark that after completing our experiments119898 ≪ 119899 in most cases since a usual value for119898 takes values ofhundreds The computation times for some examples of theMTM operations will be shown in the following section
4 Experiments
In this section we present some experiments in order to testthe trajectory-based operations The first one compares twoversions of the classical dilation versus the trajectory-basedone In Algorithm 2 we show the classical version algorithmsused in the tests
The algorithm called MM1 corresponds to a classicalmorphological dilation whereas the MM2 refers to thatclassical version operating only on the boundary of theobject Note that MM2 does not perform a valid dilationIt was only developed to test the frontier effect that is theadvantage that MTM has since it only processes boundarypixels The trajectory-based version was called MTM for theexperiments The images were evaluated on an Intel PentiumDual Core processor 28GHz and 2GB in RAMTheywereobtained on a Windows based platform
Several tests and experiments were carried out in order toobtain the computing time under different input conditionsBoth the size of the object and the size of the structuringelement were varied as well as the parameters that took partin the morphological operation
Figure 6 shows the behavior of the algorithms resultingfrom the variation of the size of the structuring element andthe size of the object respectively As a consequence from
The Scientific World Journal 7
10
100
1000
10000
0 20 40 60 80 100 120
Tim
e (m
s)
SE radius (pixels)
MTM
MM2
MM1
(a)
10
100
1000
0 10000 20000 30000Pixels of all objects
MTM
MM2
MM1
Tim
e (m
s)
(b)
Figure 6 Morphological dilation tests On the left influence of the size of the structuring element On the right influence on the size ofobjects
MM1 dilation(1) For 119901 isin 119883 do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) EndforMM2 dilation(1) For 119901 isin Fr(119883) do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) Endfor
Algorithm 2 Pseudo-code used for the 2D experiments
these experiments we can see that the computing time ofthe morphological trajectory model remains almost constantagainst the variations in the size of the structuring elementthis is logical due to the fact that we use its geometricrepresentation instead of its content as opposed to MM1and MM2 The difference between MM2 and MM1 arisesfrom the fact that the former only expands the structuringelement for the pixels in the object boundary and thereforeits computing time decreases an order of magnitude withrespect to MM1
With regard to the variation in the size of the objectsin this experiment the MTM gives better results than MM2and much better than MM1 Since MM2 works only on theobject boundary the difference with the MTM arises fromthe size chosen for the SE if this size is small enough MM2will employ less time than the MTM as it has a simplercomputational logic (see Algorithm 2) The radius of thestructuring element chosen for this test was 40 pixels whichis equivalent to an area of 5025 points In order to determinethe effect of the increase of the size of the structuring element
on a group of objects the experiments described below werecarried out
The experiment shown in Figure 7 compares the execu-tion times of theMTM andMM2 for different radii of the SEItmay be seen that the increase in the structuring element sizecauses a crossing point between MM2 and MTM for objectsizes of about 5000 pixels
Furthermore Figure 7 shows that there is no significantincrease in the computation time for the MTMwhen the sizeof the structuring element is increased since this variation isminimal compared to the classical morphological methodsThe small increase comes from the number of pixels of theobject where the distance function is definedThus if the sizeof the SE increases this number is greater and this causes theneed to make new calculations on the new points
On the other hand Figure 8 shows different results ofthe application of the morphological filters Specifically it isshown how the MTM obtains the frontier of the morpholog-ical operation made by MM1
8 The Scientific World Journal
0
20
40
60
80
100
120
140
0 10000 20000 30000 40000Pixels of all objects
MM2 R80
MM2 R40
MM2 R20
MTM R20 40 80
Tim
e (m
s)
Figure 7 Morphologic dilation Comparative study between MM2and MTMmodels for SE sizes of 20 40 and 80 pixels
Time MM1 = 9515msTime MM2 = 80msTime MTM = 110ms
Time MM1 = 8560msTime MM2 = 34msTime MTM = 96ms
(a)
Time MM1 = 4115msTime MM2 = 180msTime MTM = 131ms
Time MM1 = 1085msTime MM2 = 75msTime MTM = 65ms
(b)
Figure 8 Morphological dilation and erosion Results of differentmorphological operations used for the experiments In the upperpart two erosions and in the lower part two dilationsThe boundaryof the original object is represented in green and the result of MM1operation in black with the MTM result in red
(a) (b)
Figure 9 Erosion of figures using our morphological approach (a)Using a circle as a structuring element (b) Using a rectangle as astructuring element
Finally in Figure 9 we present some images related toother erosion experiments using several structuring elementgeometries where the result is presented in green
5 Conclusions and Discussion
In this paper we have developed a topological system result-ing from applying the conventional morphological modelby means of trajectory-based morphological operations todo this we introduced a new feature consisting of order-ing the morphological primitives As shown the proposedoperations are especially useful when large images need to beprocessed
The morphological trajectory model offers an effectivealternative to traditionalmethods for computingmorpholog-ical primitives This alternative is justified if the number ofpoints of the objects and that of the structuring elements arehigh The independency from the structuring element sizecould be interesting to apply morphological operations onhigh definition images or 3D image reconstruction Due tothe fact that the number of the group points is directly relatedto the dimension of the space in which the object and thestructuring element are defined the importance of the MTMis more relevant when the dimension of the representationspace is increased (3D 4D ) In the two dimensionalspace the application of the MTMmay be justified for high-resolution images where large size operators are applied
Other trajectory-based operations such as openingsclosings and skeletons are defined in [18] We are interestedin demonstrating their utility and efficiency by means of thistrajectory optimization
The newmodel has been presented for binary 2D imagesHowever the new paradigm is extensible for any number ofdimensions of the Euclidean space In [19ndash21] new versionsof the morphological model for color images were presentedThesemodels consider the color as a third coordinate and thecomputation is made in the 3D space Future works will befocused on extending the model in order to support efficientfiltering of color images or real 3D images
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
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[2] M van Herk ldquoA fast algorithm for local minimum and max-imum filters on rectangular and octagonal kernelsrdquo PatternRecognition Letters vol 13 no 7 pp 517ndash521 1992
[3] J Gil and M Werman ldquoComputing 2-D min median and maxfiltersrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 15 no 5 pp 504ndash507 1993
[4] J Y Gil and R Kimmel ldquoEfficient dilation erosion openingand closing algorithmsrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 24 no 12 pp 1606ndash1617 2002
The Scientific World Journal 9
[5] O Deforges N Normand and M Babel ldquoFast recursivegrayscale morphology operators from the algorithm to thepipeline architecturerdquo Journal of Real-Time Image Processingvol 8 no 2 pp 143ndash152 2013
[6] C Clienti M Bilodeau and S Beucher ldquoAn efficient hardwarearchitecture without line memories for morphological imageprocessingrdquo in Proceedings of the 10th International ConferenceonAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo08)J Blanc-Talon S Bourennane W Philips D Popescu and PScheunders Eds pp 147ndash156 Springer Heidelberg Germany2008
[7] K Sivakumar M J Patel N Kehtarnavaz Y Balagurunathanand E R Dougherty ldquoA constant-time algorithm for ero-sionsdilations with applications to morphological texture fea-ture computationrdquoReal-Time Imaging vol 6 no 3 pp 223ndash2392000
[8] P Soille and H Talbot ldquoImage Structure Orientation UsingMathematical Morphologyrdquo in Proceedings of the 14th Interna-tional Conference on Pattern Recognition A Jain S Venkateshand B Lovell Eds vol 2 pp 1467ndash1469 Brisbane AustraliaAugust 1998
[9] P Soille and H Talbot ldquoDirectional morphological filteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 23 no 11 pp 1313ndash1329 2001
[10] E R Urbach and M H F Wilkinson ldquoEfficient 2-D grayscalemorphological transformations with arbitrary flat structuringelementsrdquo IEEE Transactions on Image Processing vol 17 no 1pp 1ndash8 2008
[11] Y Zhang and L Wu ldquoRecursive structure element decompo-sition using migration fitness scaling genetic algorithmrdquo inProceedings of the 2nd International Conference on Advances inSwarm Intelligence (ICSI rsquo11) Y Tan Y Shi Y Chai and GWang Eds part 1 pp 514ndash521 Springer Heidelberg Germany2011
[12] G Anelli A Broggi and G Destri ldquoDecomposition of arbi-trarily shaped binarymorphological structuring elements usinggenetic algorithmsrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 20 no 2 pp 217ndash224 1998
[13] H Park and J Yoo ldquoStructuring element decomposition for effi-cient implementation ofmorphological filtersrdquo IEE ProceedingsVision Image and Signal Processing vol 148 no 1 pp 31ndash352001
[14] F Y Shih and Y-T Wu ldquoDecomposition of binary mor-phological structuring elements based on genetic algorithmsrdquoComputer Vision and Image Understanding vol 99 no 2 pp291ndash302 2005
[15] H Hedberg P Dokladal and V Owall ldquoBinary morphologywith spatially variant structuring elements algorithm andarchitecturerdquo IEEE Transactions on Image Processing vol 18 no3 pp 562ndash572 2009
[16] P Dokladal and E Dokladalova ldquoComputationally efficientone-pass algorithm for morphological filtersrdquo Journal of VisualCommunication and Image Representation vol 22 no 5 pp411ndash420 2011
[17] M van Droogenbroeck and M J Buckley ldquoMorphologicalerosions and openings fast algorithms based on anchorsrdquoJournal of Mathematical Imaging and Vision vol 22 no 2 pp121ndash142 2005
[18] E Cooksey and W D Withers ldquoRapid image binarizationwith morphological operatorsrdquo in Proceedings of the 15th IEEEInternational Conference on Image Processing (ICIP rsquo08) pp1017ndash1020 San Diego Calif USA October 2008
[19] J Goutsias H J A M Heijmans and K Sivakumar ldquoMorpho-logical operators for image sequencesrdquo Computer Vision andImage Understanding vol 62 no 3 pp 326ndash346 1995
[20] A Hanbury ldquoMathematical morphology in the HLS colourspacerdquo in Proceedings of the 12th British Machine Vision Con-ference (BMVC rsquo01) vol 2 pp 451ndash460 Manchester UK 2001
[21] J Angulo and J Serra ldquoMorphological color size distributionfor image classification and retrievalrdquo in Proceedings of theAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo02)pp 46ndash53 2002
On the other hand an algorithm for calculating grayscalemorphological operations with flat arbitrary-shaped struc-turing elements was presented in [10] Their approach isindependent of both the image content and the numberof necessary gray levels The use of arbitrary elements isinteresting particularly in cases where a structuring elementcannot be decomposed into smaller ones Essentially thealgorithm decomposes a structuring element into a seriesof chords that can be understood as a series of pixels ofmaximum extent and considers each chord as a horizontalstructuring element It also uses a look-up table to store theminimum intensity values of the pixels belonging to eachchord The experiments were performed with a wide varietyof elements and showed that this method improves manyothers that were developed to decompose some specific typesof structuring elements It also allows working with floatingpoint data
Another improvement in the decomposition of arbitrary-shaped structuring elements has been proposed in [11] Thedecomposition method is recursive and is optimized byusing genetic algorithms thus improving the results of otherwell-known decomposition approaches based on geneticalgorithms such as the ones described in [12ndash14]
In relation to this a mathematical morphology algorithmfor spatially variant square structuring elements was devel-oped in [15] achieving very low temporal cost and memoryrequirements In addition they proposed an efficient hard-ware implementation of morphological operations based onthis type of structuring elements this hardware architectureis presented as a hardware accelerator for the dilation anderosion operations in embedded systems
Recently in [16] a method for the development of mor-phological filters that runs in linear time with respect tothe image size and is constant in time with respect to thesize of the structuring element was proposed It is basedon the decomposition of a rectangular structuring elementinto one-dimensional segments then it eliminates redundantvalues and finally the result is encoded by calculating thedistance between every change of the valueThe authors claimthat it is possible for this method to achieve an efficientreal-time implementation having also very small memoryrequirements and supporting floating-point data
Regarding the algorithms to eliminate redundanciesthe one proposed by [17] is particularly interesting whichintroduced the concept of anchor defined as the positionin which a signal 119891 is not affected by the application of acertain operator Ψ Their approach is based on the searchfor anchors for the erosion and the closing and it allows amorphological operation to run with one-dimensional struc-turing elements about 30 faster than previous methodsHowever the main drawback is that this method requiresthe use of histograms so that its extension to two and threedimensions is not straightforward and the improvement incomputation speed is therefore minimized In addition thealgorithmdepends greatly on the image content Another sig-nificant work is described in [18] which is focused on imagebinarization using morphological operators To do this theso-called quick-closing and quick-opening are defined theseoperations have reduced computational cost and remove
the redundant comparisons in the neighborhood of everypixel The method works efficiently but only for square-shaped structuring elements
To sum up from this revision two conclusions emergefirst of all there is still a great interest from many researchgroups in order to accelerate morphological operations bothfor improving the basic morphological algorithms and fortheir hardware implementation on the other hand most ofthese investigations are based on optimizing morphologicalfilters for those cases where there is a dependence on theshape of the structuring element with the squarerectangularelements being the ones that get better results in terms ofoptimization While there are several studies to work witharbitrary-shaped elements there is still much work to do inthis field and some interesting structuring elements suchas disks or ellipses are difficult to decompose or to beapproximated by polygons
As a consequence in this work we show a new mathe-matical morphology approach the morphological trajectorymodel (MTM) which takes into account the trajectory inwhich a morphological operation is applied As it will beshown our method is independent from the structuring ele-ment size and can be easily applied to industrial systems andhigh-resolution images To complete our task in Section 2we shall define the so-called trajectory-based morpholog-ical operators Then in Section 3 the computation of thetrajectory-based filters is shown and afterwards Section 4considers some of the experiments completed to verifythat our system behaves properly Finally some importantremarks to our work as well as some future research tasksare summarized in Section 5
2 The Morphological Trajectory Model
The classical morphological model has a nondeterministicnature as it is defined over elements of a set without orderrestrictions in the access to these elements In our newapproach the morphologic operations will be restricted tosupport an order The order of the morphology operation isimportant because it will represent the structuring elementtrajectory As a result in this section an order relationbetween the elements in a set will be included so that asequence of operations could be established and therefore adeterministic component will be added to the morphologicalparadigm Let us define some terms first
21 Preliminary Definitions Let 119864 be the domain where thesets to be treated are defined Let us assume that in general119864 equiv 119877119899 Let 119883 sube 119864 be a subset of 119864 Thus in thecase of two-dimensional objects 119864 equiv 1198772 and for three-dimensional objects 119864 equiv 1198773 and consequently 119883 would bea two-dimensional or three-dimensional object respectivelyNotice that the domain is defined in a real space so themethod is suitable for any continuous domain Images can beconsidered a particular case where the domain is discretized
Let In(119883) be a function that obtains the inner part ofa set (ie an object) that is its result is object 119883 withoutits borders This function is defined as the set of positions of
The Scientific World Journal 3
(x minus cx)2 + (y minus cy)
2 = r2
(x minus cx)2
a2+
(y minus cy)2
b2+
(z minus cz)2
d2= 1
Figure 1 Examples of several possible representations of structuring elements The leftmost figures show the analytical expressions of SEson the right the corresponding classical SEs are shown
BX
crarr
(a)
XB
c
Γ(k)
rarr
(b)
XB
pc
rarr
(c)
Figure 2 Geometric description of an instant basic operation(a) Initial position (b) Transformation of object 119883 (c) Distancecomputing
the center of a solid 119899-ball of radius 120576 so that the ball is insidethe object
In (119883) = 119909 isin 119864
exist120576gt 0 119861 (119909 120576) sub 119860 (1)
where 119861(119909 120576) is a solid 119899-ball of center 119909 and radius 120576On the other hand let Fr(119883) be a function relating a set
