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Closing and Opening Based on Discrete T-Norms. Applications to Natural Image Analysis

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Closing and Opening Based on Discrete T-Norms. Applications to Natural Image Analysis EUSFLAT-LFA 2011 July 2011 Aix-les-Bains, France Closing and Opening based on discrete t-norms. Applications to natural Image Analysis M. González-Hidalgo1 S. Massanet1 1Dept. of Math. and Comp. Science, University of the Balearic Islands, 07122 Palma de Mallorca, Spain Abstract the classical properties. Initial results in edge detec- tion, showed there, improved, at least at naked eye, This paper delves deeply into the recently intro- those obtained using the nilpotent t-norms. The duced mathematical morphology based on discrete mathematical morphology has been already used in t-norms. Closing and opening operators and the medical image analysis ([9], [10]). Therefore, dis- concepts of open and closed objects are introduced. crete fuzzy mathematical morphology could have a All the properties satisfied by nilpotent t-norms, wide range of applications in this research area. The even the generalized idempotence, hold too. After use of performance measures supports this fact since that, some experimental results, using comparative it shows that the discrete framework outperforms measures, on edge detection are showed. Some ex- the currently used mathematical morphologies and perimental results concerning the Top-Hat transfor- in some cases, the classical algorithms (Canny, So- mations and the basic filters built from the opening bel, Prewitt. ). However, medical images can be and closing operators are presented. Top-Hat exper- distorted by noise and consequently, morphological iments are compared with those obtained with the filters will play a key role (see [9], [11]). In this di- umbra approach, nilpotent t-norms and uninorms, rection, the study of the algebraic properties and proving that the discrete approach provides notable characterization of the closing and opening discrete results. Moreover, different objective measures are operators and open and closed objects when using used in order to evaluate the filtered results depend- discrete t-norms is indispensable. Properties ob- ing on the amount of noise in the image. tained are similar to those obtained in [12] and in a broader context, in [13]. This theoretical back- Keywords: Discrete t-norm, mathematical mor- ground allows the construction of the so-called alter- phology, edge detection, noise reduction. nate filters and the Top-Hat transformations. The Top-Hat is used to highlight certain components of 1. Introduction the image, while the alternate filters are designed to eliminate and reduce noise. The fuzzy mathematical morphology is a general- The communication is organized as follows. In Sec- ization of the binary morphology ([1]) using con- tion 2, definitions and properties of fuzzy discrete cepts and techniques from the fuzzy sets theory morphological operators are recalled. In Section 3 ([2], [3], [4]). This theory allows a better treat- the properties related to open and closed objects ment and a representation with greater flexibility of including the generalized idempotence law are pre- the uncertainty and ambiguity present in any level sented. In Section 4, some experimental results of an image (raw sensor output and its extension on medical images edge detection are performed. to higher levels). Consequently, the component ex- Then, the Top-Hat transformation is introduced traction and the shape recognition improve drasti- and results comparing different morphological ap- cally. The four basic morphological operators are proaches are shown. The behaviour of alternate fil- dilation, erosion, closing and opening. Thanks to ters is investigated depending on the structuring el- the fact that gray-scale images can be represented ement shape and the amount of noise in the image. as fuzzy sets, fuzzy tools can be used to define fuzzy Some different objective measures together with the morphological operators. Thus, conjuntors (usually fuzzy DI-subsethood measures ([14]) are used to continuous t-norms) and its residual implicators has evaluate the filtered results. been used. Conjunctive uninorms also recently have proved useful as a special case of conjunctors (see [5], [6], [7]). 