Soft ComputDOI 10.1007/s00500-013-1204-6

METHODOLOGIES AND APPLICATION

A fuzzy mathematical morphology based on discrete t-norms:fundamentals and applications to image processing

Manuel González-Hidalgo · Sebastia Massanet

© Springer-Verlag Berlin Heidelberg 2013

Abstract In this paper, a new approach to fuzzy mathemat-ical morphology based on discrete t-norms is studied. Thediscrete t-norms that have to be used in order to preserve themost usual algebraical and morphological properties, such asmonotonicity, idempotence, scaling invariance, among oth-ers, are fully determined. In addition, the properties relatedto B-open and B-closed objects and the generalized idempo-tence are also studied. In fact, all properties satisfied by theapproach based on continuous nilpotent t-norms hold in thediscrete case. This is quite important since in practice we onlywork with discrete objects. In addition, it is proved that morediscrete t-norms satisfying all the properties are available inthis approach than in the continuous case, which reduces tothe Łukasiewicz t-norm. This morphology based on discretet-norms can be considered embedded in more general frame-works, such as L-fuzzy sets or quantale modules, but all theseframeworks have been studied only from a theoretical pointof view. Our main contribution is the practical applicationof this discrete approach to image processing. Experimentalresults on edge detection, noise removal and top-hat trans-formations for some discrete t-norms and their comparisonwith the corresponding ones obtained by the umbra approachand the continuous Łukasiewicz t-norm are included showingthat this theory can be suitable to be used in a wide range ofapplications on image processing. In particular, a new edgedetector based on the morphological gradient, non-maximasuppression and a hysteresis method is presented.

Communicated by E. Viedma.

M. González-Hidalgo · S. Massanet (B)Department of Mathematics and Computer Science, University of theBalearic Islands, Ctra. de Valldemossa, Km.7.5, 07122 Palma, Spaine-mail: s.massanet@uib.es

M. González-Hidalgoe-mail: manuel.gonzalez@uib.es

Keywords Fuzzy mathematical morphology · Discretet-norm · Edge detection · Alternate filter · Top-hattransformation

1 Introduction

The recognition of shapes and therefore of vision systemsare an indispensable step for the identification of objects,object feature extraction and anomalies detection in auto-mated industrial processes. In this context, the mathematicalmorphology is an effective tool for extracting image com-ponents that are useful in the representation and descriptionof region shapes, such as boundaries, skeletons and convexhull. It is also useful for many pre- and post-processing tech-niques, specially in edge thinning and pruning. The morpho-logical operations are the basic tools of this theory. A mor-phological operation P transforms an image A that we wantto analyse by means of a structuring element B into a newimage P(A, B). The four basic morphological operationsare dilation, erosion, closing, and opening. These operationsare based on set theory and were originally developed forbinary images and afterwards successfully extended to gray-scale images in Serra (1982,1988) and Soille (1999). Severalresearchers have introduced alternative morphological oper-ations, a detailed account can be found in Bloch and Maître(1995) and Nachtegael and Kerre (2000a).

Due to the fact that the shapes in an image are not alwayscrisply defined, the method used for a recognition or visionsystem should have a provision for representing and manip-ulating the uncertainties as well as sufficient flexibility forprocessing the uncertainty at any level. Uncertainty can arisewithin each level of image analysis and pattern recognitionbecause of the raw sensor output and the extension to higherlevels. Fuzzy set theory provides a mechanism to represent

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and manipulate uncertainty and ambiguity. Fuzzy operatorsand their properties as well as fuzzy inference rules havefound considerable applications in image analysis and pat-tern recognition (see for instance Cheng et al. 2000; Kerre andNachtegael 2000; Nachtegael et al. 2003; Hassanien 2007;Melin et al. 2010).

The fuzzy mathematical morphology is a generalization ofbinary morphology using concepts and techniques of fuzzysets (see Bloch 2009, 2011; Bloch and Maître 1995; DeBaets et al. 1995a,b; De Baets 1997; Nachtegael and Kerre2000a; Nachtegael et al. 2011). The basic idea is that gray-level images can be represented as fuzzy sets, then the fuzzytools can be used to define fuzzy morphological operators.The fuzzy operators used to build a fuzzy morphology areconjunctions (usually continuous t-norms) and their residualimplications. Recently also conjunctive uninorms, as a partic-ular case of conjunctions, have been used in this area (see DeBaets et al. 1997; González et al. 2003; González-Hidalgoet al. 2009a,b). In addition, fuzzy mathematical morphol-ogy plays an important role in many applications like amongothers segmentation and edge detection (see Su et al. 2011;Papari and Petkov 2011; Luengo-Oroz et al. 2010; González-Hidalgo et al. 2009b) and filtering (see Lerallut et al. 2007;Maragos 2009).

However, gray-scale images are not represented as RN →

[0, 1] functions in practice. Indeed, there exist two technicallimitations in order to process images on a computer:

– the images are stored as finite matrices– the gray levels belong to a finite subchain with 256 values.

Therefore, images are represented as discrete functions and,similarly as in the continuous case, we can use techniquesfrom fuzzy set theory. The advantages of this approach isthat a fuzzyfication function is not needed and consequentlythe results do not depend on that process. Meanwhile, an ade-quate treatment of uncertainty is guaranteed due to the use ofdiscrete fuzzy operators (Mayor and Torrens 2005) into themorphological operators (see Nachtegael and Kerre 2000c).This approach is relatively new. Only some related worksdevoted to some theoretical aspects about this framework areavailable in the literature, see for instance Deng and Chen(2005), Nachtegael and Kerre (1999) and Nachtegael andKerre (2000b). In the last one, M. Nachtegael and E. Kerrestudied α-cuts decomposition properties of morphologicaloperators based on discrete semi-norms, from a theoreticalpoint of view without showing experimental results. Moregeneral discrete frameworks, such as L-fuzzy sets (Russo2010) or quantale modules (Sussner et al. 2009), have beenstudied also from a theoretical and abstract point of view.Thus although the algebraical properties of the morphologywe propose could be deduced from these general results, thediscrete t-norms that have to be used in order to satisfy these

properties have to determined. In addition, the applicationof these discrete approaches have not been deeply studiedyet. The application to image processing of this morphol-ogy based on discrete t-norms will be the main contribu-tion of this paper. For other approaches, more details can befound in Deng and Heijmans (2002) and Sussner and Valle(2008).