to its border so that all the points belonging to the objectcontour are obtained
Fr (119883) = 119883 minus In (119883) (2)
As mentioned before structuring elements are an essen-tial tool to develop morphological operators For methodsthat use classical mathematical morphology the structuringelement (SE) can be seen as a group of pixels Howeverin our trajectory-based approach the structuring elementwill be defined on the basis of the geometric definition ofits frontier so that any representation of the SE that allowsthe extraction of its frontier is valid for our method Anespecially interesting case is the use of analytical expressionsto define the SE because this continuous representationgives as a result an adaptive precision and a more efficientcomputation than classical SE definitions as it will be shownin the following sections Figure 1 illustrates this concept withtwo cases of analytical SEs (left) and classical SEs (right)For instance the first row shows a circular SE which canbe analytically expressed as (119909 minus 119888
119909)2 + (119910 minus 119888
119910)2 = 1199032
corresponding to a circumference centered on point (119888119909 119888119910)
of radius 119903 instead of the classical neighborhood of pixelsdefined by the area of the circle as shown on the right Thisfact can be also extended to three-dimensional SEs as shownon the second row in Figure 1
This equation-based definition for the structuring ele-ment is used here for simplicity but notice that our approachcan be extended to any other frontier-based definitionThough the use of analytical or classical SEs in ourmodel doesnot add any restriction for computing the trajectory-basedmorphological operators (as it will be shown the method isbased on a distance calculation) the analytical expression ispreferable for efficiency and precision reasons
22 Instant BasicOperations Amorphological operationwillbe divided into a sequence of unitary or basic operationsThis sequence will guarantee the resulting order of the wholeoperation Since every basic operation will correspond to aparticular position of a structuring element along a trajectorythat is performedduring a period of timewe call them instantbasic operations
Let us define the instant basic operator⊙Γ(119896)
for any giveninstant 119896 as follows
119883⊙Γ(119896)
119861 = 119901 isin 119864 119901 = distV (119861 119883 ∙ Γ (119896)) sdot V and 119861119901cap 119883 =Oslash
(3)
where 119883 is the target object 119861 the structuring element 119861119901
are copies of the structuring element centered at every point119901when it touches the boundary of119883119864 equiv 119877119899 Γ(119896) represents anhomogeneous transformationmatrix in 119877119899+1 times119877119899+1 obtainedfor a particular real value 119896 and distV is the Euclidean distancebetween the structuring element and the transformation ofthe object 119883-obtained by postmultiplying every element ofthe set 119883 by the homogeneous transformation matrix Γ(119896)-computed in the direction addressed by vector V In otherwords this operation obtains the structuring element centerwhen it touches the boundary119883 following direction V
A graphical example of this operator is shown in Figure 2Thus an object 119883 is transformed applying a 2D rotationmatrix over its center 119888 For this case 119896 could representthe number of degrees in that transformation matrix so itsvalues are in the [0 2120587) range Once the object is transformed(Figure 2(c)) the distance between 119861 and 119883 in the directionV is applied to the center of 119861 in order to obtain the resultof the instant basic operation (ie point 119901) For differentand ordered real 119896 values (using the lt relation in 119877) we willobtain a new set of structuring element centers 119861
119901that touch
the boundary of 119883 These centers will also be ordered in thegeometric space due to the use of different rotation matrixes
4 The Scientific World Journal
The calculation of function distV is the most time-consuming operation in (2) Here the description of thestructuring element plays a crucial role We propose threemethods for obtaining the distance distV between thestructuring element and the target object in the directionaddressed by vector V
(i) In the case of having an analytical description for boththe SE and the target object an analytical expressionmay be obtained for calculating the distance as wellThis way the distance calculation is straightforwardand its computation time will be low
(ii) When no object can be described using an analyticalexpression (neither the SE nor the target object) thedistance is obtained using a discrete method bothobjects must be discretized to obtain the points intheir boundaries and the distance is obtained point-to-pointThis is theworst case and the computationalcost depends on the discretization precision
(iii) Finally a mixed method is proposed when only oneobject (the SE or the target object) can be describedusing an analytical expression In this case the dis-tance is obtained between the points in the surface ofthe discretized object and the other object as a wholeusing its analytical description The computationalcost is much lower than for the purely discretemethod
23 Trajectory-Based Morphological Operators In this sec-tion the instant basic operator ⊙
Γ(119896)is applied in order to
achieve a whole morphologic operation We are particularlyinterested in defining the two fundamental operations in themorphologic paradigm the dilation and the erosion Dueto the fact that frontiers of objects and structuring elementsare only taken into account to compute the instant basicoperations the goal is to obtain only the boundary of dilationand erosion
231 Dilation In general terms this operation is classicallydefined as the place of the center positions of the structuringelement 119861 when it touches a set119883 [18]
Derived from (3) and (5) we define the instant basicdilation which specifies a center position touching theboundary of a set119883 but from the outside
119883oplusΓ(119896)
119861 = 119901 isin 119864
119901 = distV (119861119883 ∙ Γ (119896)) sdot V and 119861119901cap 119883 =Oslash and 119861
119901cap In (119883) =Oslash
(6)
Using the instant basic dilation we define the trajectory-based dilation 119883oplus
Γ119861 as the set of points resulting from
the repeated and ordered application of the instant basicdilation for the normalized 119896 range [0 1] The trajectorydefined by Γ(119896) must cover all the surface of the objectin the normalized range Note that only boundary pointsare computed and that the frontier of the trajectory-baseddilation is expressed in (6)
119883oplusΓ119861 = ⋃119896isin[01]
(119883oplusΓ(119896)
119861)
= 119909 isin 119864 119861119909cap 119883 =Oslash and 119861
119909cap In (119883) = Oslash
(7)
Trajectory-based dilation can orientate the structuringelement in any position on the object boundary by means ofhomogeneous transformations which are a combination oftranslations and rotations This feature is not supported byclassical dilations In addition partial dilations of objects arenow also possible when a subrange of 119896 is chosen
232 Erosion Classically this operationmdashwhich is com-monly used for image filteringmdashis defined as the place of thecenter positions of the structuring element119861when it is forcedto be inside a set119883
119883Θ119861 = 119909 isin 119864 119861119909sube In (119883) (8)
In our context we are interested only in the erosionboundary which is the place of the center positions that touchthe frontier of set119883 from the inside
Fr (119883Θ119861) = 119909 isin 119864 119861119909sube In (119883) and 119861
119909cap Fr (119883) =Oslash (9)
Derived from (3) and (9) we define the instant basic ero-sion which specifies a center position touching the boundaryof a set119883 but from the inside
119883ΘΓ(119896)
119861 = 119901 isin 119864
119901=distV (119861119883 ∙ Γ (119896)) sdotV and 119861119901sube In (119883) and 119861
119901cap Fr (119883) =Oslash
(10)
Consequently we define the trajectory-based erosion119883ΘΓ(119896)
119861 as the set of points resulting from the repeatedand ordered application of the instant basic erosion for thenormalized 119896 range [0 1] As in the case of dilation thetrajectory defined by Γ(119896) must cover all the surface of theobject in the normalized rangeThe frontier of the trajectory-based erosion is
119883ΘΓ119861 = ⋃119896isin[01]
(119883ΘΓ(119896)
119861)
= 119909 isin 119864 119861119909sube In (119883) and 119861
119909cap Fr (119883) = Oslash
(11)
Figure 3 shows an example of dilation and erosion appliedto a 2D image The black part corresponds to the classicaloperation result The frontier is computed by means of theassociated trajectory-based operator
The Scientific World Journal 5
Morphological operation
Original object boundaries Dilation
Erosion
SE
Figure 3 Classical morphological operations on 2D images On the left a morphological dilation On the right a morphological erosionThe structuring element (SE)mdasha circle of 20 pixels in radiusmdashis shown at the top left corner
Table 1 Characteristics of the algorithms based on classical mathematical morphology versus the morphological trajectory model
Classical morphology Morphological trajectory model
Application space Finite group of pixels as a discretization ofEuclidean space
2D Euclidean space (extensible to 119899-D Euclideanspace)
Objects Based on a complete image Based on the boundary of each object(discretecontinuous)
SEs Group of pixels Geometric representation of the frontier
Method On each pixel it operates in a neighborhoodenvironment defined by the SE
The minimum distance of the SE center is calculatedin a direction V on a trajectory defined by Γ(119896)
Result Erosiondilation as group of pixels Frontier points of the erosiondilation operation
Figure 4 Partial morphological erosion as a subset of the completeerosion (over the subrange defined by the dotted line)
As with dilation trajectory-based erosion can orientatethe structuring element in any position on the object bound-ary and define partial erosion of objects when a subrange of119896 is chosen (see Figure 4)
Table 1 summarizes the main differences between themethods described in this section In classical mathematicalmorphology the calculation is made on a complete imagethat is the morphological operation does not distinguishwhether the pixels belong to a specific object or not it simplyapplies a calculation operation of supremum or infimum in aneighborhood environment In the morphological trajectorymodel (MTM) it is necessary to differentiate between theobjects given in the space since each object has a differentgeometric representation Furthermore this representationdefines the frontier of the object and not its interior Anotherimportant difference is the representation of the structuringelement whereas in traditional morphology it is treated asa subgroup of points (which is discretized for the case ofworking with images) the MTM considers the geometricfunction of the points that make up its frontier withoutbeing necessary to carry out a discretization of the structuringelement
3 Morphological Trajectory Computation
In this section the computation of the erosion for the MTMis presented The algorithm becomes straightforward if themorphological erosion concept defined in [1] is applied Firstof all the boundary curve 119862 of object 119883 is represented asa set of ordered points 119901 organized in collinear segments 119904For every point we compute the structuring element centerposition (1199011015840) that touches each point in the direction of acertain vector V
119901 which must be perpendicular from inside
object 119883 The center 1199011015840 will be valid only if the structuringelement placed at 1199011015840 is inside the shape (ie it will not collidewith the curve 119862) Note that for the dilation operation theprocedure will be the same but in this case the elementwill touch the boundary from the outside A pseudo codealgorithm for the erosion is presented in Algorithm 1
If a point 119901 presents a discontinuity in the first derivativewe generate a set of new vectors in order to cover the gap (seeFigure 5) From that new set we also compute new possiblestructuring element centers
Let us analyze now the computational cost of the MTMalgorithm in terms of the problem size The operator used is119874 to determine an upper limit of the computation cost Asshown in Algorithm 1 the algorithm essentially consists ofan external loop which is used to have access to every pointof the shape and two main function calls Letrsquos call 119899 to thenumber of points that represent the shape 119862 once it has beendiscretized If we use a constant step factor 119904 and the totallength of 119862 is 119871 then 119899 will be 119871119904 points
The function ObtainSECenter computes the center of thestructuring element when it touches a point119901 in the directionaddressed by vector V
119901 So this function depends on the SE
6 The Scientific World Journal
(1) For every 119901119894isin 119862 do
(2) 1199011015840119894= ObtainSECenter(119901
119894V119901119894)
(3) If not CollideSE(1199011015840119894 119862) then AddTrajectory(1199011015840
119894)
(4) Endfor
Algorithm 1 Basic pseudo-code algorithm for the morphological trajectory erosion
Table 2 Equations to calculate the intersection between a circle and a 2D segment
Segment function Circle function Segment-Circle intersection equation on 119905119909 = 119909
Dark-grey SE positions are discarded due to shape collision Notethat discontinuity at 119901
119889is solved by a vector swept generation
geometry For simple SEs such as circles rectangles andtriangles the function can be evaluated in a constant time ctEquation (12) shows this function for a circular SE of radius119877
ObtainSECenter (119901 V119901) = 119901 + 119877 sdot V
119901 (12)
The next function called CollideSE(1199011015840 119862) is true if theSE centered at point 1199011015840 is not completely inside shape 119862 andreturns false otherwise In order to evaluate this conditionthis function computes the intersection of the SE geometryand shape 119862 The cost of this function depends on therepresentation of 119862 For the experiments we have organizedthe shape into a set of contiguous segments that representsthe shape Then every segment is tested (at a constant time)and if a segment produces two or more intersections in theSE geometry then the function returns a true value Notethat in this case the discretization of shape 119862 will not be thesame as the one we used to determine the center positionsin the shape For shapes with a high degree of colinearity thenumber of segments will be reduced slightly Let us call119898 thatnumber of segments
The expressions in Table 2 show the quadratic equationused to determine the intersection between a circle cen-tered at the origin and a 2D segment defined between points
(1199091 1199101) and (119909
2 1199102) for a normalized range 119905 [0 1] As
a conclusion a double solution for variable 119905 in the range[0 1] will cause a true return in the function CollideSEotherwise the next segment will be analyzed
Finally the third function called AddTrajectory adds thenew center 1199011015840 to the list of successful centers at a constanttime so it is not considered for evaluating the cost
As a conclusion let us analyze the whole algorithm inorder to obtain an upper limit for the computational costThenext expression evaluates this cost
We must remark that after completing our experiments119898 ≪ 119899 in most cases since a usual value for119898 takes values ofhundreds The computation times for some examples of theMTM operations will be shown in the following section
4 Experiments
In this section we present some experiments in order to testthe trajectory-based operations The first one compares twoversions of the classical dilation versus the trajectory-basedone In Algorithm 2 we show the classical version algorithmsused in the tests
The algorithm called MM1 corresponds to a classicalmorphological dilation whereas the MM2 refers to thatclassical version operating only on the boundary of theobject Note that MM2 does not perform a valid dilationIt was only developed to test the frontier effect that is theadvantage that MTM has since it only processes boundarypixels The trajectory-based version was called MTM for theexperiments The images were evaluated on an Intel PentiumDual Core processor 28GHz and 2GB in RAMTheywereobtained on a Windows based platform
Several tests and experiments were carried out in order toobtain the computing time under different input conditionsBoth the size of the object and the size of the structuringelement were varied as well as the parameters that took partin the morphological operation
Figure 6 shows the behavior of the algorithms resultingfrom the variation of the size of the structuring element andthe size of the object respectively As a consequence from
The Scientific World Journal 7
10
100
1000
10000
0 20 40 60 80 100 120
Tim
e (m
s)
SE radius (pixels)
MTM
MM2
MM1
(a)
10
100
1000
0 10000 20000 30000Pixels of all objects
MTM
MM2
MM1
Tim
e (m
s)
(b)
Figure 6 Morphological dilation tests On the left influence of the size of the structuring element On the right influence on the size ofobjects
MM1 dilation(1) For 119901 isin 119883 do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) EndforMM2 dilation(1) For 119901 isin Fr(119883) do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) Endfor
Algorithm 2 Pseudo-code used for the 2D experiments
these experiments we can see that the computing time ofthe morphological trajectory model remains almost constantagainst the variations in the size of the structuring elementthis is logical due to the fact that we use its geometricrepresentation instead of its content as opposed to MM1and MM2 The difference between MM2 and MM1 arisesfrom the fact that the former only expands the structuringelement for the pixels in the object boundary and thereforeits computing time decreases an order of magnitude withrespect to MM1
With regard to the variation in the size of the objectsin this experiment the MTM gives better results than MM2and much better than MM1 Since MM2 works only on theobject boundary the difference with the MTM arises fromthe size chosen for the SE if this size is small enough MM2will employ less time than the MTM as it has a simplercomputational logic (see Algorithm 2) The radius of thestructuring element chosen for this test was 40 pixels whichis equivalent to an area of 5025 points In order to determinethe effect of the increase of the size of the structuring element