2. Fuzzy discrete morphological operators However, gray-scale images are not represented in practice as functions of Rn into [0, 1] because they We will suppose the reader to be familiar with the are stored in finite matrices whose gray levels belong basic definitions and properties of the fuzzy discrete to a finite chain of 256 values. Therefore, the images logical operators that will be used in this work, spe- are represented as discrete functions and discrete cially those related to discrete t-norms and discrete fuzzy operators can be used. In [8] it was shown residual implicators (see [8]). From now on, the fol- that it is possible to use discrete t-norms to build lowing notation will be used: L = {0, . , n} a finite a fuzzy mathematical morphology that satisfies all chain, I will denote a discrete implicator, C a dis- © 2011. The authors - Published by Atlantis Press 358 crete conjunctor, N the only strong negation on L In addition to these properties, it is worth to safe- which is given by N(x) = n − x for all x ∈ L, T guard the duality between the discrete fuzzy mor- a discrete t-norm, IT its residual implicator, A a phological operators. Therefore, discrete t-norms gray-scale image and B a gray-level structuring el- satisfying IT = IT,N are needed. This property ement that takes values on L. Our methodology is holds for the discrete t-norms enumerated in the similar to the used ones in [3, 4, 5, 12, 13]. following result (see [8]). Definition 1 The fuzzy discrete dilation DC(A, B) Proposition 3 The identity IT = IT,N is satisfied and the fuzzy discrete erosion EI (A, B) of A by B in the following cases: are the gray-scale images defined as 1. When T is the Łukasiewicz discrete t-norm, DC(A, B)(y) = max C(B(x − y),A(x)) x TL(x, y) = max{0, x + y − n}. EI (A, B)(y) = min I(B(x − y),A(x)). 2. When T is the nilpotent minimum given by the x following expression Definition 2 The fuzzy discrete closing 0 if x + y ≤ n CC,I (A, B) and the fuzzy discrete opening TnM (x, y) = min{x, y} otherwise OC,I (A, B) of A by B are the gray-scale im- ages defined as 3. When T is an ordinal sum (with only one sum- CC I(A, B)(y) = EI (DC(A, B), −B)(y) mand) of the Łukasiewicz t-norm in a square , 2 OC,I(A, B)(y) = DC(EI (A, B), −B)(y). [a, n − a] , a ∈ L with a ≤ n − a, truncated by 0, given by the expression The reflection −B of an n-dimensional fuzzy set n B is defined as −B(x) = B(−x), for all x ∈ Z . TnMa(x, y) = 0 if x + y ≤ n Obviously a discrete t-norm is a conjunctor. if x + y > n and x + y − (n − a) Thus, these operators and their residual implicators a < x, y ≤ n − a can be used to define fuzzy discrete morphological min{x, y} otherwise operators using the previous definitions guarantee- ing the adjunction property. In [8], the discrete t- 3. Closed and open fuzzy objects norms that have to be used in order to preserve the morphological and algebraic properties that satisfy The idempotence properties of fuzzy opening and the classical morphological operators were fully de- closing when T is a discrete t-norm and IT its resid- termined. The algebraic properties of the fuzzy dis- ual implicator motivate, as in the classical mathe- crete morphological operators that will be used in matical morphology, the following definitions. the next section are recalled. Definition 4 Let A and B be two gray-scale im- • The fuzzy dilation DT is increasing in both ar- ages. It is said that A is B-closed (resp. B-open) if guments, the fuzzy erosion EIT is increasing in CT,IT (A, B) = A (resp. OT,IT (A, B) = A). the first argument and decreasing in the sec- ond one, the fuzzy closing CT,IT and the fuzzy It is important to note that due to the idempo- opening OT,IT are both increasing in the first tence of the closing and opening, the closing is B- argument. closed and the opening is B-open. Almost all the • All the usual properties (those satisfied in the results presented in this section are analogous to the morphology based on left-continuous t-norms) respective in the [0, 1]-framework and the proofs are respect to the interactions with Zadeh’s union similar and therefore, they are not included (see [12] and intersection of discrete images are also sat- and [7]). The only difference worth to mention is re- isfied. lated to the left-continuity or the right-continuity of • If B(0) = n we have: EIT (A, B) ⊆ A ⊆ the conjunctor or the implicator. These properties DT (A, B). are necessary in the [0, 1]-framework in order to en- • In addition, OT,IT (A, B) ⊆ A ⊆ CT,IT (A, B). sure that both operators preserve the infimum and • The fuzzy closing and the fuzzy opening are the supremum adequately. However, in the discrete idempotent, i.e.: CT,IT (CT,IT (A, B),B) = approach they are not necessary since we work with CT,IT (A, B) and OT,IT (OT,IT (A, B),B) = maxima and minima. First of all, each B-open and OT,IT (A, B). B-closed objects are the opening or closing of some • If B(0) = n, then EIT (A, B) ⊆ OT,IT (A, B) ⊆ image, respectively. A ⊆ CT,IT (A, B) ⊆ DT (A, B). • As in classical morphology, the difference be- Proposition 5 Let T be a discrete t-norm and IT tween the fuzzy dilation and the fuzzy erosion its residual implicator.
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