In this paper, we follow the general framework for fuzzymathematical morphology constructed by De Baets in DeBaets (1997) where he uses conjunctions and implicationsin order to define fuzzy erosion and fuzzy dilation, withoutforcing duality relationships between these operators, andobtaining good properties for the corresponding fuzzy clos-ing and fuzzy opening operators. Furthermore, the study ofthe algebraic properties and characterization of the closingand opening discrete operators and open and closed objectswhen using discrete t-norms is indispensable. This theoreti-cal background allows the construction of the so-called alter-nate filters (compositions alternating openings and closings)and the top-hat transformations (residues between openingand closing, a morphological gradient) too. The top-hat isused to highlight certain components of the image, while thealternate filters are designed to eliminate and reduce noise.Our goal is, on one hand, to provide some similar theoret-ical framework in the context of discrete operators and, onother hand, to show experimental results of its applicationto several images, and to provide evidences that their use isappropriate for image processing and shape recognition. Wecompare the results obtained by the discrete approach withthe ones obtained from the classical umbra approach and thecontinuous Łukasiewicz t-norm.

The paper is organized as follows. In the next section wereview the basic definitions and properties of fuzzy logicaloperators needed in the subsequent sections in the contextof finite chains. In Sect. 3, we present the general frame-work initiated by De Baets in De Baets (1997) and we willdiscuss then, the algebraical and morphological propertiessatisfied by the fuzzy morphological operators based on dis-crete t-norms, following a similar structure than the one usedin Nachtegael and Kerre (2000a). In addition, the propertiesrelated to open and closed objects including the generalizedidempotence law are presented. Section 5 is devoted to dis-play the results of the implementation of our approach, usingseveral discrete t-norms, on different images. Several appli-cations are presented: edge detection, noise removal and top-hat transformations. The paper ends with some conclusionsand future work.

2 Preliminaries

Let us recall the fuzzy discrete logical operators that we willuse throughout the paper. More details on these operators can

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be found for instance in Mayor and Torrens (2005). From nowon, consider L a finite chain L = {0, . . . , n}.Definition 1 A strong negation N on L is a non-increasinginvolution from L to L . That is, a non-increasing functionN : L → L satisfying N (N (x)) = x for all x ∈ L .

There is one, and only one, strong negation on L which isgiven (see Mayor and Torrens 2005) by

N (x) = n − x for all x ∈ L .

Definition 2 An increasing binary operator C : L × L → Lis called a discrete conjunction if it satisfies

C(0, n) = C(n, 0) = 0 and C(n, n) = n.

Definition 3 A binary operator I : L × L → L is called adiscrete implication if it is nonincreasing in the first variable,nondecreasing in the second one and it satisfies

I (0, 0) = I (n, n) = n and I (n, 0) = 0.

A discrete implication I is called a border implication on Lif it satisfies I (n, x) = x for all x ∈ L .

One can construct conjunctions and implications fromeach other. On one hand, given an implication I the binaryoperator defined by

CI,N (a, b) = N (I (a, N (b))) = n − I (a, N (b))

is a conjunction. On the other hand, given a conjunction Cthe binary operator defined by

IC,N (a, b) = N (C(a, N (b))) = n − C(a, N (b))

is an implication. Another way to construct implications fromconjunctions is by residuation. Given a conjunction C thebinary operator

IC (a, b) = max{c ∈ L | C(a, c) ≤ b}is an implication called the residual implication of C . Recip-rocally, a discrete conjunction can be defined in terms of adiscrete implication as follows using adjunction:

CI (a, b) = min{c ∈ L | b ≤ I (a, c)}.The most well-known kind of conjunctions in the discrete

framework, as well as on the [0, 1], is the class of t-norms.

Definition 4 A conjunction T on L is called a discretet-norm when it is commutative, associative and satisfiesT (x, n) = x for all x ∈ L .

Details and properties on discrete t-norms and their resid-ual implications can be found in Mayor and Torrens (2005)and Mas et al. (2004).

3 Fuzzy discrete mathematical morphologyand its properties

3.1 Fuzzy discrete morphological operators

From now on, we will use the following notation: I willdenote a discrete implication, C a discrete conjunction, Nthe only strong negation in L , T a discrete t-norm, IT itsresidual implication, A a gray-scale image, and B a gray-scale structuring element.

In order to extend the definition of classical erosion anddilation given in Serra (1982,1988), De Baets fuzzified theBoolean conjunction and the Boolean implication to obtaina successful fuzzification (see De Baets 1997). However, themain difference is that here an N -dimensional gray-scaleimage is modelled by a Z

N → L function instead of a fuzzyset in R

N . Thus, we have the following definitions.

Definition 5 The discrete fuzzy dilation DC (A, B) and dis-crete fuzzy erosion EI (A, B) of A by B are the gray-scaleimages defined by

DC (A, B)(y) = maxx

C(B(x − y), A(x)),

EI (A, B)(y) = minx

I (B(x − y), A(x)).

Definition 6 The discrete fuzzy closing CC,I (A, B) and dis-crete fuzzy opening OC,I (A, B) of A by B are the gray-scaleimages defined by

CC,I (A, B)(y)

= EI (DC (A, B), B̌)(y)

= minx

I (B(y − x),maxz

C(B(z − x), A(z))),

OC,I (A, B)(y)

= DC (EI (A, B), B̌)(y)

= maxx

C(B(y − x),minz

I (B(z − x), A(z))).

Note that the reflection B̌ of a N -dimensional fuzzy set Bis defined by B̌(x) = B(−x), for all x ∈ Z

N .It is obvious that a discrete t-norm is a conjunction. So,

we can use these operators and related implications to definefuzzy discrete morphological operators following the previ-ous definitions.