on a group of objects the experiments described below werecarried out
The experiment shown in Figure 7 compares the execu-tion times of theMTM andMM2 for different radii of the SEItmay be seen that the increase in the structuring element sizecauses a crossing point between MM2 and MTM for objectsizes of about 5000 pixels
Furthermore Figure 7 shows that there is no significantincrease in the computation time for the MTMwhen the sizeof the structuring element is increased since this variation isminimal compared to the classical morphological methodsThe small increase comes from the number of pixels of theobject where the distance function is definedThus if the sizeof the SE increases this number is greater and this causes theneed to make new calculations on the new points
On the other hand Figure 8 shows different results ofthe application of the morphological filters Specifically it isshown how the MTM obtains the frontier of the morpholog-ical operation made by MM1
8 The Scientific World Journal
0
20
40
60
80
100
120
140
0 10000 20000 30000 40000Pixels of all objects
MM2 R80
MM2 R40
MM2 R20
MTM R20 40 80
Tim
e (m
s)
Figure 7 Morphologic dilation Comparative study between MM2and MTMmodels for SE sizes of 20 40 and 80 pixels
Time MM1 = 9515msTime MM2 = 80msTime MTM = 110ms
Time MM1 = 8560msTime MM2 = 34msTime MTM = 96ms
(a)
Time MM1 = 4115msTime MM2 = 180msTime MTM = 131ms
Time MM1 = 1085msTime MM2 = 75msTime MTM = 65ms
(b)
Figure 8 Morphological dilation and erosion Results of differentmorphological operations used for the experiments In the upperpart two erosions and in the lower part two dilationsThe boundaryof the original object is represented in green and the result of MM1operation in black with the MTM result in red
(a) (b)
Figure 9 Erosion of figures using our morphological approach (a)Using a circle as a structuring element (b) Using a rectangle as astructuring element
Finally in Figure 9 we present some images related toother erosion experiments using several structuring elementgeometries where the result is presented in green
5 Conclusions and Discussion
In this paper we have developed a topological system result-ing from applying the conventional morphological modelby means of trajectory-based morphological operations todo this we introduced a new feature consisting of order-ing the morphological primitives As shown the proposedoperations are especially useful when large images need to beprocessed
The morphological trajectory model offers an effectivealternative to traditionalmethods for computingmorpholog-ical primitives This alternative is justified if the number ofpoints of the objects and that of the structuring elements arehigh The independency from the structuring element sizecould be interesting to apply morphological operations onhigh definition images or 3D image reconstruction Due tothe fact that the number of the group points is directly relatedto the dimension of the space in which the object and thestructuring element are defined the importance of the MTMis more relevant when the dimension of the representationspace is increased (3D 4D ) In the two dimensionalspace the application of the MTMmay be justified for high-resolution images where large size operators are applied
Other trajectory-based operations such as openingsclosings and skeletons are defined in [18] We are interestedin demonstrating their utility and efficiency by means of thistrajectory optimization
The newmodel has been presented for binary 2D imagesHowever the new paradigm is extensible for any number ofdimensions of the Euclidean space In [19ndash21] new versionsof the morphological model for color images were presentedThesemodels consider the color as a third coordinate and thecomputation is made in the 3D space Future works will befocused on extending the model in order to support efficientfiltering of color images or real 3D images
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Serra Image Analysis and Mathematical Morphology Aca-demic Press London UK 1982
[2] M van Herk ldquoA fast algorithm for local minimum and max-imum filters on rectangular and octagonal kernelsrdquo PatternRecognition Letters vol 13 no 7 pp 517ndash521 1992
[3] J Gil and M Werman ldquoComputing 2-D min median and maxfiltersrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 15 no 5 pp 504ndash507 1993
[4] J Y Gil and R Kimmel ldquoEfficient dilation erosion openingand closing algorithmsrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 24 no 12 pp 1606ndash1617 2002
The Scientific World Journal 9
[5] O Deforges N Normand and M Babel ldquoFast recursivegrayscale morphology operators from the algorithm to thepipeline architecturerdquo Journal of Real-Time Image Processingvol 8 no 2 pp 143ndash152 2013
[6] C Clienti M Bilodeau and S Beucher ldquoAn efficient hardwarearchitecture without line memories for morphological imageprocessingrdquo in Proceedings of the 10th International ConferenceonAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo08)J Blanc-Talon S Bourennane W Philips D Popescu and PScheunders Eds pp 147ndash156 Springer Heidelberg Germany2008
[7] K Sivakumar M J Patel N Kehtarnavaz Y Balagurunathanand E R Dougherty ldquoA constant-time algorithm for ero-sionsdilations with applications to morphological texture fea-ture computationrdquoReal-Time Imaging vol 6 no 3 pp 223ndash2392000
[8] P Soille and H Talbot ldquoImage Structure Orientation UsingMathematical Morphologyrdquo in Proceedings of the 14th Interna-tional Conference on Pattern Recognition A Jain S Venkateshand B Lovell Eds vol 2 pp 1467ndash1469 Brisbane AustraliaAugust 1998
[9] P Soille and H Talbot ldquoDirectional morphological filteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 23 no 11 pp 1313ndash1329 2001
[10] E R Urbach and M H F Wilkinson ldquoEfficient 2-D grayscalemorphological transformations with arbitrary flat structuringelementsrdquo IEEE Transactions on Image Processing vol 17 no 1pp 1ndash8 2008
[11] Y Zhang and L Wu ldquoRecursive structure element decompo-sition using migration fitness scaling genetic algorithmrdquo inProceedings of the 2nd International Conference on Advances inSwarm Intelligence (ICSI rsquo11) Y Tan Y Shi Y Chai and GWang Eds part 1 pp 514ndash521 Springer Heidelberg Germany2011
[12] G Anelli A Broggi and G Destri ldquoDecomposition of arbi-trarily shaped binarymorphological structuring elements usinggenetic algorithmsrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 20 no 2 pp 217ndash224 1998
[13] H Park and J Yoo ldquoStructuring element decomposition for effi-cient implementation ofmorphological filtersrdquo IEE ProceedingsVision Image and Signal Processing vol 148 no 1 pp 31ndash352001
[14] F Y Shih and Y-T Wu ldquoDecomposition of binary mor-phological structuring elements based on genetic algorithmsrdquoComputer Vision and Image Understanding vol 99 no 2 pp291ndash302 2005
[15] H Hedberg P Dokladal and V Owall ldquoBinary morphologywith spatially variant structuring elements algorithm andarchitecturerdquo IEEE Transactions on Image Processing vol 18 no3 pp 562ndash572 2009
[16] P Dokladal and E Dokladalova ldquoComputationally efficientone-pass algorithm for morphological filtersrdquo Journal of VisualCommunication and Image Representation vol 22 no 5 pp411ndash420 2011
[17] M van Droogenbroeck and M J Buckley ldquoMorphologicalerosions and openings fast algorithms based on anchorsrdquoJournal of Mathematical Imaging and Vision vol 22 no 2 pp121ndash142 2005
[18] E Cooksey and W D Withers ldquoRapid image binarizationwith morphological operatorsrdquo in Proceedings of the 15th IEEEInternational Conference on Image Processing (ICIP rsquo08) pp1017ndash1020 San Diego Calif USA October 2008
[19] J Goutsias H J A M Heijmans and K Sivakumar ldquoMorpho-logical operators for image sequencesrdquo Computer Vision andImage Understanding vol 62 no 3 pp 326ndash346 1995
[20] A Hanbury ldquoMathematical morphology in the HLS colourspacerdquo in Proceedings of the 12th British Machine Vision Con-ference (BMVC rsquo01) vol 2 pp 451ndash460 Manchester UK 2001
[21] J Angulo and J Serra ldquoMorphological color size distributionfor image classification and retrievalrdquo in Proceedings of theAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo02)pp 46ndash53 2002
Figure 1 Examples of several possible representations of structuring elements The leftmost figures show the analytical expressions of SEson the right the corresponding classical SEs are shown
BX
crarr
(a)
XB
c
Γ(k)
rarr
(b)
XB
pc
rarr
(c)
Figure 2 Geometric description of an instant basic operation(a) Initial position (b) Transformation of object 119883 (c) Distancecomputing
the center of a solid 119899-ball of radius 120576 so that the ball is insidethe object
In (119883) = 119909 isin 119864
exist120576gt 0 119861 (119909 120576) sub 119860 (1)
where 119861(119909 120576) is a solid 119899-ball of center 119909 and radius 120576On the other hand let Fr(119883) be a function relating a set
to its border so that all the points belonging to the objectcontour are obtained
Fr (119883) = 119883 minus In (119883) (2)
As mentioned before structuring elements are an essen-tial tool to develop morphological operators For methodsthat use classical mathematical morphology the structuringelement (SE) can be seen as a group of pixels Howeverin our trajectory-based approach the structuring elementwill be defined on the basis of the geometric definition ofits frontier so that any representation of the SE that allowsthe extraction of its frontier is valid for our method Anespecially interesting case is the use of analytical expressionsto define the SE because this continuous representationgives as a result an adaptive precision and a more efficientcomputation than classical SE definitions as it will be shownin the following sections Figure 1 illustrates this concept withtwo cases of analytical SEs (left) and classical SEs (right)For instance the first row shows a circular SE which canbe analytically expressed as (119909 minus 119888
119909)2 + (119910 minus 119888
119910)2 = 1199032
corresponding to a circumference centered on point (119888119909 119888119910)
of radius 119903 instead of the classical neighborhood of pixelsdefined by the area of the circle as shown on the right Thisfact can be also extended to three-dimensional SEs as shownon the second row in Figure 1
This equation-based definition for the structuring ele-ment is used here for simplicity but notice that our approachcan be extended to any other frontier-based definitionThough the use of analytical or classical SEs in ourmodel doesnot add any restriction for computing the trajectory-basedmorphological operators (as it will be shown the method isbased on a distance calculation) the analytical expression ispreferable for efficiency and precision reasons
22 Instant BasicOperations Amorphological operationwillbe divided into a sequence of unitary or basic operationsThis sequence will guarantee the resulting order of the wholeoperation Since every basic operation will correspond to aparticular position of a structuring element along a trajectorythat is performedduring a period of timewe call them instantbasic operations
Let us define the instant basic operator⊙Γ(119896)
for any giveninstant 119896 as follows
119883⊙Γ(119896)
119861 = 119901 isin 119864 119901 = distV (119861 119883 ∙ Γ (119896)) sdot V and 119861119901cap 119883 =Oslash
(3)
where 119883 is the target object 119861 the structuring element 119861119901
are copies of the structuring element centered at every point119901when it touches the boundary of119883119864 equiv 119877119899 Γ(119896) represents anhomogeneous transformationmatrix in 119877119899+1 times119877119899+1 obtainedfor a particular real value 119896 and distV is the Euclidean distancebetween the structuring element and the transformation ofthe object 119883-obtained by postmultiplying every element ofthe set 119883 by the homogeneous transformation matrix Γ(119896)-computed in the direction addressed by vector V In otherwords this operation obtains the structuring element centerwhen it touches the boundary119883 following direction V
A graphical example of this operator is shown in Figure 2Thus an object 119883 is transformed applying a 2D rotationmatrix over its center 119888 For this case 119896 could representthe number of degrees in that transformation matrix so itsvalues are in the [0 2120587) range Once the object is transformed(Figure 2(c)) the distance between 119861 and 119883 in the directionV is applied to the center of 119861 in order to obtain the resultof the instant basic operation (ie point 119901) For differentand ordered real 119896 values (using the lt relation in 119877) we willobtain a new set of structuring element centers 119861
119901that touch
the boundary of 119883 These centers will also be ordered in thegeometric space due to the use of different rotation matrixes
4 The Scientific World Journal
The calculation of function distV is the most time-consuming operation in (2) Here the description of thestructuring element plays a crucial role We propose threemethods for obtaining the distance distV between thestructuring element and the target object in the directionaddressed by vector V
(i) In the case of having an analytical description for boththe SE and the target object an analytical expressionmay be obtained for calculating the distance as wellThis way the distance calculation is straightforwardand its computation time will be low
(ii) When no object can be described using an analyticalexpression (neither the SE nor the target object) thedistance is obtained using a discrete method bothobjects must be discretized to obtain the points intheir boundaries and the distance is obtained point-to-pointThis is theworst case and the computationalcost depends on the discretization precision
(iii) Finally a mixed method is proposed when only oneobject (the SE or the target object) can be describedusing an analytical expression In this case the dis-tance is obtained between the points in the surface ofthe discretized object and the other object as a wholeusing its analytical description The computationalcost is much lower than for the purely discretemethod
23 Trajectory-Based Morphological Operators In this sec-tion the instant basic operator ⊙
Γ(119896)is applied in order to
achieve a whole morphologic operation We are particularlyinterested in defining the two fundamental operations in themorphologic paradigm the dilation and the erosion Dueto the fact that frontiers of objects and structuring elementsare only taken into account to compute the instant basicoperations the goal is to obtain only the boundary of dilationand erosion
231 Dilation In general terms this operation is classicallydefined as the place of the center positions of the structuringelement 119861 when it touches a set119883 [18]
Derived from (3) and (5) we define the instant basicdilation which specifies a center position touching theboundary of a set119883 but from the outside
119883oplusΓ(119896)
119861 = 119901 isin 119864
119901 = distV (119861119883 ∙ Γ (119896)) sdot V and 119861119901cap 119883 =Oslash and 119861
119901cap In (119883) =Oslash
(6)
Using the instant basic dilation we define the trajectory-based dilation 119883oplus
Γ119861 as the set of points resulting from
the repeated and ordered application of the instant basicdilation for the normalized 119896 range [0 1] The trajectorydefined by Γ(119896) must cover all the surface of the objectin the normalized range Note that only boundary pointsare computed and that the frontier of the trajectory-baseddilation is expressed in (6)
119883oplusΓ119861 = ⋃119896isin[01]
(119883oplusΓ(119896)
119861)
= 119909 isin 119864 119861119909cap 119883 =Oslash and 119861
119909cap In (119883) = Oslash
(7)
Trajectory-based dilation can orientate the structuringelement in any position on the object boundary by means ofhomogeneous transformations which are a combination oftranslations and rotations This feature is not supported byclassical dilations In addition partial dilations of objects arenow also possible when a subrange of 119896 is chosen
232 Erosion Classically this operationmdashwhich is com-monly used for image filteringmdashis defined as the place of thecenter positions of the structuring element119861when it is forcedto be inside a set119883
119883Θ119861 = 119909 isin 119864 119861119909sube In (119883) (8)
In our context we are interested only in the erosionboundary which is the place of the center positions that touchthe frontier of set119883 from the inside
Fr (119883Θ119861) = 119909 isin 119864 119861119909sube In (119883) and 119861
119909cap Fr (119883) =Oslash (9)
Derived from (3) and (9) we define the instant basic ero-sion which specifies a center position touching the boundaryof a set119883 but from the inside
119883ΘΓ(119896)
119861 = 119901 isin 119864
119901=distV (119861119883 ∙ Γ (119896)) sdotV and 119861119901sube In (119883) and 119861
119901cap Fr (119883) =Oslash
(10)
Consequently we define the trajectory-based erosion119883ΘΓ(119896)
119861 as the set of points resulting from the repeatedand ordered application of the instant basic erosion for thenormalized 119896 range [0 1] As in the case of dilation thetrajectory defined by Γ(119896) must cover all the surface of theobject in the normalized rangeThe frontier of the trajectory-based erosion is
119883ΘΓ119861 = ⋃119896isin[01]
(119883ΘΓ(119896)
119861)
= 119909 isin 119864 119861119909sube In (119883) and 119861
119909cap Fr (119883) = Oslash
(11)
Figure 3 shows an example of dilation and erosion appliedto a 2D image The black part corresponds to the classicaloperation result The frontier is computed by means of theassociated trajectory-based operator
The Scientific World Journal 5
Morphological operation
Original object boundaries Dilation
Erosion
SE
Figure 3 Classical morphological operations on 2D images On the left a morphological dilation On the right a morphological erosionThe structuring element (SE)mdasha circle of 20 pixels in radiusmdashis shown at the top left corner
Table 1 Characteristics of the algorithms based on classical mathematical morphology versus the morphological trajectory model
Classical morphology Morphological trajectory model
Application space Finite group of pixels as a discretization ofEuclidean space
2D Euclidean space (extensible to 119899-D Euclideanspace)
Objects Based on a complete image Based on the boundary of each object(discretecontinuous)
SEs Group of pixels Geometric representation of the frontier
Method On each pixel it operates in a neighborhoodenvironment defined by the SE
The minimum distance of the SE center is calculatedin a direction V on a trajectory defined by Γ(119896)
Result Erosiondilation as group of pixels Frontier points of the erosiondilation operation
Figure 4 Partial morphological erosion as a subset of the completeerosion (over the subrange defined by the dotted line)
As with dilation trajectory-based erosion can orientatethe structuring element in any position on the object bound-ary and define partial erosion of objects when a subrange of119896 is chosen (see Figure 4)
Table 1 summarizes the main differences between themethods described in this section In classical mathematicalmorphology the calculation is made on a complete imagethat is the morphological operation does not distinguishwhether the pixels belong to a specific object or not it simplyapplies a calculation operation of supremum or infimum in aneighborhood environment In the morphological trajectorymodel (MTM) it is necessary to differentiate between theobjects given in the space since each object has a differentgeometric representation Furthermore this representationdefines the frontier of the object and not its interior Anotherimportant difference is the representation of the structuringelement whereas in traditional morphology it is treated asa subgroup of points (which is discretized for the case ofworking with images) the MTM considers the geometricfunction of the points that make up its frontier withoutbeing necessary to carry out a discretization of the structuringelement
3 Morphological Trajectory Computation
In this section the computation of the erosion for the MTMis presented The algorithm becomes straightforward if themorphological erosion concept defined in [1] is applied Firstof all the boundary curve 119862 of object 119883 is represented asa set of ordered points 119901 organized in collinear segments 119904For every point we compute the structuring element centerposition (1199011015840) that touches each point in the direction of acertain vector V
119901 which must be perpendicular from inside
object 119883 The center 1199011015840 will be valid only if the structuringelement placed at 1199011015840 is inside the shape (ie it will not collidewith the curve 119862) Note that for the dilation operation theprocedure will be the same but in this case the elementwill touch the boundary from the outside A pseudo codealgorithm for the erosion is presented in Algorithm 1
If a point 119901 presents a discontinuity in the first derivativewe generate a set of new vectors in order to cover the gap (seeFigure 5) From that new set we also compute new possiblestructuring element centers
Let us analyze now the computational cost of the MTMalgorithm in terms of the problem size The operator used is119874 to determine an upper limit of the computation cost Asshown in Algorithm 1 the algorithm essentially consists ofan external loop which is used to have access to every pointof the shape and two main function calls Letrsquos call 119899 to thenumber of points that represent the shape 119862 once it has beendiscretized If we use a constant step factor 119904 and the totallength of 119862 is 119871 then 119899 will be 119871119904 points
The function ObtainSECenter computes the center of thestructuring element when it touches a point119901 in the directionaddressed by vector V
119901 So this function depends on the SE
6 The Scientific World Journal
(1) For every 119901119894isin 119862 do
(2) 1199011015840119894= ObtainSECenter(119901
119894V119901119894)
(3) If not CollideSE(1199011015840119894 119862) then AddTrajectory(1199011015840
119894)
(4) Endfor
Algorithm 1 Basic pseudo-code algorithm for the morphological trajectory erosion
Table 2 Equations to calculate the intersection between a circle and a 2D segment
Segment function Circle function Segment-Circle intersection equation on 119905119909 = 119909
Dark-grey SE positions are discarded due to shape collision Notethat discontinuity at 119901
119889is solved by a vector swept generation
geometry For simple SEs such as circles rectangles andtriangles the function can be evaluated in a constant time ctEquation (12) shows this function for a circular SE of radius119877
ObtainSECenter (119901 V119901) = 119901 + 119877 sdot V
119901 (12)
The next function called CollideSE(1199011015840 119862) is true if theSE centered at point 1199011015840 is not completely inside shape 119862 andreturns false otherwise In order to evaluate this conditionthis function computes the intersection of the SE geometryand shape 119862 The cost of this function depends on therepresentation of 119862 For the experiments we have organizedthe shape into a set of contiguous segments that representsthe shape Then every segment is tested (at a constant time)and if a segment produces two or more intersections in theSE geometry then the function returns a true value Notethat in this case the discretization of shape 119862 will not be thesame as the one we used to determine the center positionsin the shape For shapes with a high degree of colinearity thenumber of segments will be reduced slightly Let us call119898 thatnumber of segments
The expressions in Table 2 show the quadratic equationused to determine the intersection between a circle cen-tered at the origin and a 2D segment defined between points
(1199091 1199101) and (119909
2 1199102) for a normalized range 119905 [0 1] As
a conclusion a double solution for variable 119905 in the range[0 1] will cause a true return in the function CollideSEotherwise the next segment will be analyzed
Finally the third function called AddTrajectory adds thenew center 1199011015840 to the list of successful centers at a constanttime so it is not considered for evaluating the cost
As a conclusion let us analyze the whole algorithm inorder to obtain an upper limit for the computational costThenext expression evaluates this cost
We must remark that after completing our experiments119898 ≪ 119899 in most cases since a usual value for119898 takes values ofhundreds The computation times for some examples of theMTM operations will be shown in the following section
4 Experiments
In this section we present some experiments in order to testthe trajectory-based operations The first one compares twoversions of the classical dilation versus the trajectory-basedone In Algorithm 2 we show the classical version algorithmsused in the tests
The algorithm called MM1 corresponds to a classicalmorphological dilation whereas the MM2 refers to thatclassical version operating only on the boundary of theobject Note that MM2 does not perform a valid dilationIt was only developed to test the frontier effect that is theadvantage that MTM has since it only processes boundarypixels The trajectory-based version was called MTM for theexperiments The images were evaluated on an Intel PentiumDual Core processor 28GHz and 2GB in RAMTheywereobtained on a Windows based platform
Several tests and experiments were carried out in order toobtain the computing time under different input conditionsBoth the size of the object and the size of the structuringelement were varied as well as the parameters that took partin the morphological operation
Figure 6 shows the behavior of the algorithms resultingfrom the variation of the size of the structuring element andthe size of the object respectively As a consequence from
The Scientific World Journal 7
10
100
1000
10000
0 20 40 60 80 100 120
Tim
e (m
s)
SE radius (pixels)
MTM
MM2
MM1
(a)
10
100
1000
0 10000 20000 30000Pixels of all objects
MTM
MM2
MM1
Tim
e (m
s)
(b)
Figure 6 Morphological dilation tests On the left influence of the size of the structuring element On the right influence on the size ofobjects
MM1 dilation(1) For 119901 isin 119883 do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) EndforMM2 dilation(1) For 119901 isin Fr(119883) do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) Endfor
Algorithm 2 Pseudo-code used for the 2D experiments
these experiments we can see that the computing time ofthe morphological trajectory model remains almost constantagainst the variations in the size of the structuring elementthis is logical due to the fact that we use its geometricrepresentation instead of its content as opposed to MM1and MM2 The difference between MM2 and MM1 arisesfrom the fact that the former only expands the structuringelement for the pixels in the object boundary and thereforeits computing time decreases an order of magnitude withrespect to MM1
With regard to the variation in the size of the objectsin this experiment the MTM gives better results than MM2and much better than MM1 Since MM2 works only on theobject boundary the difference with the MTM arises fromthe size chosen for the SE if this size is small enough MM2will employ less time than the MTM as it has a simplercomputational logic (see Algorithm 2) The radius of thestructuring element chosen for this test was 40 pixels whichis equivalent to an area of 5025 points In order to determinethe effect of the increase of the size of the structuring element
on a group of objects the experiments described below werecarried out
The experiment shown in Figure 7 compares the execu-tion times of theMTM andMM2 for different radii of the SEItmay be seen that the increase in the structuring element sizecauses a crossing point between MM2 and MTM for objectsizes of about 5000 pixels
Furthermore Figure 7 shows that there is no significantincrease in the computation time for the MTMwhen the sizeof the structuring element is increased since this variation isminimal compared to the classical morphological methodsThe small increase comes from the number of pixels of theobject where the distance function is definedThus if the sizeof the SE increases this number is greater and this causes theneed to make new calculations on the new points
On the other hand Figure 8 shows different results ofthe application of the morphological filters Specifically it isshown how the MTM obtains the frontier of the morpholog-ical operation made by MM1
8 The Scientific World Journal
0
20
40
60
80
100
120
140
0 10000 20000 30000 40000Pixels of all objects
MM2 R80
MM2 R40
MM2 R20
MTM R20 40 80
Tim
e (m
s)
Figure 7 Morphologic dilation Comparative study between MM2and MTMmodels for SE sizes of 20 40 and 80 pixels
Time MM1 = 9515msTime MM2 = 80msTime MTM = 110ms
Time MM1 = 8560msTime MM2 = 34msTime MTM = 96ms
(a)
Time MM1 = 4115msTime MM2 = 180msTime MTM = 131ms
Time MM1 = 1085msTime MM2 = 75msTime MTM = 65ms
(b)
Figure 8 Morphological dilation and erosion Results of differentmorphological operations used for the experiments In the upperpart two erosions and in the lower part two dilationsThe boundaryof the original object is represented in green and the result of MM1operation in black with the MTM result in red
(a) (b)
Figure 9 Erosion of figures using our morphological approach (a)Using a circle as a structuring element (b) Using a rectangle as astructuring element
Finally in Figure 9 we present some images related toother erosion experiments using several structuring elementgeometries where the result is presented in green
5 Conclusions and Discussion
In this paper we have developed a topological system result-ing from applying the conventional morphological modelby means of trajectory-based morphological operations todo this we introduced a new feature consisting of order-ing the morphological primitives As shown the proposedoperations are especially useful when large images need to beprocessed
The morphological trajectory model offers an effectivealternative to traditionalmethods for computingmorpholog-ical primitives This alternative is justified if the number ofpoints of the objects and that of the structuring elements arehigh The independency from the structuring element sizecould be interesting to apply morphological operations onhigh definition images or 3D image reconstruction Due tothe fact that the number of the group points is directly relatedto the dimension of the space in which the object and thestructuring element are defined the importance of the MTMis more relevant when the dimension of the representationspace is increased (3D 4D ) In the two dimensionalspace the application of the MTMmay be justified for high-resolution images where large size operators are applied
Other trajectory-based operations such as openingsclosings and skeletons are defined in [18] We are interestedin demonstrating their utility and efficiency by means of thistrajectory optimization
The newmodel has been presented for binary 2D imagesHowever the new paradigm is extensible for any number ofdimensions of the Euclidean space In [19ndash21] new versionsof the morphological model for color images were presentedThesemodels consider the color as a third coordinate and thecomputation is made in the 3D space Future works will befocused on extending the model in order to support efficientfiltering of color images or real 3D images
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Serra Image Analysis and Mathematical Morphology Aca-demic Press London UK 1982
[2] M van Herk ldquoA fast algorithm for local minimum and max-imum filters on rectangular and octagonal kernelsrdquo PatternRecognition Letters vol 13 no 7 pp 517ndash521 1992
[3] J Gil and M Werman ldquoComputing 2-D min median and maxfiltersrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 15 no 5 pp 504ndash507 1993
[4] J Y Gil and R Kimmel ldquoEfficient dilation erosion openingand closing algorithmsrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 24 no 12 pp 1606ndash1617 2002
The Scientific World Journal 9
[5] O Deforges N Normand and M Babel ldquoFast recursivegrayscale morphology operators from the algorithm to thepipeline architecturerdquo Journal of Real-Time Image Processingvol 8 no 2 pp 143ndash152 2013
[6] C Clienti M Bilodeau and S Beucher ldquoAn efficient hardwarearchitecture without line memories for morphological imageprocessingrdquo in Proceedings of the 10th International ConferenceonAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo08)J Blanc-Talon S Bourennane W Philips D Popescu and PScheunders Eds pp 147ndash156 Springer Heidelberg Germany2008
[7] K Sivakumar M J Patel N Kehtarnavaz Y Balagurunathanand E R Dougherty ldquoA constant-time algorithm for ero-sionsdilations with applications to morphological texture fea-ture computationrdquoReal-Time Imaging vol 6 no 3 pp 223ndash2392000
[8] P Soille and H Talbot ldquoImage Structure Orientation UsingMathematical Morphologyrdquo in Proceedings of the 14th Interna-tional Conference on Pattern Recognition A Jain S Venkateshand B Lovell Eds vol 2 pp 1467ndash1469 Brisbane AustraliaAugust 1998
[9] P Soille and H Talbot ldquoDirectional morphological filteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 23 no 11 pp 1313ndash1329 2001
[10] E R Urbach and M H F Wilkinson ldquoEfficient 2-D grayscalemorphological transformations with arbitrary flat structuringelementsrdquo IEEE Transactions on Image Processing vol 17 no 1pp 1ndash8 2008
[11] Y Zhang and L Wu ldquoRecursive structure element decompo-sition using migration fitness scaling genetic algorithmrdquo inProceedings of the 2nd International Conference on Advances inSwarm Intelligence (ICSI rsquo11) Y Tan Y Shi Y Chai and GWang Eds part 1 pp 514ndash521 Springer Heidelberg Germany2011
[12] G Anelli A Broggi and G Destri ldquoDecomposition of arbi-trarily shaped binarymorphological structuring elements usinggenetic algorithmsrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 20 no 2 pp 217ndash224 1998
[13] H Park and J Yoo ldquoStructuring element decomposition for effi-cient implementation ofmorphological filtersrdquo IEE ProceedingsVision Image and Signal Processing vol 148 no 1 pp 31ndash352001
[14] F Y Shih and Y-T Wu ldquoDecomposition of binary mor-phological structuring elements based on genetic algorithmsrdquoComputer Vision and Image Understanding vol 99 no 2 pp291ndash302 2005
[15] H Hedberg P Dokladal and V Owall ldquoBinary morphologywith spatially variant structuring elements algorithm andarchitecturerdquo IEEE Transactions on Image Processing vol 18 no3 pp 562ndash572 2009
[16] P Dokladal and E Dokladalova ldquoComputationally efficientone-pass algorithm for morphological filtersrdquo Journal of VisualCommunication and Image Representation vol 22 no 5 pp411ndash420 2011
[17] M van Droogenbroeck and M J Buckley ldquoMorphologicalerosions and openings fast algorithms based on anchorsrdquoJournal of Mathematical Imaging and Vision vol 22 no 2 pp121ndash142 2005
[18] E Cooksey and W D Withers ldquoRapid image binarizationwith morphological operatorsrdquo in Proceedings of the 15th IEEEInternational Conference on Image Processing (ICIP rsquo08) pp1017ndash1020 San Diego Calif USA October 2008
[19] J Goutsias H J A M Heijmans and K Sivakumar ldquoMorpho-logical operators for image sequencesrdquo Computer Vision andImage Understanding vol 62 no 3 pp 326ndash346 1995
[20] A Hanbury ldquoMathematical morphology in the HLS colourspacerdquo in Proceedings of the 12th British Machine Vision Con-ference (BMVC rsquo01) vol 2 pp 451ndash460 Manchester UK 2001
[21] J Angulo and J Serra ldquoMorphological color size distributionfor image classification and retrievalrdquo in Proceedings of theAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo02)pp 46ndash53 2002
The calculation of function distV is the most time-consuming operation in (2) Here the description of thestructuring element plays a crucial role We propose threemethods for obtaining the distance distV between thestructuring element and the target object in the directionaddressed by vector V
(i) In the case of having an analytical description for boththe SE and the target object an analytical expressionmay be obtained for calculating the distance as wellThis way the distance calculation is straightforwardand its computation time will be low
(ii) When no object can be described using an analyticalexpression (neither the SE nor the target object) thedistance is obtained using a discrete method bothobjects must be discretized to obtain the points intheir boundaries and the distance is obtained point-to-pointThis is theworst case and the computationalcost depends on the discretization precision