3.2 Algebraic properties of discrete morphologicaloperators using discrete t-norms

In this section, we will give sufficient and/or necessary condi-tions on the discrete t-norm in order to guarantee similar prop-erties as in the classical binary and gray-scale mathematicalmorphologies. Thus, we investigate which discrete t-normsneed to be chosen in order to preserve duality, monotonic-ity, interaction with union and intersection, invariance undertranslating and scaling, extensivity and idempotence, inclu-sion properties, commutativity and associativity of the fuzzy

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dilation, combinations of dilation and erosion, local knowl-edge property and adjunction property.

All results presented in this section are immediate coun-terparts of the [0, 1] case. In order not to break the develop-ment of the exposition we will not include the proofs and werefer to Massanet (2012). In De Baets (1997) and Gonzálezet al. (2003), the corresponding proofs for the [0, 1] frame-work are included. A remarkable difference is related to theleft-continuity or right-continuity of the conjunction or theimplication. These properties are needed in the frameworkof [0, 1] in order to ensure that conjunctions and implica-tions preserve infimum or supremum adequately. However,in our framework this is clear because infimum and supre-mum are in fact minimum and maximum as we have alreadycommented. On the other hand, some of these results can bededuced from more general results on L-fuzzy sets, quantalemodules or complete lattices. We will only concentrate onthe results which will be useful in the applications presentedin Sect. 5.

Proposition 1 Let T be a discrete t-norm and I a discreteimplication. It holds: the fuzzy dilation DT is increasing inboth arguments, the fuzzy erosion EI is increasing in the firstargument and decreasing in the second one, the fuzzy closingCT,I and the fuzzy opening OT,I are both increasing in thefirst argument.

Now, for an arbitrary family (Ai )i∈I of discrete images wecan define the Zadeh union and intersection by:

⋃i∈I Ai (x)

= supi∈I Ai (x) = maxi∈I Ai (x) and⋂

i∈I Ai (x) =inf i∈I Ai (x) = mini∈I Ai (x). Our approach has an optimalbehaviour with unions and intersections for any discrete t-norm T and implication I . For example,

DT

(⋃

i∈I

Ai , B

)

=⋃

i∈I

DT (Ai , B),

EI

(⋂

i∈I

Ai , B

)

=⋂

i∈I

EI (Ai , B).

At this point let us study the translation and the scaling ofa discrete image. The translation Tv(A) of a discrete imageA by v ∈ Z

N , is defined by Tv(A)(x) = A(x − v), while thescaling Hλ−1 of a discrete image A by λ > 0, λ ∈ Z (notethat in our case we only can do scaling by an integer factor), isdefined by Hλ−1(A)(x) = A(λx). Thus, the usual propertiesrelated to these two transformations hold. For example,

DT (Tv(A), B) = Tv(DT (A, B)),

CT,I (Tv(A), Tv(B)) = Tv(CT,I (A, B)),

DT (Hλ−1(A), Hλ−1(B)) = Hλ−1(DT (A, B)),

CT,I (Hλ−1(A), Hλ−1(B)) = Hλ−1(CT,I (A, B)).

The properties in the proposition above are also verified fornegative integers, although they have no practical interpre-

tation in general. However for λ = −1, we obtain the spe-cial case concerning the reflection of an image. A detailedaccount of these properties in the general framework of DeBaets can be found in Nachtegael and Kerre (2000a) and inthe uninorm framework in González-Hidalgo et al. (2009a).

The extensivity of the fuzzy dilation and the anti-extensivity of the fuzzy erosion are ensured by the next propo-sition.

Proposition 2 Let T be a discrete t-norm, let I be a discreteborder implication, let B a gray-scale structuring elementsuch that B(0) = n. Then the following inclusions hold:

EI (A, B) ⊆ A ⊆ DT (A, B)

Thus, as in classical morphology, the difference betweenthe fuzzy dilation and the fuzzy erosion of a gray-scale image,DT (A, B)\EIT (A, B), called the fuzzy gradient operator,can be used in edge detection.

Proposition 3 Let T be a discrete t-norm and I a discreteborder implication, let A be a gray-scale image and let B bea gray-scale structuring element, then it holds

1. If the following property holds: for all x, y ∈ L, y ≤I (x, T (x, y)) then the fuzzy closing CT,I is extensive:A ⊆ CT,I (A, B).

2. If the following property holds: for all x, y ∈ L,T (x, I (x, y)) ≤ y then the fuzzy opening OT,I is anti-extensive: OT,IU (A, B) ⊆ A.

3. If the following property holds: for all x, y ∈ L,T (x, I (x, y)) ≤ y ≤ I (x, T (x, y)) then the fuzzy closingand the fuzzy opening are idempotent:

CT,I (CT,I (A, B), B) = CT,I (A, B),

OT,I (OT,I (A, B), B) = OT,I (A, B).

The idempotence of the discrete fuzzy opening and the dis-crete fuzzy closing is going to be essential on the generationof the so-called alternate filters, used in noise reduction. Next,by joining some of the results in the propositions above, wehave the following result.

Proposition 4 Let T be a discrete t-norm, let I be a discreteborder implicator. Let A be a gray scale-image and let B bea gray-scale structuring element such that B(0) = n. If Tand I satisfy

∀x, y ∈ L , T (x, I(x, y)) ≤ y ≤ I(x, T (x, y))

then it holds that:

EI(A, B)⊆ OT,I(A, B) ⊆ A ⊆ CT,I(A, B) ⊆ DT (A, B).

Finally, it can be proved easily that the local knowledgeproperty and the usual properties concerning combinations

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of dilations and erosions among many others hold in ourdiscrete approach (see Massanet 2012). Again in Nachtegaeland Kerre (2000a) and González-Hidalgo et al. (2009a) theproofs for the [0, 1] and uninorm frameworks respectivelyare also available.