(iii) Finally a mixed method is proposed when only oneobject (the SE or the target object) can be describedusing an analytical expression In this case the dis-tance is obtained between the points in the surface ofthe discretized object and the other object as a wholeusing its analytical description The computationalcost is much lower than for the purely discretemethod
23 Trajectory-Based Morphological Operators In this sec-tion the instant basic operator ⊙
Γ(119896)is applied in order to
achieve a whole morphologic operation We are particularlyinterested in defining the two fundamental operations in themorphologic paradigm the dilation and the erosion Dueto the fact that frontiers of objects and structuring elementsare only taken into account to compute the instant basicoperations the goal is to obtain only the boundary of dilationand erosion
231 Dilation In general terms this operation is classicallydefined as the place of the center positions of the structuringelement 119861 when it touches a set119883 [18]
Derived from (3) and (5) we define the instant basicdilation which specifies a center position touching theboundary of a set119883 but from the outside
119883oplusΓ(119896)
119861 = 119901 isin 119864
119901 = distV (119861119883 ∙ Γ (119896)) sdot V and 119861119901cap 119883 =Oslash and 119861
119901cap In (119883) =Oslash
(6)
Using the instant basic dilation we define the trajectory-based dilation 119883oplus
Γ119861 as the set of points resulting from
the repeated and ordered application of the instant basicdilation for the normalized 119896 range [0 1] The trajectorydefined by Γ(119896) must cover all the surface of the objectin the normalized range Note that only boundary pointsare computed and that the frontier of the trajectory-baseddilation is expressed in (6)
119883oplusΓ119861 = ⋃119896isin[01]
(119883oplusΓ(119896)
119861)
= 119909 isin 119864 119861119909cap 119883 =Oslash and 119861
119909cap In (119883) = Oslash
(7)
Trajectory-based dilation can orientate the structuringelement in any position on the object boundary by means ofhomogeneous transformations which are a combination oftranslations and rotations This feature is not supported byclassical dilations In addition partial dilations of objects arenow also possible when a subrange of 119896 is chosen
232 Erosion Classically this operationmdashwhich is com-monly used for image filteringmdashis defined as the place of thecenter positions of the structuring element119861when it is forcedto be inside a set119883
119883Θ119861 = 119909 isin 119864 119861119909sube In (119883) (8)
In our context we are interested only in the erosionboundary which is the place of the center positions that touchthe frontier of set119883 from the inside
Fr (119883Θ119861) = 119909 isin 119864 119861119909sube In (119883) and 119861
119909cap Fr (119883) =Oslash (9)
Derived from (3) and (9) we define the instant basic ero-sion which specifies a center position touching the boundaryof a set119883 but from the inside
119883ΘΓ(119896)
119861 = 119901 isin 119864
119901=distV (119861119883 ∙ Γ (119896)) sdotV and 119861119901sube In (119883) and 119861
119901cap Fr (119883) =Oslash
(10)
Consequently we define the trajectory-based erosion119883ΘΓ(119896)
119861 as the set of points resulting from the repeatedand ordered application of the instant basic erosion for thenormalized 119896 range [0 1] As in the case of dilation thetrajectory defined by Γ(119896) must cover all the surface of theobject in the normalized rangeThe frontier of the trajectory-based erosion is
119883ΘΓ119861 = ⋃119896isin[01]
(119883ΘΓ(119896)
119861)
= 119909 isin 119864 119861119909sube In (119883) and 119861
119909cap Fr (119883) = Oslash
(11)
Figure 3 shows an example of dilation and erosion appliedto a 2D image The black part corresponds to the classicaloperation result The frontier is computed by means of theassociated trajectory-based operator
The Scientific World Journal 5
Morphological operation
Original object boundaries Dilation
Erosion
SE
Figure 3 Classical morphological operations on 2D images On the left a morphological dilation On the right a morphological erosionThe structuring element (SE)mdasha circle of 20 pixels in radiusmdashis shown at the top left corner
Table 1 Characteristics of the algorithms based on classical mathematical morphology versus the morphological trajectory model
Classical morphology Morphological trajectory model
Application space Finite group of pixels as a discretization ofEuclidean space
2D Euclidean space (extensible to 119899-D Euclideanspace)
Objects Based on a complete image Based on the boundary of each object(discretecontinuous)
SEs Group of pixels Geometric representation of the frontier
Method On each pixel it operates in a neighborhoodenvironment defined by the SE
The minimum distance of the SE center is calculatedin a direction V on a trajectory defined by Γ(119896)
Result Erosiondilation as group of pixels Frontier points of the erosiondilation operation
Figure 4 Partial morphological erosion as a subset of the completeerosion (over the subrange defined by the dotted line)
As with dilation trajectory-based erosion can orientatethe structuring element in any position on the object bound-ary and define partial erosion of objects when a subrange of119896 is chosen (see Figure 4)
Table 1 summarizes the main differences between themethods described in this section In classical mathematicalmorphology the calculation is made on a complete imagethat is the morphological operation does not distinguishwhether the pixels belong to a specific object or not it simplyapplies a calculation operation of supremum or infimum in aneighborhood environment In the morphological trajectorymodel (MTM) it is necessary to differentiate between theobjects given in the space since each object has a differentgeometric representation Furthermore this representationdefines the frontier of the object and not its interior Anotherimportant difference is the representation of the structuringelement whereas in traditional morphology it is treated asa subgroup of points (which is discretized for the case ofworking with images) the MTM considers the geometricfunction of the points that make up its frontier withoutbeing necessary to carry out a discretization of the structuringelement
3 Morphological Trajectory Computation
In this section the computation of the erosion for the MTMis presented The algorithm becomes straightforward if themorphological erosion concept defined in [1] is applied Firstof all the boundary curve 119862 of object 119883 is represented asa set of ordered points 119901 organized in collinear segments 119904For every point we compute the structuring element centerposition (1199011015840) that touches each point in the direction of acertain vector V
119901 which must be perpendicular from inside
object 119883 The center 1199011015840 will be valid only if the structuringelement placed at 1199011015840 is inside the shape (ie it will not collidewith the curve 119862) Note that for the dilation operation theprocedure will be the same but in this case the elementwill touch the boundary from the outside A pseudo codealgorithm for the erosion is presented in Algorithm 1
If a point 119901 presents a discontinuity in the first derivativewe generate a set of new vectors in order to cover the gap (seeFigure 5) From that new set we also compute new possiblestructuring element centers
Let us analyze now the computational cost of the MTMalgorithm in terms of the problem size The operator used is119874 to determine an upper limit of the computation cost Asshown in Algorithm 1 the algorithm essentially consists ofan external loop which is used to have access to every pointof the shape and two main function calls Letrsquos call 119899 to thenumber of points that represent the shape 119862 once it has beendiscretized If we use a constant step factor 119904 and the totallength of 119862 is 119871 then 119899 will be 119871119904 points
The function ObtainSECenter computes the center of thestructuring element when it touches a point119901 in the directionaddressed by vector V
119901 So this function depends on the SE
6 The Scientific World Journal
(1) For every 119901119894isin 119862 do
(2) 1199011015840119894= ObtainSECenter(119901
119894V119901119894)
(3) If not CollideSE(1199011015840119894 119862) then AddTrajectory(1199011015840
119894)
(4) Endfor
Algorithm 1 Basic pseudo-code algorithm for the morphological trajectory erosion
Table 2 Equations to calculate the intersection between a circle and a 2D segment
Segment function Circle function Segment-Circle intersection equation on 119905119909 = 119909
Dark-grey SE positions are discarded due to shape collision Notethat discontinuity at 119901
119889is solved by a vector swept generation
geometry For simple SEs such as circles rectangles andtriangles the function can be evaluated in a constant time ctEquation (12) shows this function for a circular SE of radius119877
ObtainSECenter (119901 V119901) = 119901 + 119877 sdot V
119901 (12)
The next function called CollideSE(1199011015840 119862) is true if theSE centered at point 1199011015840 is not completely inside shape 119862 andreturns false otherwise In order to evaluate this conditionthis function computes the intersection of the SE geometryand shape 119862 The cost of this function depends on therepresentation of 119862 For the experiments we have organizedthe shape into a set of contiguous segments that representsthe shape Then every segment is tested (at a constant time)and if a segment produces two or more intersections in theSE geometry then the function returns a true value Notethat in this case the discretization of shape 119862 will not be thesame as the one we used to determine the center positionsin the shape For shapes with a high degree of colinearity thenumber of segments will be reduced slightly Let us call119898 thatnumber of segments
The expressions in Table 2 show the quadratic equationused to determine the intersection between a circle cen-tered at the origin and a 2D segment defined between points
(1199091 1199101) and (119909
2 1199102) for a normalized range 119905 [0 1] As
a conclusion a double solution for variable 119905 in the range[0 1] will cause a true return in the function CollideSEotherwise the next segment will be analyzed
Finally the third function called AddTrajectory adds thenew center 1199011015840 to the list of successful centers at a constanttime so it is not considered for evaluating the cost
As a conclusion let us analyze the whole algorithm inorder to obtain an upper limit for the computational costThenext expression evaluates this cost
We must remark that after completing our experiments119898 ≪ 119899 in most cases since a usual value for119898 takes values ofhundreds The computation times for some examples of theMTM operations will be shown in the following section
4 Experiments
In this section we present some experiments in order to testthe trajectory-based operations The first one compares twoversions of the classical dilation versus the trajectory-basedone In Algorithm 2 we show the classical version algorithmsused in the tests
The algorithm called MM1 corresponds to a classicalmorphological dilation whereas the MM2 refers to thatclassical version operating only on the boundary of theobject Note that MM2 does not perform a valid dilationIt was only developed to test the frontier effect that is theadvantage that MTM has since it only processes boundarypixels The trajectory-based version was called MTM for theexperiments The images were evaluated on an Intel PentiumDual Core processor 28GHz and 2GB in RAMTheywereobtained on a Windows based platform
Several tests and experiments were carried out in order toobtain the computing time under different input conditionsBoth the size of the object and the size of the structuringelement were varied as well as the parameters that took partin the morphological operation
Figure 6 shows the behavior of the algorithms resultingfrom the variation of the size of the structuring element andthe size of the object respectively As a consequence from
The Scientific World Journal 7
10
100
1000
10000
0 20 40 60 80 100 120
Tim
e (m
s)
SE radius (pixels)
MTM
MM2
MM1
(a)
10
100
1000
0 10000 20000 30000Pixels of all objects
MTM
MM2
MM1
Tim
e (m
s)
(b)
Figure 6 Morphological dilation tests On the left influence of the size of the structuring element On the right influence on the size ofobjects
MM1 dilation(1) For 119901 isin 119883 do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) EndforMM2 dilation(1) For 119901 isin Fr(119883) do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) Endfor
Algorithm 2 Pseudo-code used for the 2D experiments
these experiments we can see that the computing time ofthe morphological trajectory model remains almost constantagainst the variations in the size of the structuring elementthis is logical due to the fact that we use its geometricrepresentation instead of its content as opposed to MM1and MM2 The difference between MM2 and MM1 arisesfrom the fact that the former only expands the structuringelement for the pixels in the object boundary and thereforeits computing time decreases an order of magnitude withrespect to MM1
With regard to the variation in the size of the objectsin this experiment the MTM gives better results than MM2and much better than MM1 Since MM2 works only on theobject boundary the difference with the MTM arises fromthe size chosen for the SE if this size is small enough MM2will employ less time than the MTM as it has a simplercomputational logic (see Algorithm 2) The radius of thestructuring element chosen for this test was 40 pixels whichis equivalent to an area of 5025 points In order to determinethe effect of the increase of the size of the structuring element
on a group of objects the experiments described below werecarried out
The experiment shown in Figure 7 compares the execu-tion times of theMTM andMM2 for different radii of the SEItmay be seen that the increase in the structuring element sizecauses a crossing point between MM2 and MTM for objectsizes of about 5000 pixels
Furthermore Figure 7 shows that there is no significantincrease in the computation time for the MTMwhen the sizeof the structuring element is increased since this variation isminimal compared to the classical morphological methodsThe small increase comes from the number of pixels of theobject where the distance function is definedThus if the sizeof the SE increases this number is greater and this causes theneed to make new calculations on the new points
On the other hand Figure 8 shows different results ofthe application of the morphological filters Specifically it isshown how the MTM obtains the frontier of the morpholog-ical operation made by MM1
8 The Scientific World Journal
0
20
40
60
80
100
120
140
0 10000 20000 30000 40000Pixels of all objects
MM2 R80
MM2 R40
MM2 R20
MTM R20 40 80
Tim
e (m
s)
Figure 7 Morphologic dilation Comparative study between MM2and MTMmodels for SE sizes of 20 40 and 80 pixels
Time MM1 = 9515msTime MM2 = 80msTime MTM = 110ms
Time MM1 = 8560msTime MM2 = 34msTime MTM = 96ms
(a)
Time MM1 = 4115msTime MM2 = 180msTime MTM = 131ms
Time MM1 = 1085msTime MM2 = 75msTime MTM = 65ms
(b)
Figure 8 Morphological dilation and erosion Results of differentmorphological operations used for the experiments In the upperpart two erosions and in the lower part two dilationsThe boundaryof the original object is represented in green and the result of MM1operation in black with the MTM result in red
(a) (b)
Figure 9 Erosion of figures using our morphological approach (a)Using a circle as a structuring element (b) Using a rectangle as astructuring element
Finally in Figure 9 we present some images related toother erosion experiments using several structuring elementgeometries where the result is presented in green
5 Conclusions and Discussion
In this paper we have developed a topological system result-ing from applying the conventional morphological modelby means of trajectory-based morphological operations todo this we introduced a new feature consisting of order-ing the morphological primitives As shown the proposedoperations are especially useful when large images need to beprocessed
The morphological trajectory model offers an effectivealternative to traditionalmethods for computingmorpholog-ical primitives This alternative is justified if the number ofpoints of the objects and that of the structuring elements arehigh The independency from the structuring element sizecould be interesting to apply morphological operations onhigh definition images or 3D image reconstruction Due tothe fact that the number of the group points is directly relatedto the dimension of the space in which the object and thestructuring element are defined the importance of the MTMis more relevant when the dimension of the representationspace is increased (3D 4D ) In the two dimensionalspace the application of the MTMmay be justified for high-resolution images where large size operators are applied
Other trajectory-based operations such as openingsclosings and skeletons are defined in [18] We are interestedin demonstrating their utility and efficiency by means of thistrajectory optimization
The newmodel has been presented for binary 2D imagesHowever the new paradigm is extensible for any number ofdimensions of the Euclidean space In [19ndash21] new versionsof the morphological model for color images were presentedThesemodels consider the color as a third coordinate and thecomputation is made in the 3D space Future works will befocused on extending the model in order to support efficientfiltering of color images or real 3D images
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Serra Image Analysis and Mathematical Morphology Aca-demic Press London UK 1982
[2] M van Herk ldquoA fast algorithm for local minimum and max-imum filters on rectangular and octagonal kernelsrdquo PatternRecognition Letters vol 13 no 7 pp 517ndash521 1992
[3] J Gil and M Werman ldquoComputing 2-D min median and maxfiltersrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 15 no 5 pp 504ndash507 1993
[4] J Y Gil and R Kimmel ldquoEfficient dilation erosion openingand closing algorithmsrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 24 no 12 pp 1606ndash1617 2002
The Scientific World Journal 9
[5] O Deforges N Normand and M Babel ldquoFast recursivegrayscale morphology operators from the algorithm to thepipeline architecturerdquo Journal of Real-Time Image Processingvol 8 no 2 pp 143ndash152 2013