In order to guarantee the fulfilment of all the above prop-erties we will consider a discrete t-norm T and its residualimplication IT . Recall that the residual implication IT ofa discrete t-norm is a border implication that satisfies theexchange principle and the following properties used in sev-eral of the previous results (see Mas et al. 2004):

– ∀x, y ∈ L , T (x, I (x, y)) ≤ y ≤ I (x, T (x, y))– ∀x, y, z ∈ L , T (x, z) ≤ y ⇔ I (x, y) ≥ z.

In addition, we define by (co A)(x) = N (A(x)) = n − A(x)the complement co A of a fuzzy set A. Two fuzzy mor-phological operations P and Q are called dual if for anytwo gray-scale objects A and B it holds that P(A, B) =co Q(co A, B). It holds that the fuzzy dilation and fuzzy ero-sion are dual if and only if I = IC,N . Moreover, if the fuzzydilation and fuzzy erosion are dual, then also the fuzzy clos-ing and fuzzy opening are dual (see De Baets 1997). Thusto have duality between our fuzzy discrete morphologicaloperators, we need to use discrete t-norms satisfying

IT = IT,N .

This property holds for some discrete t-norms given in thefollowing proposition (see Mas et al. 2004).

Proposition 5 The identity

IT = IT,N

is satisfied in each of the following situations

1. When T is the Łukasiewicz discrete t-norm that is

TL(x, y) = max{0, x + y − n}

2. When T is the nilpotent minimum given by the followingexpression

TnM (x, y) ={

0 if x + y ≤ n,

min{x, y} otherwise.

3. When T is an ordinal sum (with only one summand) of theŁukasiewicz t-norm in a square [a, n − a]2, a ∈ L witha ≤ n − a, truncated by 0, given by the expression

TnMa(x, y) =

⎧⎪⎪⎨

⎪⎪⎩

0 if x + y ≤ n,x + y − (n − a) if x + y > n and

a < x, y ≤ n − a,min{x, y} otherwise.

Fig. 1 The nilpotent minimum TnM (left) and its generalization TnMa(right)

In Fig. 1, the structure of some of the t-norms of Proposi-tion 5 is displayed. Note that cases 1) and 2) in the previousproposition are the extreme cases of 3) when a = 0 and ais the floor of n/2, respectively. The residual implicationsof this family of discrete t-norms can be found in Mas etal. (2004). Consequently the choice of any of the above dis-crete t-norms jointly with its residual implication guaranteeall the studied properties for the corresponding mathematicalmorphology.

Remark 1 Note that given a discrete t-norm and its resid-ual implication, the duality with respect to adjunction holdsstraightforwardly. This fact naturally yields morphologicaloperators with many desired properties (see Heijmans 1994;Serra 1982,1988).

Remark 2 Duality in this approach could be implementedthrough a negation N ′ : L −→ L non necessarily strong,in order to avoid the problem of having one and only onenegation. However, duality properties between morpholog-ical operators can fail in this case even using the t-normsin the previous proposition substituting the strong negationN (x) = n − x by any negation N ′ : L −→ L not strong.

Remark 3 Note that the same family of t-norms in theframework of [0, 1] can be considered. All these t-normsare left-continuous and satisfy the property IT = IT,N

with N (x) = 1 − x . Thus, a mathematical morphol-ogy could be also implemented with these t-norms, butrequiring a fuzzification and a defuzzification steps in theprocess. On the contrary, in our framework these steps areavoided.

4 Closed and open fuzzy objects

The idempotence properties of fuzzy opening and closingwhen T is a discrete t-norm and IT its residual implicationmotivate, as in the classical mathematical morphology, thefollowing definitions.

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Definition 7 Let A and B be two gray-scale images. It issaid that A is B-closed (resp. B-open) if CT,IT (A, B) = A(resp. OT,IT (A, B) = A).

It is important to note that due to the idempotence of theclosing and opening, the closing is B-closed and the openingis B-open. Almost all the results presented in this section areanalogous to the respective in the [0, 1]-framework and theproofs are quite similar and therefore, they are not included(see Massanet 2012; De Baets 1995; González-Hidalgo et al.2009a). Again a difference worth to mention is related to theleft-continuity or the right-continuity of the conjunction orthe implication. These properties are necessary in the [0, 1]-framework in order to ensure that both operators preservethe infimum and the supremum adequately. However, in thediscrete approach they are not necessary since we work withmaxima and minima.

First of all, all B-open and B-closed objects are the open-ing or closing of some image, respectively.

Proposition 6 Let T be a discrete t-norm and IT its resid-ual implication. Then it holds that A is B-open (resp. B-closed) if, and only if, it exists a fuzzy object F such thatA = DT (F,−B) (resp. A = EIT (F,−B)).

In Bodenhofer (2003), it is stated that the closing and open-ing operators only make sense if the opening always gives anopen object, and the closing gives a closed object. Further-more, it is recommended that they have extrema properties.The following result shows that these requisites hold in thistheory.

Proposition 7 Let T be a discrete t-norm and IT its residualimplication. Then OT,IT (A, B) is the largest B-open subsetgreater than A and CT,IT (A, B) is the smallest B-closedsubset that contains A.

The intersections and unions preserve the properties ofB-open and B-closed objects, respectively.

Proposition 8 Let T be a discrete t-norm and A1 and A2

two gray-scale images. Then if A1 and A2 are both B-open(resp. B-closed) then A1 ∪ A2 is B-open (resp. A1 ∩ A2 isB-closed).

Up to now any discrete t-norm satisfies all the proper-ties. However, if the duality between fuzzy open and closedobjects is required, we must resort to the discrete t-norms ofProposition 5.

Proposition 9 Let T be a discrete t-norm of Proposition 5.Then, A is B-open if, and only if, co A is B-closed, where(co A)(x) = n − A(x).

One of the most important properties that the morphologi-cal operators can satisfy is the so-called generalized idempo-tence. This property is satisfied by the classical opening and

closing and the fuzzy ones when we consider continuous t-norms and their residual implication. The following proposi-tions are previous steps to establish this property when usingthe discrete t-norms of Proposition 5.