[6] C Clienti M Bilodeau and S Beucher ldquoAn efficient hardwarearchitecture without line memories for morphological imageprocessingrdquo in Proceedings of the 10th International ConferenceonAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo08)J Blanc-Talon S Bourennane W Philips D Popescu and PScheunders Eds pp 147ndash156 Springer Heidelberg Germany2008
[7] K Sivakumar M J Patel N Kehtarnavaz Y Balagurunathanand E R Dougherty ldquoA constant-time algorithm for ero-sionsdilations with applications to morphological texture fea-ture computationrdquoReal-Time Imaging vol 6 no 3 pp 223ndash2392000
[8] P Soille and H Talbot ldquoImage Structure Orientation UsingMathematical Morphologyrdquo in Proceedings of the 14th Interna-tional Conference on Pattern Recognition A Jain S Venkateshand B Lovell Eds vol 2 pp 1467ndash1469 Brisbane AustraliaAugust 1998
[9] P Soille and H Talbot ldquoDirectional morphological filteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 23 no 11 pp 1313ndash1329 2001
[10] E R Urbach and M H F Wilkinson ldquoEfficient 2-D grayscalemorphological transformations with arbitrary flat structuringelementsrdquo IEEE Transactions on Image Processing vol 17 no 1pp 1ndash8 2008
[11] Y Zhang and L Wu ldquoRecursive structure element decompo-sition using migration fitness scaling genetic algorithmrdquo inProceedings of the 2nd International Conference on Advances inSwarm Intelligence (ICSI rsquo11) Y Tan Y Shi Y Chai and GWang Eds part 1 pp 514ndash521 Springer Heidelberg Germany2011
[12] G Anelli A Broggi and G Destri ldquoDecomposition of arbi-trarily shaped binarymorphological structuring elements usinggenetic algorithmsrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 20 no 2 pp 217ndash224 1998
[13] H Park and J Yoo ldquoStructuring element decomposition for effi-cient implementation ofmorphological filtersrdquo IEE ProceedingsVision Image and Signal Processing vol 148 no 1 pp 31ndash352001
[14] F Y Shih and Y-T Wu ldquoDecomposition of binary mor-phological structuring elements based on genetic algorithmsrdquoComputer Vision and Image Understanding vol 99 no 2 pp291ndash302 2005
[15] H Hedberg P Dokladal and V Owall ldquoBinary morphologywith spatially variant structuring elements algorithm andarchitecturerdquo IEEE Transactions on Image Processing vol 18 no3 pp 562ndash572 2009
[16] P Dokladal and E Dokladalova ldquoComputationally efficientone-pass algorithm for morphological filtersrdquo Journal of VisualCommunication and Image Representation vol 22 no 5 pp411ndash420 2011
[17] M van Droogenbroeck and M J Buckley ldquoMorphologicalerosions and openings fast algorithms based on anchorsrdquoJournal of Mathematical Imaging and Vision vol 22 no 2 pp121ndash142 2005
[18] E Cooksey and W D Withers ldquoRapid image binarizationwith morphological operatorsrdquo in Proceedings of the 15th IEEEInternational Conference on Image Processing (ICIP rsquo08) pp1017ndash1020 San Diego Calif USA October 2008
[19] J Goutsias H J A M Heijmans and K Sivakumar ldquoMorpho-logical operators for image sequencesrdquo Computer Vision andImage Understanding vol 62 no 3 pp 326ndash346 1995
[20] A Hanbury ldquoMathematical morphology in the HLS colourspacerdquo in Proceedings of the 12th British Machine Vision Con-ference (BMVC rsquo01) vol 2 pp 451ndash460 Manchester UK 2001
[21] J Angulo and J Serra ldquoMorphological color size distributionfor image classification and retrievalrdquo in Proceedings of theAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo02)pp 46ndash53 2002
Figure 3 Classical morphological operations on 2D images On the left a morphological dilation On the right a morphological erosionThe structuring element (SE)mdasha circle of 20 pixels in radiusmdashis shown at the top left corner
Table 1 Characteristics of the algorithms based on classical mathematical morphology versus the morphological trajectory model
Classical morphology Morphological trajectory model
Application space Finite group of pixels as a discretization ofEuclidean space
2D Euclidean space (extensible to 119899-D Euclideanspace)
Objects Based on a complete image Based on the boundary of each object(discretecontinuous)
SEs Group of pixels Geometric representation of the frontier
Method On each pixel it operates in a neighborhoodenvironment defined by the SE
The minimum distance of the SE center is calculatedin a direction V on a trajectory defined by Γ(119896)
Result Erosiondilation as group of pixels Frontier points of the erosiondilation operation
Figure 4 Partial morphological erosion as a subset of the completeerosion (over the subrange defined by the dotted line)
As with dilation trajectory-based erosion can orientatethe structuring element in any position on the object bound-ary and define partial erosion of objects when a subrange of119896 is chosen (see Figure 4)
Table 1 summarizes the main differences between themethods described in this section In classical mathematicalmorphology the calculation is made on a complete imagethat is the morphological operation does not distinguishwhether the pixels belong to a specific object or not it simplyapplies a calculation operation of supremum or infimum in aneighborhood environment In the morphological trajectorymodel (MTM) it is necessary to differentiate between theobjects given in the space since each object has a differentgeometric representation Furthermore this representationdefines the frontier of the object and not its interior Anotherimportant difference is the representation of the structuringelement whereas in traditional morphology it is treated asa subgroup of points (which is discretized for the case ofworking with images) the MTM considers the geometricfunction of the points that make up its frontier withoutbeing necessary to carry out a discretization of the structuringelement
3 Morphological Trajectory Computation
In this section the computation of the erosion for the MTMis presented The algorithm becomes straightforward if themorphological erosion concept defined in [1] is applied Firstof all the boundary curve 119862 of object 119883 is represented asa set of ordered points 119901 organized in collinear segments 119904For every point we compute the structuring element centerposition (1199011015840) that touches each point in the direction of acertain vector V
119901 which must be perpendicular from inside
object 119883 The center 1199011015840 will be valid only if the structuringelement placed at 1199011015840 is inside the shape (ie it will not collidewith the curve 119862) Note that for the dilation operation theprocedure will be the same but in this case the elementwill touch the boundary from the outside A pseudo codealgorithm for the erosion is presented in Algorithm 1
If a point 119901 presents a discontinuity in the first derivativewe generate a set of new vectors in order to cover the gap (seeFigure 5) From that new set we also compute new possiblestructuring element centers
Let us analyze now the computational cost of the MTMalgorithm in terms of the problem size The operator used is119874 to determine an upper limit of the computation cost Asshown in Algorithm 1 the algorithm essentially consists ofan external loop which is used to have access to every pointof the shape and two main function calls Letrsquos call 119899 to thenumber of points that represent the shape 119862 once it has beendiscretized If we use a constant step factor 119904 and the totallength of 119862 is 119871 then 119899 will be 119871119904 points
The function ObtainSECenter computes the center of thestructuring element when it touches a point119901 in the directionaddressed by vector V
119901 So this function depends on the SE
6 The Scientific World Journal
(1) For every 119901119894isin 119862 do
(2) 1199011015840119894= ObtainSECenter(119901
119894V119901119894)
(3) If not CollideSE(1199011015840119894 119862) then AddTrajectory(1199011015840
119894)
(4) Endfor
Algorithm 1 Basic pseudo-code algorithm for the morphological trajectory erosion
Table 2 Equations to calculate the intersection between a circle and a 2D segment
Segment function Circle function Segment-Circle intersection equation on 119905119909 = 119909
Dark-grey SE positions are discarded due to shape collision Notethat discontinuity at 119901
119889is solved by a vector swept generation
geometry For simple SEs such as circles rectangles andtriangles the function can be evaluated in a constant time ctEquation (12) shows this function for a circular SE of radius119877
ObtainSECenter (119901 V119901) = 119901 + 119877 sdot V
119901 (12)
The next function called CollideSE(1199011015840 119862) is true if theSE centered at point 1199011015840 is not completely inside shape 119862 andreturns false otherwise In order to evaluate this conditionthis function computes the intersection of the SE geometryand shape 119862 The cost of this function depends on therepresentation of 119862 For the experiments we have organizedthe shape into a set of contiguous segments that representsthe shape Then every segment is tested (at a constant time)and if a segment produces two or more intersections in theSE geometry then the function returns a true value Notethat in this case the discretization of shape 119862 will not be thesame as the one we used to determine the center positionsin the shape For shapes with a high degree of colinearity thenumber of segments will be reduced slightly Let us call119898 thatnumber of segments
The expressions in Table 2 show the quadratic equationused to determine the intersection between a circle cen-tered at the origin and a 2D segment defined between points
(1199091 1199101) and (119909
2 1199102) for a normalized range 119905 [0 1] As
a conclusion a double solution for variable 119905 in the range[0 1] will cause a true return in the function CollideSEotherwise the next segment will be analyzed
Finally the third function called AddTrajectory adds thenew center 1199011015840 to the list of successful centers at a constanttime so it is not considered for evaluating the cost
As a conclusion let us analyze the whole algorithm inorder to obtain an upper limit for the computational costThenext expression evaluates this cost
We must remark that after completing our experiments119898 ≪ 119899 in most cases since a usual value for119898 takes values ofhundreds The computation times for some examples of theMTM operations will be shown in the following section
4 Experiments
In this section we present some experiments in order to testthe trajectory-based operations The first one compares twoversions of the classical dilation versus the trajectory-basedone In Algorithm 2 we show the classical version algorithmsused in the tests
The algorithm called MM1 corresponds to a classicalmorphological dilation whereas the MM2 refers to thatclassical version operating only on the boundary of theobject Note that MM2 does not perform a valid dilationIt was only developed to test the frontier effect that is theadvantage that MTM has since it only processes boundarypixels The trajectory-based version was called MTM for theexperiments The images were evaluated on an Intel PentiumDual Core processor 28GHz and 2GB in RAMTheywereobtained on a Windows based platform
Several tests and experiments were carried out in order toobtain the computing time under different input conditionsBoth the size of the object and the size of the structuringelement were varied as well as the parameters that took partin the morphological operation
Figure 6 shows the behavior of the algorithms resultingfrom the variation of the size of the structuring element andthe size of the object respectively As a consequence from
The Scientific World Journal 7
10
100
1000
10000
0 20 40 60 80 100 120
Tim
e (m
s)
SE radius (pixels)
MTM
MM2
MM1
(a)
10
100
1000
0 10000 20000 30000Pixels of all objects
MTM
MM2
MM1
Tim
e (m
s)
(b)
Figure 6 Morphological dilation tests On the left influence of the size of the structuring element On the right influence on the size ofobjects
MM1 dilation(1) For 119901 isin 119883 do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) EndforMM2 dilation(1) For 119901 isin Fr(119883) do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) Endfor
Algorithm 2 Pseudo-code used for the 2D experiments
these experiments we can see that the computing time ofthe morphological trajectory model remains almost constantagainst the variations in the size of the structuring elementthis is logical due to the fact that we use its geometricrepresentation instead of its content as opposed to MM1and MM2 The difference between MM2 and MM1 arisesfrom the fact that the former only expands the structuringelement for the pixels in the object boundary and thereforeits computing time decreases an order of magnitude withrespect to MM1
With regard to the variation in the size of the objectsin this experiment the MTM gives better results than MM2and much better than MM1 Since MM2 works only on theobject boundary the difference with the MTM arises fromthe size chosen for the SE if this size is small enough MM2will employ less time than the MTM as it has a simplercomputational logic (see Algorithm 2) The radius of thestructuring element chosen for this test was 40 pixels whichis equivalent to an area of 5025 points In order to determinethe effect of the increase of the size of the structuring element
on a group of objects the experiments described below werecarried out
The experiment shown in Figure 7 compares the execu-tion times of theMTM andMM2 for different radii of the SEItmay be seen that the increase in the structuring element sizecauses a crossing point between MM2 and MTM for objectsizes of about 5000 pixels
Furthermore Figure 7 shows that there is no significantincrease in the computation time for the MTMwhen the sizeof the structuring element is increased since this variation isminimal compared to the classical morphological methodsThe small increase comes from the number of pixels of theobject where the distance function is definedThus if the sizeof the SE increases this number is greater and this causes theneed to make new calculations on the new points
On the other hand Figure 8 shows different results ofthe application of the morphological filters Specifically it isshown how the MTM obtains the frontier of the morpholog-ical operation made by MM1
8 The Scientific World Journal
0
20
40
60
80
100
120
140
0 10000 20000 30000 40000Pixels of all objects
MM2 R80
MM2 R40
MM2 R20
MTM R20 40 80
Tim
e (m
s)
Figure 7 Morphologic dilation Comparative study between MM2and MTMmodels for SE sizes of 20 40 and 80 pixels
Time MM1 = 9515msTime MM2 = 80msTime MTM = 110ms
Time MM1 = 8560msTime MM2 = 34msTime MTM = 96ms
(a)
Time MM1 = 4115msTime MM2 = 180msTime MTM = 131ms
Time MM1 = 1085msTime MM2 = 75msTime MTM = 65ms
(b)
Figure 8 Morphological dilation and erosion Results of differentmorphological operations used for the experiments In the upperpart two erosions and in the lower part two dilationsThe boundaryof the original object is represented in green and the result of MM1operation in black with the MTM result in red
(a) (b)
Figure 9 Erosion of figures using our morphological approach (a)Using a circle as a structuring element (b) Using a rectangle as astructuring element
Finally in Figure 9 we present some images related toother erosion experiments using several structuring elementgeometries where the result is presented in green
5 Conclusions and Discussion
In this paper we have developed a topological system result-ing from applying the conventional morphological modelby means of trajectory-based morphological operations todo this we introduced a new feature consisting of order-ing the morphological primitives As shown the proposedoperations are especially useful when large images need to beprocessed
The morphological trajectory model offers an effectivealternative to traditionalmethods for computingmorpholog-ical primitives This alternative is justified if the number ofpoints of the objects and that of the structuring elements arehigh The independency from the structuring element sizecould be interesting to apply morphological operations onhigh definition images or 3D image reconstruction Due tothe fact that the number of the group points is directly relatedto the dimension of the space in which the object and thestructuring element are defined the importance of the MTMis more relevant when the dimension of the representationspace is increased (3D 4D ) In the two dimensionalspace the application of the MTMmay be justified for high-resolution images where large size operators are applied
Other trajectory-based operations such as openingsclosings and skeletons are defined in [18] We are interestedin demonstrating their utility and efficiency by means of thistrajectory optimization
The newmodel has been presented for binary 2D imagesHowever the new paradigm is extensible for any number ofdimensions of the Euclidean space In [19ndash21] new versionsof the morphological model for color images were presentedThesemodels consider the color as a third coordinate and thecomputation is made in the 3D space Future works will befocused on extending the model in order to support efficientfiltering of color images or real 3D images
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Serra Image Analysis and Mathematical Morphology Aca-demic Press London UK 1982
[2] M van Herk ldquoA fast algorithm for local minimum and max-imum filters on rectangular and octagonal kernelsrdquo PatternRecognition Letters vol 13 no 7 pp 517ndash521 1992
[3] J Gil and M Werman ldquoComputing 2-D min median and maxfiltersrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 15 no 5 pp 504ndash507 1993
[4] J Y Gil and R Kimmel ldquoEfficient dilation erosion openingand closing algorithmsrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 24 no 12 pp 1606ndash1617 2002
The Scientific World Journal 9
[5] O Deforges N Normand and M Babel ldquoFast recursivegrayscale morphology operators from the algorithm to thepipeline architecturerdquo Journal of Real-Time Image Processingvol 8 no 2 pp 143ndash152 2013
[6] C Clienti M Bilodeau and S Beucher ldquoAn efficient hardwarearchitecture without line memories for morphological imageprocessingrdquo in Proceedings of the 10th International ConferenceonAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo08)J Blanc-Talon S Bourennane W Philips D Popescu and PScheunders Eds pp 147ndash156 Springer Heidelberg Germany2008
[7] K Sivakumar M J Patel N Kehtarnavaz Y Balagurunathanand E R Dougherty ldquoA constant-time algorithm for ero-sionsdilations with applications to morphological texture fea-ture computationrdquoReal-Time Imaging vol 6 no 3 pp 223ndash2392000