Proposition 10 Let T be a discrete t-norm and IT its resid-ual implication. Then

(i) ∀x, y ∈ L, T (x, IT (x, y)) ≤ y.(ii) ∀x, y, z ∈ L, T (IT (x, y), IT (y, z)) ≤ IT (x, z).

(iii) ∀x, y, z ∈ L, IT (x, T (y, z)) ≥ T (IT (x, y), z).

Proposition 11 Let T be a discrete t-norm and IT its resid-ual implication. For all a, b, c, d, e, f ∈ L, if T (a, IT (b, c))≥ d and T (e, IT ( f, b)) ≥ a then T (e, IT ( f, c)) ≥ d.

Proposition 12 Let T be a discrete t-norm and IT itsresidual implication. If for all a, b, c, d, e, f, g, h ∈ L,T (a, IT (b, c)) ≥ d, T (c, IT (e, f )) ≤ g and T (d, IT (e, f ))≥ h are satisfied, then T (a, IT (b, g)) ≥ h.

Now, using the discrete t-norms of Proposition 5 and theprevious inequalities, the generalized idempotence of the dis-crete fuzzy opening and closing is proved in a similar way tothe [0, 1] case (see De Baets 2000 and an improvement with-out assuming finite ranges in González-Hidalgo et al. 2009a).It is obvious that in the discrete case, there is no restrictionon the ranges of A and B.

Proposition 13 Let T be a discrete t-norm of those of Propo-sition 5 and IT its residual implication. If A is B-open, thenfor all fuzzy object F,

OT,IT (F, A) ⊆ OT,IT (F, B) ⊆ F

holds and by duality, F ⊆ CT,IT (F, B) ⊆ CT,IT (F, A).

Proposition 14 (Generalized idempotence) Let T be a dis-crete t-norm of those of Proposition 5 and IT its residualimplication. If A is B-open, then for all fuzzy object F,

OT,IT (OT,IT (F, B), A) = OT,IT (OT,IT (F, A), B)

= OT,IT (F, A)

holds and dually for the closing.

Remark 4 The two previous propositions are valid for anydiscrete t-norm. However, due to the fact that duality is onlysatisfied when using the discrete t-norms of Proposition 5,these are the only ones that satisfy all the algebraic propertiesto guarantee a good mathematical morphology.

5 Applications

In this section, some experiments using the discrete t-normsenumerated in Proposition 5 are presented. These examples

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illustrate the influence of the choice of the pair (T, IT ) usingthe three types described before. During these experimentalresults, the structuring elements B used for the fuzzy discreteoperators are represented by the matrices

B1 =⎛

⎝219 219 219219 255 219219 219 219

⎞

⎠ B2 =⎛

⎝0 255 0255 255 2550 255 0

⎞

⎠ .

Note that the structuring element B1 is the discrete versionof the one used in Nachtegael and Kerre (2000a) and B2 isa common binary disk. In order to ensure the goodness ofthis method, the results obtained in this section by the dis-crete approach will be compared with the ones obtained bytwo classical algorithms, the umbra approach and the con-tinuous Łukasiewicz t-norm, TL(x, y) = max{0, x + y − 1}for all x, y ∈ [0, 1]. Recall that the pair (TL , ITL ) is therepresentative of the only class of t-norms (nilpotent ones)that guarantees the fulfilment of all the properties in orderto have a good fuzzy mathematical morphology, includingduality and generalized idempotence (see González-Hidalgoet al. 2009a). When the fuzzy approach based on nilpotentt-norms is used, the continuous counterpart of the structuringelements B are applied.

First of all, in Fig. 2 we can observe, from left to right,the original image, the fuzzy erosion and the fuzzy dilationobtained using the discrete nilpotent minimum t-norm and thestructuring element B1. The expected effects on the objectsfrom both operators are clearly manifested on the letters, theumbrella and the cat’s whiskers.

On the other hand, the opening and closing are the mostelementary morphological filters, which are called basic fil-ters. As we have seen in Sect. 3, the opening is an antiex-tensive morphological filter and the closing is an extensivemorphological filter. So, in a first step, they can be used toremove non desired objects in an image. We can observe thatthe opening of a gray level image by a structuring elementremoves the light zones of less size than the element andmakes the light objects darker. The morphological closinghelps to remove dark structures of less size than the struc-turing element, toning down the dark objects. The size andshape of the used structuring element in the opening mustagree with the image structures that we want to remove. Theseeffects are shown in Fig. 3. Note the reflections of the light inthe women’s hair and the button of the blouse of the secondwoman.

5.1 Edge detection

One of the first applications that we have implemented is thefuzzy morphological gradient

DT (A, B)\EIT (A, B)

(a) (b) (c)

Fig. 2 Effects of the fuzzy discrete erosion and dilation using TnM andB1. a Original, b erosion, c dilation

which is a useful tool for edge detection because of Propo-sition 2. Edge detection is a fundamental low-level imageprocessing operation, which is essential to carry out severalhigher level operations such as image segmentation, com-puter vision, motion and feature analysis and recognition.Its performance is crucial for the final results of the imageprocessing techniques. A lot of edge detection algorithmshave been developed over the last decades. These differentapproaches vary from the classical ones (Pratt 2007) andtheir improved versions (Lopez-Molina et al. 2013b; Medina-Carnicer et al. 2011) based on a set of convolution masks,to the new techniques based on fuzzy sets (Bustince et al.2009).

We will present the results following the approach ofNachtegael and Kerre in Nachtegael and Kerre (2000a). Fol-lowing the definitions given in that work, we compare theedge images with those obtained using the classical gray-scale morphology based on the umbra approach, and thoseobtained by the fuzzy approach based on the Łukasiewiczcontinuous t-norm. From Figs. 4, 5, 6 this gradient has beenapplied using three of the discrete t-norms of Proposition 5

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(a) (b) (c)

Fig. 3 Effects of the fuzzy discrete closing and opening using TnM andB1. a Original, b closing, c opening

(a) (b) (c)

(d) (e) (f)

Fig. 4 Fuzzy edge images of the “cameraman” image obtained by sev-eral approaches using B1. a Original, b umbra approach, c (TL , ITL ), d(TL , ITL ) discrete, e (TnM , ITnM ), f (TnM25, ITnM25 )

and the results are displayed jointly with the ones obtainedby the umbra approach and using the nilpotent t-norms. Aswe can see, the results using discrete t-norms are equal orbetter than those obtained from the continuous Łukasiewiczt-norm and from the umbra approach, from a visual point ofview.