[8] P Soille and H Talbot ldquoImage Structure Orientation UsingMathematical Morphologyrdquo in Proceedings of the 14th Interna-tional Conference on Pattern Recognition A Jain S Venkateshand B Lovell Eds vol 2 pp 1467ndash1469 Brisbane AustraliaAugust 1998
[9] P Soille and H Talbot ldquoDirectional morphological filteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 23 no 11 pp 1313ndash1329 2001
[10] E R Urbach and M H F Wilkinson ldquoEfficient 2-D grayscalemorphological transformations with arbitrary flat structuringelementsrdquo IEEE Transactions on Image Processing vol 17 no 1pp 1ndash8 2008
[11] Y Zhang and L Wu ldquoRecursive structure element decompo-sition using migration fitness scaling genetic algorithmrdquo inProceedings of the 2nd International Conference on Advances inSwarm Intelligence (ICSI rsquo11) Y Tan Y Shi Y Chai and GWang Eds part 1 pp 514ndash521 Springer Heidelberg Germany2011
[12] G Anelli A Broggi and G Destri ldquoDecomposition of arbi-trarily shaped binarymorphological structuring elements usinggenetic algorithmsrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 20 no 2 pp 217ndash224 1998
[13] H Park and J Yoo ldquoStructuring element decomposition for effi-cient implementation ofmorphological filtersrdquo IEE ProceedingsVision Image and Signal Processing vol 148 no 1 pp 31ndash352001
[14] F Y Shih and Y-T Wu ldquoDecomposition of binary mor-phological structuring elements based on genetic algorithmsrdquoComputer Vision and Image Understanding vol 99 no 2 pp291ndash302 2005
[15] H Hedberg P Dokladal and V Owall ldquoBinary morphologywith spatially variant structuring elements algorithm andarchitecturerdquo IEEE Transactions on Image Processing vol 18 no3 pp 562ndash572 2009
[16] P Dokladal and E Dokladalova ldquoComputationally efficientone-pass algorithm for morphological filtersrdquo Journal of VisualCommunication and Image Representation vol 22 no 5 pp411ndash420 2011
[17] M van Droogenbroeck and M J Buckley ldquoMorphologicalerosions and openings fast algorithms based on anchorsrdquoJournal of Mathematical Imaging and Vision vol 22 no 2 pp121ndash142 2005
[18] E Cooksey and W D Withers ldquoRapid image binarizationwith morphological operatorsrdquo in Proceedings of the 15th IEEEInternational Conference on Image Processing (ICIP rsquo08) pp1017ndash1020 San Diego Calif USA October 2008
[19] J Goutsias H J A M Heijmans and K Sivakumar ldquoMorpho-logical operators for image sequencesrdquo Computer Vision andImage Understanding vol 62 no 3 pp 326ndash346 1995
[20] A Hanbury ldquoMathematical morphology in the HLS colourspacerdquo in Proceedings of the 12th British Machine Vision Con-ference (BMVC rsquo01) vol 2 pp 451ndash460 Manchester UK 2001
[21] J Angulo and J Serra ldquoMorphological color size distributionfor image classification and retrievalrdquo in Proceedings of theAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo02)pp 46ndash53 2002
Dark-grey SE positions are discarded due to shape collision Notethat discontinuity at 119901
119889is solved by a vector swept generation
geometry For simple SEs such as circles rectangles andtriangles the function can be evaluated in a constant time ctEquation (12) shows this function for a circular SE of radius119877
ObtainSECenter (119901 V119901) = 119901 + 119877 sdot V
119901 (12)
The next function called CollideSE(1199011015840 119862) is true if theSE centered at point 1199011015840 is not completely inside shape 119862 andreturns false otherwise In order to evaluate this conditionthis function computes the intersection of the SE geometryand shape 119862 The cost of this function depends on therepresentation of 119862 For the experiments we have organizedthe shape into a set of contiguous segments that representsthe shape Then every segment is tested (at a constant time)and if a segment produces two or more intersections in theSE geometry then the function returns a true value Notethat in this case the discretization of shape 119862 will not be thesame as the one we used to determine the center positionsin the shape For shapes with a high degree of colinearity thenumber of segments will be reduced slightly Let us call119898 thatnumber of segments
The expressions in Table 2 show the quadratic equationused to determine the intersection between a circle cen-tered at the origin and a 2D segment defined between points
(1199091 1199101) and (119909
2 1199102) for a normalized range 119905 [0 1] As
a conclusion a double solution for variable 119905 in the range[0 1] will cause a true return in the function CollideSEotherwise the next segment will be analyzed
Finally the third function called AddTrajectory adds thenew center 1199011015840 to the list of successful centers at a constanttime so it is not considered for evaluating the cost
As a conclusion let us analyze the whole algorithm inorder to obtain an upper limit for the computational costThenext expression evaluates this cost
We must remark that after completing our experiments119898 ≪ 119899 in most cases since a usual value for119898 takes values ofhundreds The computation times for some examples of theMTM operations will be shown in the following section
4 Experiments
In this section we present some experiments in order to testthe trajectory-based operations The first one compares twoversions of the classical dilation versus the trajectory-basedone In Algorithm 2 we show the classical version algorithmsused in the tests
The algorithm called MM1 corresponds to a classicalmorphological dilation whereas the MM2 refers to thatclassical version operating only on the boundary of theobject Note that MM2 does not perform a valid dilationIt was only developed to test the frontier effect that is theadvantage that MTM has since it only processes boundarypixels The trajectory-based version was called MTM for theexperiments The images were evaluated on an Intel PentiumDual Core processor 28GHz and 2GB in RAMTheywereobtained on a Windows based platform
Several tests and experiments were carried out in order toobtain the computing time under different input conditionsBoth the size of the object and the size of the structuringelement were varied as well as the parameters that took partin the morphological operation
Figure 6 shows the behavior of the algorithms resultingfrom the variation of the size of the structuring element andthe size of the object respectively As a consequence from
The Scientific World Journal 7
10
100
1000
10000
0 20 40 60 80 100 120
Tim
e (m
s)
SE radius (pixels)
MTM
MM2
MM1
(a)
10
100
1000
0 10000 20000 30000Pixels of all objects
MTM
MM2
MM1
Tim
e (m
s)
(b)
Figure 6 Morphological dilation tests On the left influence of the size of the structuring element On the right influence on the size ofobjects
MM1 dilation(1) For 119901 isin 119883 do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) EndforMM2 dilation(1) For 119901 isin Fr(119883) do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) Endfor
Algorithm 2 Pseudo-code used for the 2D experiments
these experiments we can see that the computing time ofthe morphological trajectory model remains almost constantagainst the variations in the size of the structuring elementthis is logical due to the fact that we use its geometricrepresentation instead of its content as opposed to MM1and MM2 The difference between MM2 and MM1 arisesfrom the fact that the former only expands the structuringelement for the pixels in the object boundary and thereforeits computing time decreases an order of magnitude withrespect to MM1
With regard to the variation in the size of the objectsin this experiment the MTM gives better results than MM2and much better than MM1 Since MM2 works only on theobject boundary the difference with the MTM arises fromthe size chosen for the SE if this size is small enough MM2will employ less time than the MTM as it has a simplercomputational logic (see Algorithm 2) The radius of thestructuring element chosen for this test was 40 pixels whichis equivalent to an area of 5025 points In order to determinethe effect of the increase of the size of the structuring element
on a group of objects the experiments described below werecarried out
The experiment shown in Figure 7 compares the execu-tion times of theMTM andMM2 for different radii of the SEItmay be seen that the increase in the structuring element sizecauses a crossing point between MM2 and MTM for objectsizes of about 5000 pixels
Furthermore Figure 7 shows that there is no significantincrease in the computation time for the MTMwhen the sizeof the structuring element is increased since this variation isminimal compared to the classical morphological methodsThe small increase comes from the number of pixels of theobject where the distance function is definedThus if the sizeof the SE increases this number is greater and this causes theneed to make new calculations on the new points
On the other hand Figure 8 shows different results ofthe application of the morphological filters Specifically it isshown how the MTM obtains the frontier of the morpholog-ical operation made by MM1
8 The Scientific World Journal
0
20
40
60
80
100
120
140
0 10000 20000 30000 40000Pixels of all objects
MM2 R80
MM2 R40
MM2 R20
MTM R20 40 80
Tim
e (m
s)
Figure 7 Morphologic dilation Comparative study between MM2and MTMmodels for SE sizes of 20 40 and 80 pixels
Time MM1 = 9515msTime MM2 = 80msTime MTM = 110ms
Time MM1 = 8560msTime MM2 = 34msTime MTM = 96ms
(a)
Time MM1 = 4115msTime MM2 = 180msTime MTM = 131ms
Time MM1 = 1085msTime MM2 = 75msTime MTM = 65ms
(b)
Figure 8 Morphological dilation and erosion Results of differentmorphological operations used for the experiments In the upperpart two erosions and in the lower part two dilationsThe boundaryof the original object is represented in green and the result of MM1operation in black with the MTM result in red
(a) (b)
Figure 9 Erosion of figures using our morphological approach (a)Using a circle as a structuring element (b) Using a rectangle as astructuring element
Finally in Figure 9 we present some images related toother erosion experiments using several structuring elementgeometries where the result is presented in green
5 Conclusions and Discussion
In this paper we have developed a topological system result-ing from applying the conventional morphological modelby means of trajectory-based morphological operations todo this we introduced a new feature consisting of order-ing the morphological primitives As shown the proposedoperations are especially useful when large images need to beprocessed
The morphological trajectory model offers an effectivealternative to traditionalmethods for computingmorpholog-ical primitives This alternative is justified if the number ofpoints of the objects and that of the structuring elements arehigh The independency from the structuring element sizecould be interesting to apply morphological operations onhigh definition images or 3D image reconstruction Due tothe fact that the number of the group points is directly relatedto the dimension of the space in which the object and thestructuring element are defined the importance of the MTMis more relevant when the dimension of the representationspace is increased (3D 4D ) In the two dimensionalspace the application of the MTMmay be justified for high-resolution images where large size operators are applied
Other trajectory-based operations such as openingsclosings and skeletons are defined in [18] We are interestedin demonstrating their utility and efficiency by means of thistrajectory optimization
The newmodel has been presented for binary 2D imagesHowever the new paradigm is extensible for any number ofdimensions of the Euclidean space In [19ndash21] new versionsof the morphological model for color images were presentedThesemodels consider the color as a third coordinate and thecomputation is made in the 3D space Future works will befocused on extending the model in order to support efficientfiltering of color images or real 3D images
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Serra Image Analysis and Mathematical Morphology Aca-demic Press London UK 1982
[2] M van Herk ldquoA fast algorithm for local minimum and max-imum filters on rectangular and octagonal kernelsrdquo PatternRecognition Letters vol 13 no 7 pp 517ndash521 1992
[3] J Gil and M Werman ldquoComputing 2-D min median and maxfiltersrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 15 no 5 pp 504ndash507 1993
[4] J Y Gil and R Kimmel ldquoEfficient dilation erosion openingand closing algorithmsrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 24 no 12 pp 1606ndash1617 2002
The Scientific World Journal 9
[5] O Deforges N Normand and M Babel ldquoFast recursivegrayscale morphology operators from the algorithm to thepipeline architecturerdquo Journal of Real-Time Image Processingvol 8 no 2 pp 143ndash152 2013
[6] C Clienti M Bilodeau and S Beucher ldquoAn efficient hardwarearchitecture without line memories for morphological imageprocessingrdquo in Proceedings of the 10th International ConferenceonAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo08)J Blanc-Talon S Bourennane W Philips D Popescu and PScheunders Eds pp 147ndash156 Springer Heidelberg Germany2008
[7] K Sivakumar M J Patel N Kehtarnavaz Y Balagurunathanand E R Dougherty ldquoA constant-time algorithm for ero-sionsdilations with applications to morphological texture fea-ture computationrdquoReal-Time Imaging vol 6 no 3 pp 223ndash2392000
[8] P Soille and H Talbot ldquoImage Structure Orientation UsingMathematical Morphologyrdquo in Proceedings of the 14th Interna-tional Conference on Pattern Recognition A Jain S Venkateshand B Lovell Eds vol 2 pp 1467ndash1469 Brisbane AustraliaAugust 1998
[9] P Soille and H Talbot ldquoDirectional morphological filteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 23 no 11 pp 1313ndash1329 2001
[10] E R Urbach and M H F Wilkinson ldquoEfficient 2-D grayscalemorphological transformations with arbitrary flat structuringelementsrdquo IEEE Transactions on Image Processing vol 17 no 1pp 1ndash8 2008
[11] Y Zhang and L Wu ldquoRecursive structure element decompo-sition using migration fitness scaling genetic algorithmrdquo inProceedings of the 2nd International Conference on Advances inSwarm Intelligence (ICSI rsquo11) Y Tan Y Shi Y Chai and GWang Eds part 1 pp 514ndash521 Springer Heidelberg Germany2011
[12] G Anelli A Broggi and G Destri ldquoDecomposition of arbi-trarily shaped binarymorphological structuring elements usinggenetic algorithmsrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 20 no 2 pp 217ndash224 1998
[13] H Park and J Yoo ldquoStructuring element decomposition for effi-cient implementation ofmorphological filtersrdquo IEE ProceedingsVision Image and Signal Processing vol 148 no 1 pp 31ndash352001
[14] F Y Shih and Y-T Wu ldquoDecomposition of binary mor-phological structuring elements based on genetic algorithmsrdquoComputer Vision and Image Understanding vol 99 no 2 pp291ndash302 2005
[15] H Hedberg P Dokladal and V Owall ldquoBinary morphologywith spatially variant structuring elements algorithm andarchitecturerdquo IEEE Transactions on Image Processing vol 18 no3 pp 562ndash572 2009
[16] P Dokladal and E Dokladalova ldquoComputationally efficientone-pass algorithm for morphological filtersrdquo Journal of VisualCommunication and Image Representation vol 22 no 5 pp411ndash420 2011
[17] M van Droogenbroeck and M J Buckley ldquoMorphologicalerosions and openings fast algorithms based on anchorsrdquoJournal of Mathematical Imaging and Vision vol 22 no 2 pp121ndash142 2005
[18] E Cooksey and W D Withers ldquoRapid image binarizationwith morphological operatorsrdquo in Proceedings of the 15th IEEEInternational Conference on Image Processing (ICIP rsquo08) pp1017ndash1020 San Diego Calif USA October 2008
[19] J Goutsias H J A M Heijmans and K Sivakumar ldquoMorpho-logical operators for image sequencesrdquo Computer Vision andImage Understanding vol 62 no 3 pp 326ndash346 1995
[20] A Hanbury ldquoMathematical morphology in the HLS colourspacerdquo in Proceedings of the 12th British Machine Vision Con-ference (BMVC rsquo01) vol 2 pp 451ndash460 Manchester UK 2001
[21] J Angulo and J Serra ldquoMorphological color size distributionfor image classification and retrievalrdquo in Proceedings of theAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo02)pp 46ndash53 2002
Figure 6 Morphological dilation tests On the left influence of the size of the structuring element On the right influence on the size ofobjects
MM1 dilation(1) For 119901 isin 119883 do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) EndforMM2 dilation(1) For 119901 isin Fr(119883) do(2) For 119890 isin SE do(3) Image(119901 + 119890) = 1(4) Endfor(5) Endfor
Algorithm 2 Pseudo-code used for the 2D experiments
these experiments we can see that the computing time ofthe morphological trajectory model remains almost constantagainst the variations in the size of the structuring elementthis is logical due to the fact that we use its geometricrepresentation instead of its content as opposed to MM1and MM2 The difference between MM2 and MM1 arisesfrom the fact that the former only expands the structuringelement for the pixels in the object boundary and thereforeits computing time decreases an order of magnitude withrespect to MM1
With regard to the variation in the size of the objectsin this experiment the MTM gives better results than MM2and much better than MM1 Since MM2 works only on theobject boundary the difference with the MTM arises fromthe size chosen for the SE if this size is small enough MM2will employ less time than the MTM as it has a simplercomputational logic (see Algorithm 2) The radius of thestructuring element chosen for this test was 40 pixels whichis equivalent to an area of 5025 points In order to determinethe effect of the increase of the size of the structuring element
on a group of objects the experiments described below werecarried out
The experiment shown in Figure 7 compares the execu-tion times of theMTM andMM2 for different radii of the SEItmay be seen that the increase in the structuring element sizecauses a crossing point between MM2 and MTM for objectsizes of about 5000 pixels
Furthermore Figure 7 shows that there is no significantincrease in the computation time for the MTMwhen the sizeof the structuring element is increased since this variation isminimal compared to the classical morphological methodsThe small increase comes from the number of pixels of theobject where the distance function is definedThus if the sizeof the SE increases this number is greater and this causes theneed to make new calculations on the new points