In Fig. 4 we display some of the results obtained using theclassical cameraman image. It can be observed that, although

(b)

(d)

(a) (c)

(e) (f)

Fig. 5 Fuzzy edge images of the “erythrocytes” image obtained byseveral approaches using B1. a Original, b umbra approach, c (TL , ITL ),d (TL , ITL ) discrete, e (TnM , ITnM ), f (TnM25, ITnM25 )

(a) (b) (c)

(d) (e) (f)

Fig. 6 Fuzzy edge images of the “pyramid” image obtained by severalapproaches using B1. a Original, b umbra approach, c (TL , ITL ), d(TL , ITL ) discrete, e (TnM , ITnM ), f (TnM25, ITnM25 )

the hard edges (see the person in the foreground and somebuildings in the background) are detected very well in all thecases, the gradient obtained using the nilpotent minimum dis-crete t-norm detects some soft edges better than the gradientobtained with (TL , ITL ) and the umbra approach (see someof the buildings in the background).

As we can observe, in Fig. 5 our discrete approach keepsthe boundaries and structures of the red blood cells and inFig. 6, the pyramids can only be discerned in the fuzzy edgeimages obtained using TnM or TnM25. Note that the furthestpyramid is completely missing in the images obtained by theclassical algorithms.

Analysing the fuzzy edge images obtained by the dis-crete approach, one can observe that the gradient obtainedby TnM25 seems to be an intermediate step from the imageobtained by the discrete Łukasiewicz t-norm to the oneobtained by the nilpotent minimum. In fact, as we can see inFigs. 7 and 8, the parameter a of the discrete t-norm TnMa

allows to control the similarity of the fuzzy edge image tothe ones obtained by the discrete TŁ and TnM . This is quiteimportant since usually the discrete Łukasiewicz t-norm fails

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A fuzzy mathematical morphology based on discrete t-norms

(c)

(d) (e) (f)

(g) (h)

(a) (b)

(i)

Fig. 7 Fuzzy edge images of the “Lutsk castle tower” image usingseveral fuzzy discrete gradients with B1. a Original, b (TL , ITL ), c(TnM15, ITnM15 ), d (TnM20, ITnM20 ), e (TnM25, ITnM25 ), f (TnM30, ITnM30 ),g (TnM35, ITnM35 ), h (TnM40, ITnM40 ), i (TnM , ITnM )

to detect some important edges while the nilpotent minimumoften detects too much short edges or textures. Thus, we canapply the value of parameter a which best fits a particularimage.

However, the fuzzy edge image does not satisfy the restric-tions imposed by Canny (1986). These restrictions force arepresentation of the edges as binary images with edges ofone pixel wide. So, in order to satisfy the Canny’s restrictions,the fuzzy edge image has to be thinned and binarized. First ofall, the fuzzy edge image becomes a fuzzy thin edge imagewhere all the edges have one pixel width. The fuzzy edgeimage will contain large values where there is a strong imagegradient, but to identify edges the broad regions present inareas where the slope is large must be thinned so that only themagnitudes at those points which are local maxima remain.Non-Maxima Suppression (NMS) performs this by suppress-ing all values along the line of the gradient that are not peakvalues (see Canny 1986). NMS has been performed usingP. Kovesis’ implementation in Matlab (Kovesi 2000). After

(c)

(d) (e) (f)

(g) (h)

(a) (b)

(i)

Fig. 8 Fuzzy edge images of the “lone palm” image using several fuzzydiscrete gradients with B1. a Original, b (TL , ITL ), c (TnM15, ITnM15 ), d(TnM20, ITnM20 ), e (TnM25, ITnM25 ), f (TnM30, ITnM30 ), g (TnM35, ITnM35 ),h (TnM40, ITnM40 ), i (TnM , ITnM )

that, a binarization technique is applied. We perform an unsu-pervised hysteresis based on the determination of the insta-bility zone on the histogram in order to find the low andthe high thresholds (see Medina-Carnicer et al. 2011). Themethod needs some initial set of candidates for the thresholdvalues. In this case, {0.01, . . . , 0.25} has been introduced,the same set used in Medina-Carnicer et al. (2011).

At this point, given a binary edge image with a pixel widthedges, it is possible to compare quantitatively the obtainedresults from the different considered approaches. In the lit-erature, there exist several performance measures on edgedetection to compare outputs of edge detectors (see Lopez-Molina et al. 2013a). These measures need, in addition tothe binary thin edge image (DE) obtained by the algorithm, aground truth edge image (GT) that is a binary thin edge imagecontaining the true edges of the original image, i.e., the ref-erence edge image. In this work, we will use the followingquantitative performance measures:

– Pratt’s figure of merit (Pratt 2007) defined as

FoM = 1

max{card{DE}, card{GT }} ·∑

x∈DE

1

1 + ad2 ,

where card is the number of edge points of the image, ais a scaling constant and d is the separation distance ofan actual edge point to the ideal edge points. In our case,we considered a = 1 and the Euclidean distance d.

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– The error metrics of Baddeley (1992), inspired on Haus-dorff distances is given by

�kw(DE,GT )

=⎡

⎣ 1

|P|∑

p∈P

|w(d(p,DE))− w(d(p,GT ))|k⎤

⎦

1k

,

where P is the set of all the pixels of the image, k ∈R, k > 0, d(p, A) represents the distance between theposition p and the nearest position p′ of A such that p′is an edge of A and w : R → R is any concave function.In our case, we considered k = 2, the Euclidean distanced and w, the identity function.

– The ρ-coefficient (Grigorescu et al. 2003), defined as

ρ = card(E)

card(E)+ card(EF N )+ card(EF P ),

where E is the set of well-detected edge pixels, EF N isthe set of ground truth edges missed by the edge detectorand EF P is the set of edge pixels detected but with nocounterpart on the ground truth image.

– The F-measure (Rijsbergen 1979), defined as the har-monic mean between precision and recall,

F = 2 · PR · RE

PR + RE,

where P R and RE denote precision and recall respec-tively with the following expressions:

PR = card(E)

card(E)+ card(EF P ),

RE = card(E)

card(E)+ card(EF N ).

Larger values of FoM , ρ and F (0 ≤ FoM, ρ, F ≤ 1) andlower values of �k

w are indicators of better capabilities foredge detection.

Consequently, we need a dataset of images with theirground truth edge images (edges specifications) in order tocompare the outputs obtained by the different algorithms.So, some of the images and their edge specifications from thepublic dataset of the University of South Florida1 (Bowyer etal. 1999) have been used. In particular in Fig. 9 the images andtheir ground truth edge images are displayed. The obtainedresults by the different approaches are showed in Figs. 10 and11. The values of the quantitative performance measures col-lected in Table 1 ensure the competitiveness of the discreteapproach compared with the other frameworks.

1 This image dataset can be downloaded from ftp://figment.csee.usf.edu/pub/ROC/edge_comparison_dataset.tar.gz.

(a) (b)

Fig. 9 Some original images and their ground truth edge images con-sidered in the experiments. a Original, b ground truth

(d) (f)

(a) (b) (c)

(e)

Fig. 10 Binary thin edge images of the “van” image obtained by sev-eral approaches using B1. a Original, b umbra approach, c (TL , ITL ), d(TL , ITL ) discrete, e (TnM , ITnM ), f (TnM25, ITnM25 )

5.2 Noise reduction and top-hat transformations

The opening and closing are morphological transformationssatisfying Propositions 3 and 4 and because of that they areuseful to compute their associated residuals, known as top-hat transformations. These transformations find structureswhich have been removed by the opening and closing filtersand the residual between the original image and the filteredimage increases notably the contrast of the erased regions(see Serra 1982,1988). So, the top-hat transformations aredefined as follows

ρT,IT (A, B) = A\OT,IT (A, B) (top-hat)

ρdT,IT

(A, B) = CT,IT (A, B)\A (dual top-hat).

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A fuzzy mathematical morphology based on discrete t-norms

(d) (f)

(a) (b) (c)

(e)

Fig. 11 Binary thin edge images of the “fan” image obtained by severalapproaches using B1. a Original, b umbra approach, c (TL , ITL ), d(TL , ITL ) discrete, e (TnM , ITnM ), f (TnM25, ITnM25 )

The top-hat transformation enhances the light objects thathave been removed by the opening and it is used to extractcontrasted components from the background, while thedual top-hat extracts the dark components which have beenremoved by the closing. Usually, these transformations deletethe soft trends.

In Fig. 12, a comparison of both transformations for thedifferent morphological approaches is displayed. We com-pare the results obtained by the discrete approach with thoseobtained by the umbra approach and by the Łukasiewicz con-tinuous t-norm. In the results, the well-known Otsu’s thresh-old method (Otsu 1979) has been applied in order to bet-ter visualize them. As we can observe, the discrete t-normsoutperform at naked eye the results obtained by the othermorphologies.

As we have already commented, the closing and the open-ing are the basic filters of the fuzzy mathematical mor-phology. The composition or combination of these opera-tors between them is the most used way to build new filters(Serra 1982,1988). The initial filters that one can build fromthe opening and closing are the alternate filters. Let ξ(A, B)

and ψ(A, B) be the opening and closing, respectively, ofan image A by an structuring element B using a discretet-norm as conjunction and its residual implication. Now, wecan build four idempotent and increasing filters: ξψ , ψξ ,ψξψ and ξψξ . Due to the idempotence property, the com-position of more than three operators does not provide usnew filters.

In Fig. 13, we have a chest tomography original image, thesame image with added salt and pepper noise with parameter0.02 and another corrupted one with added Gaussian noisewith parameter 0.001. The noise was added using the stan-dard functions of Matlab R2008a. Figures 14 and 15 showthe effect of the choice of the different discrete t-norms anddifferent structuring elements in the results. In addition, thediscrete approach gives better or comparable results to theones obtained by the umbra approach and the nilpotent t-norms. Note that the results improve using a more adequatestructuring element for this type of noise as B2. For binarystructuring elements as B2 all discrete t-norms give the sameresult due to the boundary properties of a discrete t-norm

T (x, n) = x and T (0, x) = 0 for all x ∈ L .

All these observations can be stressed using some quantita-tive performance measures on noise reduction. Let O1 andF2 be two images of dimensions M × N . We suppose that O1

is the original noise-free image and F2 is the restored imagefor which some filter has been applied. We will use MSE(Pratt 2007) and SSIM (Wang et al. 2004) that are given by:

MSE(F2, O1) = 1

M N

M∑

i=1

N∑

j=1

(O1(i, j)− F2(i, j))2,

SSIM(F2, O1) = (2μ1μ2 + C1)

(μ21 + μ2

2 + C1)· (2σ12 + C2)

(σ 21 + σ 2

2 + C2),

where μk , k = 1, 2 is the mean of the image O1 and F2,respectively, σ 2

k is the variance of each image, σ12 is thecovariance between the two images, C1 = (0.01 · 255)2 andC2 = (0.03 ·255)2 (see Wang et al. 2004 for details). Smallervalues of MSE and larger values of SSIM (0 ≤ SSI M ≤ 1)

Table 1 Values of the differentperformance measures on edgedetection for the results obtainedfor each algorithm for eachimage displayed in Figs. 10 and11