On the other hand Figure 8 shows different results ofthe application of the morphological filters Specifically it isshown how the MTM obtains the frontier of the morpholog-ical operation made by MM1
8 The Scientific World Journal
0
20
40
60
80
100
120
140
0 10000 20000 30000 40000Pixels of all objects
MM2 R80
MM2 R40
MM2 R20
MTM R20 40 80
Tim
e (m
s)
Figure 7 Morphologic dilation Comparative study between MM2and MTMmodels for SE sizes of 20 40 and 80 pixels
Time MM1 = 9515msTime MM2 = 80msTime MTM = 110ms
Time MM1 = 8560msTime MM2 = 34msTime MTM = 96ms
(a)
Time MM1 = 4115msTime MM2 = 180msTime MTM = 131ms
Time MM1 = 1085msTime MM2 = 75msTime MTM = 65ms
(b)
Figure 8 Morphological dilation and erosion Results of differentmorphological operations used for the experiments In the upperpart two erosions and in the lower part two dilationsThe boundaryof the original object is represented in green and the result of MM1operation in black with the MTM result in red
(a) (b)
Figure 9 Erosion of figures using our morphological approach (a)Using a circle as a structuring element (b) Using a rectangle as astructuring element
Finally in Figure 9 we present some images related toother erosion experiments using several structuring elementgeometries where the result is presented in green
5 Conclusions and Discussion
In this paper we have developed a topological system result-ing from applying the conventional morphological modelby means of trajectory-based morphological operations todo this we introduced a new feature consisting of order-ing the morphological primitives As shown the proposedoperations are especially useful when large images need to beprocessed
The morphological trajectory model offers an effectivealternative to traditionalmethods for computingmorpholog-ical primitives This alternative is justified if the number ofpoints of the objects and that of the structuring elements arehigh The independency from the structuring element sizecould be interesting to apply morphological operations onhigh definition images or 3D image reconstruction Due tothe fact that the number of the group points is directly relatedto the dimension of the space in which the object and thestructuring element are defined the importance of the MTMis more relevant when the dimension of the representationspace is increased (3D 4D ) In the two dimensionalspace the application of the MTMmay be justified for high-resolution images where large size operators are applied
Other trajectory-based operations such as openingsclosings and skeletons are defined in [18] We are interestedin demonstrating their utility and efficiency by means of thistrajectory optimization
The newmodel has been presented for binary 2D imagesHowever the new paradigm is extensible for any number ofdimensions of the Euclidean space In [19ndash21] new versionsof the morphological model for color images were presentedThesemodels consider the color as a third coordinate and thecomputation is made in the 3D space Future works will befocused on extending the model in order to support efficientfiltering of color images or real 3D images
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Serra Image Analysis and Mathematical Morphology Aca-demic Press London UK 1982
[2] M van Herk ldquoA fast algorithm for local minimum and max-imum filters on rectangular and octagonal kernelsrdquo PatternRecognition Letters vol 13 no 7 pp 517ndash521 1992
[3] J Gil and M Werman ldquoComputing 2-D min median and maxfiltersrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 15 no 5 pp 504ndash507 1993
[4] J Y Gil and R Kimmel ldquoEfficient dilation erosion openingand closing algorithmsrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 24 no 12 pp 1606ndash1617 2002
The Scientific World Journal 9
[5] O Deforges N Normand and M Babel ldquoFast recursivegrayscale morphology operators from the algorithm to thepipeline architecturerdquo Journal of Real-Time Image Processingvol 8 no 2 pp 143ndash152 2013
[6] C Clienti M Bilodeau and S Beucher ldquoAn efficient hardwarearchitecture without line memories for morphological imageprocessingrdquo in Proceedings of the 10th International ConferenceonAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo08)J Blanc-Talon S Bourennane W Philips D Popescu and PScheunders Eds pp 147ndash156 Springer Heidelberg Germany2008
[7] K Sivakumar M J Patel N Kehtarnavaz Y Balagurunathanand E R Dougherty ldquoA constant-time algorithm for ero-sionsdilations with applications to morphological texture fea-ture computationrdquoReal-Time Imaging vol 6 no 3 pp 223ndash2392000
[8] P Soille and H Talbot ldquoImage Structure Orientation UsingMathematical Morphologyrdquo in Proceedings of the 14th Interna-tional Conference on Pattern Recognition A Jain S Venkateshand B Lovell Eds vol 2 pp 1467ndash1469 Brisbane AustraliaAugust 1998
[9] P Soille and H Talbot ldquoDirectional morphological filteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 23 no 11 pp 1313ndash1329 2001
[10] E R Urbach and M H F Wilkinson ldquoEfficient 2-D grayscalemorphological transformations with arbitrary flat structuringelementsrdquo IEEE Transactions on Image Processing vol 17 no 1pp 1ndash8 2008
[11] Y Zhang and L Wu ldquoRecursive structure element decompo-sition using migration fitness scaling genetic algorithmrdquo inProceedings of the 2nd International Conference on Advances inSwarm Intelligence (ICSI rsquo11) Y Tan Y Shi Y Chai and GWang Eds part 1 pp 514ndash521 Springer Heidelberg Germany2011
[12] G Anelli A Broggi and G Destri ldquoDecomposition of arbi-trarily shaped binarymorphological structuring elements usinggenetic algorithmsrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 20 no 2 pp 217ndash224 1998
[13] H Park and J Yoo ldquoStructuring element decomposition for effi-cient implementation ofmorphological filtersrdquo IEE ProceedingsVision Image and Signal Processing vol 148 no 1 pp 31ndash352001
[14] F Y Shih and Y-T Wu ldquoDecomposition of binary mor-phological structuring elements based on genetic algorithmsrdquoComputer Vision and Image Understanding vol 99 no 2 pp291ndash302 2005
[15] H Hedberg P Dokladal and V Owall ldquoBinary morphologywith spatially variant structuring elements algorithm andarchitecturerdquo IEEE Transactions on Image Processing vol 18 no3 pp 562ndash572 2009
[16] P Dokladal and E Dokladalova ldquoComputationally efficientone-pass algorithm for morphological filtersrdquo Journal of VisualCommunication and Image Representation vol 22 no 5 pp411ndash420 2011
[17] M van Droogenbroeck and M J Buckley ldquoMorphologicalerosions and openings fast algorithms based on anchorsrdquoJournal of Mathematical Imaging and Vision vol 22 no 2 pp121ndash142 2005
[18] E Cooksey and W D Withers ldquoRapid image binarizationwith morphological operatorsrdquo in Proceedings of the 15th IEEEInternational Conference on Image Processing (ICIP rsquo08) pp1017ndash1020 San Diego Calif USA October 2008
[19] J Goutsias H J A M Heijmans and K Sivakumar ldquoMorpho-logical operators for image sequencesrdquo Computer Vision andImage Understanding vol 62 no 3 pp 326ndash346 1995
[20] A Hanbury ldquoMathematical morphology in the HLS colourspacerdquo in Proceedings of the 12th British Machine Vision Con-ference (BMVC rsquo01) vol 2 pp 451ndash460 Manchester UK 2001
[21] J Angulo and J Serra ldquoMorphological color size distributionfor image classification and retrievalrdquo in Proceedings of theAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo02)pp 46ndash53 2002
Figure 7 Morphologic dilation Comparative study between MM2and MTMmodels for SE sizes of 20 40 and 80 pixels
Time MM1 = 9515msTime MM2 = 80msTime MTM = 110ms
Time MM1 = 8560msTime MM2 = 34msTime MTM = 96ms
(a)
Time MM1 = 4115msTime MM2 = 180msTime MTM = 131ms
Time MM1 = 1085msTime MM2 = 75msTime MTM = 65ms
(b)
Figure 8 Morphological dilation and erosion Results of differentmorphological operations used for the experiments In the upperpart two erosions and in the lower part two dilationsThe boundaryof the original object is represented in green and the result of MM1operation in black with the MTM result in red
(a) (b)
Figure 9 Erosion of figures using our morphological approach (a)Using a circle as a structuring element (b) Using a rectangle as astructuring element
Finally in Figure 9 we present some images related toother erosion experiments using several structuring elementgeometries where the result is presented in green
5 Conclusions and Discussion
In this paper we have developed a topological system result-ing from applying the conventional morphological modelby means of trajectory-based morphological operations todo this we introduced a new feature consisting of order-ing the morphological primitives As shown the proposedoperations are especially useful when large images need to beprocessed
The morphological trajectory model offers an effectivealternative to traditionalmethods for computingmorpholog-ical primitives This alternative is justified if the number ofpoints of the objects and that of the structuring elements arehigh The independency from the structuring element sizecould be interesting to apply morphological operations onhigh definition images or 3D image reconstruction Due tothe fact that the number of the group points is directly relatedto the dimension of the space in which the object and thestructuring element are defined the importance of the MTMis more relevant when the dimension of the representationspace is increased (3D 4D ) In the two dimensionalspace the application of the MTMmay be justified for high-resolution images where large size operators are applied
Other trajectory-based operations such as openingsclosings and skeletons are defined in [18] We are interestedin demonstrating their utility and efficiency by means of thistrajectory optimization
The newmodel has been presented for binary 2D imagesHowever the new paradigm is extensible for any number ofdimensions of the Euclidean space In [19ndash21] new versionsof the morphological model for color images were presentedThesemodels consider the color as a third coordinate and thecomputation is made in the 3D space Future works will befocused on extending the model in order to support efficientfiltering of color images or real 3D images
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Serra Image Analysis and Mathematical Morphology Aca-demic Press London UK 1982
[2] M van Herk ldquoA fast algorithm for local minimum and max-imum filters on rectangular and octagonal kernelsrdquo PatternRecognition Letters vol 13 no 7 pp 517ndash521 1992
[3] J Gil and M Werman ldquoComputing 2-D min median and maxfiltersrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 15 no 5 pp 504ndash507 1993
[4] J Y Gil and R Kimmel ldquoEfficient dilation erosion openingand closing algorithmsrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 24 no 12 pp 1606ndash1617 2002
The Scientific World Journal 9
[5] O Deforges N Normand and M Babel ldquoFast recursivegrayscale morphology operators from the algorithm to thepipeline architecturerdquo Journal of Real-Time Image Processingvol 8 no 2 pp 143ndash152 2013
[6] C Clienti M Bilodeau and S Beucher ldquoAn efficient hardwarearchitecture without line memories for morphological imageprocessingrdquo in Proceedings of the 10th International ConferenceonAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo08)J Blanc-Talon S Bourennane W Philips D Popescu and PScheunders Eds pp 147ndash156 Springer Heidelberg Germany2008
[7] K Sivakumar M J Patel N Kehtarnavaz Y Balagurunathanand E R Dougherty ldquoA constant-time algorithm for ero-sionsdilations with applications to morphological texture fea-ture computationrdquoReal-Time Imaging vol 6 no 3 pp 223ndash2392000
[8] P Soille and H Talbot ldquoImage Structure Orientation UsingMathematical Morphologyrdquo in Proceedings of the 14th Interna-tional Conference on Pattern Recognition A Jain S Venkateshand B Lovell Eds vol 2 pp 1467ndash1469 Brisbane AustraliaAugust 1998
[9] P Soille and H Talbot ldquoDirectional morphological filteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 23 no 11 pp 1313ndash1329 2001
[10] E R Urbach and M H F Wilkinson ldquoEfficient 2-D grayscalemorphological transformations with arbitrary flat structuringelementsrdquo IEEE Transactions on Image Processing vol 17 no 1pp 1ndash8 2008
[11] Y Zhang and L Wu ldquoRecursive structure element decompo-sition using migration fitness scaling genetic algorithmrdquo inProceedings of the 2nd International Conference on Advances inSwarm Intelligence (ICSI rsquo11) Y Tan Y Shi Y Chai and GWang Eds part 1 pp 514ndash521 Springer Heidelberg Germany2011
[12] G Anelli A Broggi and G Destri ldquoDecomposition of arbi-trarily shaped binarymorphological structuring elements usinggenetic algorithmsrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 20 no 2 pp 217ndash224 1998
[13] H Park and J Yoo ldquoStructuring element decomposition for effi-cient implementation ofmorphological filtersrdquo IEE ProceedingsVision Image and Signal Processing vol 148 no 1 pp 31ndash352001
[14] F Y Shih and Y-T Wu ldquoDecomposition of binary mor-phological structuring elements based on genetic algorithmsrdquoComputer Vision and Image Understanding vol 99 no 2 pp291ndash302 2005
[15] H Hedberg P Dokladal and V Owall ldquoBinary morphologywith spatially variant structuring elements algorithm andarchitecturerdquo IEEE Transactions on Image Processing vol 18 no3 pp 562ndash572 2009
[16] P Dokladal and E Dokladalova ldquoComputationally efficientone-pass algorithm for morphological filtersrdquo Journal of VisualCommunication and Image Representation vol 22 no 5 pp411ndash420 2011
[17] M van Droogenbroeck and M J Buckley ldquoMorphologicalerosions and openings fast algorithms based on anchorsrdquoJournal of Mathematical Imaging and Vision vol 22 no 2 pp121ndash142 2005
[18] E Cooksey and W D Withers ldquoRapid image binarizationwith morphological operatorsrdquo in Proceedings of the 15th IEEEInternational Conference on Image Processing (ICIP rsquo08) pp1017ndash1020 San Diego Calif USA October 2008
[19] J Goutsias H J A M Heijmans and K Sivakumar ldquoMorpho-logical operators for image sequencesrdquo Computer Vision andImage Understanding vol 62 no 3 pp 326ndash346 1995
[20] A Hanbury ldquoMathematical morphology in the HLS colourspacerdquo in Proceedings of the 12th British Machine Vision Con-ference (BMVC rsquo01) vol 2 pp 451ndash460 Manchester UK 2001
[21] J Angulo and J Serra ldquoMorphological color size distributionfor image classification and retrievalrdquo in Proceedings of theAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo02)pp 46ndash53 2002
[5] O Deforges N Normand and M Babel ldquoFast recursivegrayscale morphology operators from the algorithm to thepipeline architecturerdquo Journal of Real-Time Image Processingvol 8 no 2 pp 143ndash152 2013
[6] C Clienti M Bilodeau and S Beucher ldquoAn efficient hardwarearchitecture without line memories for morphological imageprocessingrdquo in Proceedings of the 10th International ConferenceonAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo08)J Blanc-Talon S Bourennane W Philips D Popescu and PScheunders Eds pp 147ndash156 Springer Heidelberg Germany2008
[7] K Sivakumar M J Patel N Kehtarnavaz Y Balagurunathanand E R Dougherty ldquoA constant-time algorithm for ero-sionsdilations with applications to morphological texture fea-ture computationrdquoReal-Time Imaging vol 6 no 3 pp 223ndash2392000
[8] P Soille and H Talbot ldquoImage Structure Orientation UsingMathematical Morphologyrdquo in Proceedings of the 14th Interna-tional Conference on Pattern Recognition A Jain S Venkateshand B Lovell Eds vol 2 pp 1467ndash1469 Brisbane AustraliaAugust 1998
[9] P Soille and H Talbot ldquoDirectional morphological filteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 23 no 11 pp 1313ndash1329 2001
[10] E R Urbach and M H F Wilkinson ldquoEfficient 2-D grayscalemorphological transformations with arbitrary flat structuringelementsrdquo IEEE Transactions on Image Processing vol 17 no 1pp 1ndash8 2008
[11] Y Zhang and L Wu ldquoRecursive structure element decompo-sition using migration fitness scaling genetic algorithmrdquo inProceedings of the 2nd International Conference on Advances inSwarm Intelligence (ICSI rsquo11) Y Tan Y Shi Y Chai and GWang Eds part 1 pp 514ndash521 Springer Heidelberg Germany2011
[12] G Anelli A Broggi and G Destri ldquoDecomposition of arbi-trarily shaped binarymorphological structuring elements usinggenetic algorithmsrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 20 no 2 pp 217ndash224 1998
[13] H Park and J Yoo ldquoStructuring element decomposition for effi-cient implementation ofmorphological filtersrdquo IEE ProceedingsVision Image and Signal Processing vol 148 no 1 pp 31ndash352001
[14] F Y Shih and Y-T Wu ldquoDecomposition of binary mor-phological structuring elements based on genetic algorithmsrdquoComputer Vision and Image Understanding vol 99 no 2 pp291ndash302 2005
[15] H Hedberg P Dokladal and V Owall ldquoBinary morphologywith spatially variant structuring elements algorithm andarchitecturerdquo IEEE Transactions on Image Processing vol 18 no3 pp 562ndash572 2009
[16] P Dokladal and E Dokladalova ldquoComputationally efficientone-pass algorithm for morphological filtersrdquo Journal of VisualCommunication and Image Representation vol 22 no 5 pp411ndash420 2011
[17] M van Droogenbroeck and M J Buckley ldquoMorphologicalerosions and openings fast algorithms based on anchorsrdquoJournal of Mathematical Imaging and Vision vol 22 no 2 pp121ndash142 2005
[18] E Cooksey and W D Withers ldquoRapid image binarizationwith morphological operatorsrdquo in Proceedings of the 15th IEEEInternational Conference on Image Processing (ICIP rsquo08) pp1017ndash1020 San Diego Calif USA October 2008
[19] J Goutsias H J A M Heijmans and K Sivakumar ldquoMorpho-logical operators for image sequencesrdquo Computer Vision andImage Understanding vol 62 no 3 pp 326ndash346 1995
[20] A Hanbury ldquoMathematical morphology in the HLS colourspacerdquo in Proceedings of the 12th British Machine Vision Con-ference (BMVC rsquo01) vol 2 pp 451ndash460 Manchester UK 2001
[21] J Angulo and J Serra ldquoMorphological color size distributionfor image classification and retrievalrdquo in Proceedings of theAdvanced Concepts for Intelligent Vision Systems (ACIVS rsquo02)pp 46ndash53 2002