Measure Image Umbra (TL , ITL ) (TL , ITL )d. TnM TnM25

FoM Van 0.4387 0.3829 0.3829 0.4888 0.5192

Fan 0.3619 0.2949 0.2949 0.5371 0.4838

�kw Van 13.3123 15.4977 15.4977 12.4726 12.4024

Fan 10.502 20.5232 20.5232 5.0816 5.1712

ρ Van 0.5176 0.4685 0.4685 0.6041 0.5968

Fan 0.5659 0.4734 0.4734 0.7602 0.7317

F Van 0.6858 0.6446 0.6446 0.7541 0.7503

Fan 0.745 0.6779 0.6779 0.8698 0.8578

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M. González-Hidalgo, S. Massanet

(b) (d)

(a)

(c) (e) (f)

Fig. 12 Top-hat (top) and dual top-hat (bottom) using several approaches with B1. a Original, b (TL , ITL ) discrete, c (TnM , ITnM ), d (TnM25, ITnM25 ),e (TL , ITL ), f umbra approach

(a) (c)(b)

Fig. 13 Original and noisy images used in the noise reduction exper-iments. a Original, b salt and pepper noisy image, c Gaussian noisyimage

are indicators of better capabilities for noise reduction andimage recovery. The obtained values can be viewed in Tables2 and 3.

We are aware that the proposed alternate filters onlysmooth the image and more complex filters are needed inorder to remove notably the noise. Consequently, in the futurework we want to develop sequential alternate filters whichwill improve the results.

6 Conclusions and future work

We have proved that it is possible to use discrete t-normsin order to construct a fuzzy mathematical morphology sat-isfying the same properties than in binary and gray-scaleclassical mathematical morphology. Moreover, the proper-ties needed in order to obtain a “good” morphology, includ-ing duality between morphological operators, are satisfied bythe t-norms presented in Proposition 5. Specially importantis the adjunction property satisfied by the choice of a dis-crete t-norm and a residual implication in the morphologicaloperators.

In particular using this kind of discrete t-norms we haveduality of the fuzzy morphological operators, we have anti-extensivity of the fuzzy opening and extensivity of the fuzzyclosing. Thus, it follows that DT (A, B)\EIT (A, B) servesas an edge detector of the image A (as a fuzzy morphologi-cal gradient) with notable results in practice, specially withthe discrete nilpotent minimum. Also, the fuzzy morpho-

(a) (b) (c) (d) (e)

Fig. 14 Filtered images using theψξψ filter with B1 (top) and B2 (bottom) of the corrupted image with salt and pepper noise of Fig. 13. a (TL , ITL )discrete, b (TnM , ITnM ), c (TnM25, ITnM25 ), d (TL , ITL ), e umbra approach

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A fuzzy mathematical morphology based on discrete t-norms

(a) (b) (c) (d) (e)

Fig. 15 Filtered images using the ψξψ filter with B1 (top) and B2 (bottom) of the corrupted image with Gaussian noise of Fig. 13. a (TL , ITL )discrete, b (TnM , ITnM ), c (TnM25, ITnM25 ), d (TL , ITL ), e umbra approach

Table 2 Values of performance for the results obtained for the imagecorrupted with salt and pepper noise in Fig. 13

Approach SE MSE SSIM

TL B1 148.08 0.989

B2 26.682 0.9981

TnM B1 128.83 0.9905

B2 26.682 0.9981

TnM25 B1 127.495 0.9906

B2 26.682 0.9981

(TL )cont. B1 148.08 0.989

B2 26.682 0.9981

Umbra B1 148.08 0.989

B2 26.682 0.9981

Table 3 Values of performance for the results obtained for the imagecorrupted with gaussian noise in Fig. 13

Approach SE MSE SSIM

TL B1 414.452 0.9618

B2 244.288 0.975

TnM B1 638.334 0.9385

B2 244.288 0.975

TnM25 B1 615.304 0.9404

B2 244.288 0.975

(TL )cont. B1 414.452 0.9618

B2 244.288 0.975

Umbra B1 776.396 0.9269

B2 278.833 0.9713

logical operators are invariant under translation and scaling.Invariance under translation implies that the fuzzy morpho-logical operations are independent of the choice of the origin,while invariance under scaling means that these operations

are independent of the used scale. Thus, the structuring ele-ment only depends on its shape and this can be exploded fordirectional transformation, which are useful in granulome-try.

On the other hand, the study of closed and open objects,as well as the generalized idempotence property allows theconstruction of the basic alternate filters. Recall that in the[0, 1] case generalized idempotence is satisfied only whenthe used t-norm is continuous (left-continuity is not enough),and consequently the t-norms on [0, 1] analogous to thosepresented in Proposition 5—2) and 3) are not suitable tobe used in this direction. In the discrete case generalizedidempotence is satisfied when using any of the t-norms inProposition 5.

The next steps are headed to go deeply into the applica-tions of this new framework:

– Edge detection First of all, in order to confirm the pre-liminary results obtained in this application, we wantto extend the objective comparison between the differ-ent edge detectors to more images. In addition, anotherinteresting line of research is related to determine thebest way to transform the discrete fuzzy edge image to abinary thin edge image through comparing different tech-niques like non-maxima suppression and several thresh-olding/hysteresis methods.

– Noise reduction The filtered images can be also com-pared through other measures like SNR or the fuzzy DI -subsethood measure E QσDI (see Bustince et al. 2007).In addition, the results could be improved with a propernoise identification function in order to apply the filteronly in those pixels corrupted by the noise or the gen-eration of sequential alternate filters. Moreover, the useof dynamic structuring elements with a variable size andshape could improve drastically the results.

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Moreover, as noted in Remark 2, the discrete approach pre-sented in this paper could be extended by considering non-strong negations N ′ : L −→ L in order not to restrict thechoice of the negation to N (x) = n−x . However, a unavoid-able step in this direction must be to determine which discretet-norms can be used to obtain duality between the morpholog-ical operators. Finally, we want to consider also other kindsof discrete conjunctions in our approach, specially discreteuninorms and discrete copulas.

Acknowledgments This paper has been partially supported by theSpanish Grant MTM2009-10320 with FEDER support.

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