Fuzzy mathematical morphology

JOURNAL

OF VISUAL

COMMUNICATION

AND

IMAGE

REPRESENTATION

Vol. 3; No. 3, September, pp. 286-302, 1992

Fuzzy Mathematical Morphology DIVYENDU SINHA Department

of Computer Science, College of Staten Island, City University of New York, Staten Island, New York 10301

AND EDWARD R. DOUGHERTY Center for Imaging Science, Rochester Institute of Technology, Rochester, New York 14623 Received June 25, 1991; accepted November 19, 1991

The original extension of binary mathematical morphology to the gray scale is based upon the lattice-theoretic supremum and infimum operations, its geometric genesis being framed in terms of the umbra transform. Abstract formulation of the mathematical theory is set in the context of complete lattices; nonetheless, as applied to the Euclidean gray scale, it remains true to the umbra formulation. In distinction to the ordinary extension of the binary theory to the gray scale, the present paper provides a generalization based on fuzzy set theory. Images are modeled as fuzzy subsetsof the Euclidean plane or Cartesian grid, and the morphological operations are defined in terms of a fuzzy index function. This approach leads to a general algebraic paradigm for fuzzy morphological algebras. More specifically, the paper investigates in depth a fuzzy morphology grounded on a fuzzy fitting characterization. Although the resulting algebras reduce to ordinary binary morphology when sets are crisp, the extension is not equivalent to the umbra-modeled approach, and binary morphology is embedded within fuzzy morphology by treating images as (0, l}-valued rather than C-m, O}-valued. As opposed to the usual gray-scale extension, the fuzzy extension closely maintains the notion of erosion being a marker, albeit a fuzzy marker. The present paper discussesfuzzy modeling (via a suitable index function), the fundamental fuzzy morphological operations, and the corresponding fuzzy Minkowski algebra. o 1992 Academic PW, IX.

1. INTRODUCTION

Binary Mathematical Morphology, as formulated by Matheron [ 11, depends upon the intuitive notion of “fitting” a structuring element. The most fundamental operation is erosion, and erosion is defined in terms of set translations that are subsets of a given input set (which is the image under investigation). As classically treated, sets are crisp and the usual crisp subset relation is em-

ployed. Gray-scale

morphology,

as originally

formulated

by

Sternberg [2-41, extends binary morphology by treating the space beneath a gray-scale image as a binary image, 286 1047-3203192 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.

called the umbra. The umbra of an n-dimensional image is an (n + 1)-dimensional set. The customary binary operations are applied to the umbrae of both the image and the structuring element (which, in this case, is also a grayscale image), and the output image is then reconstructed by taking the “surface” of the resulting (n + l)-dimensional set. In fact, the umbra is an artifice, and gray-scale morphology can be rigorously formulated directly in terms of suprema and infima, this being the approach taken by Giardina and Dougherty [5]. As recognized by Serra [6, 71, the appropriate framework for mathematical morphology is the set of images whose gray-values are drawn from a complete lattice. This encompasses both the original Boolean formulation and the supremum-infimum gray-scale theory. Serra [8] and Matheron [9] place the theory of morphological filters into complete lattices, and subsequent algebraic developments due to Heijmans and Ronse [lo, 111 have continued within the lattice framework. Yet, from the perspective of the current paper, the key point is that all gray-scale approaches have remained in the spirit of Sternberg’s approach. Not only has the current gray-scale approach yielded fruitful applications, it has successfully extended Matheron’s binary formulations-the elements of which were derived from Hadwiger [12]. Moreover, the binary Minkowski algebra is lifted into the lattice framework and Matheron’s key representation theorems concerning increasing, translation-invariant mappings and T-openings appear without essential alteration in the latticebased theory [7-l I]. In the present study, we present an alternate form of mathematical morphology based upon the fuzzy set theory of Zadeh [13, 141. The morphological operations will be modeled on a “fuzzy” notion of fitting. Whereas morphology has been interpreted using fuzzy formalisms (for example, see Goetcherian [ 151, Sinha [ 161, and Giardina and Sinha [17]), these did not introduce an intrinsically

FUZZY

MATHEMATICAL

fuzzy mathematical morphology. In these efforts, fuzziness was introduced only in the modeling of gray-scale images, and not in the operations. Consequently, one obtained the fuzzy morphological operations by simply replacing ordinary set theoretic operations by their fuzzy counterparts. There are two salient points regarding the present study. First, it provides a different generalization of binary morphology, one that directly preserves the role of erosion as an operator marking spatial locations at which a structuring element fits within a binary image. Herein, images are modeled as fuzzy subsets of the Euclidean plane or the Cartesian grid, the notion of fitting is intrinsically fuzzy, and the result is an erosion operation that measures (by means of fuzzy membership functions) the degree to which a fuzzy structuring element fits within a fuzzy image. The result is a theory providing a framework to deal with a fuzzy foreground and fuzzy structuring element in “binary” images. A second key aspect of the paper is the manner in which generalization proceeds: to introduce fuzziness, we define an index function for subset inclusion and then define erosion in terms of the index function. It is this index function that characterizes fuzzy fitting, and it does so in such a way that if all images are crisp, then the traditional binary morphology results. More generally, since the erosion definition only involves spatial translation and the index function, other fuzzy morphological algebras can be devised by simply changing the definition of the index function. Our study takes the following course: articulate the properties an index function should satisfy to achieve the kind of propositions that are fundamental to the traditional binary Minkowski algebra; define an index function that models the type of fuzzy fitting we desire; show that the particular index function satisfies many of the properties satisfied by the traditional binary-subset index function; define the fuzzy morphological operations by means of the index function; study the resulting Minkowski algebra. l

l

l

287

MORPHOLOGY

ticular fuzzy realization we develop, it should not be surprising that the extension could have been formulated in a nonfuzzy manner. However, proceeding in a nonfuzzy manner would have missed the basic points, that of modeling fuzzy fitting and of providing an index-functionbased framework for general fuzzy extensions of mathematical morphology. 2.

PRELIMINARIES

In this section we formally define the various operations and their properties that are needed in the subsequent discussions. Our discussions of the various operations are self-contained. 2.1.

Set-Theoretic Operations

Let the universe of discourse be %. So far as this paper is concerned, % is either the Euclidean plane 97 x 9? or the Cartesian grid % x 2.. Note that 5! and PJ!denote the set of all integers and reals, respectively. The characteristic function of a crisp set A, denoted as PA: % -+ (0, l}, is defined to be 1 ifxEA PA(x)

=

(1)

1 0 otherwise.

The membership function if a fuzzy set A, denoted as PA: Q + [0, 11, is defined in such a way that pLA(x) denotes the degree to which x “belongs” to the set A. The higher the value of ~,~(x), the more x belongs to A; and conversely, the smaller the value of PA(x), the less likelihood of x being, in the set A. For further details, see Zadeh [13, 141. The union, intersection, difference, and complement operations on crisp as well as fuzzy sets can now be defined in terms of their characteristic/membership functions: pAUB(X)

=

max[pA(-h

k?(x)I,

(2)

wAd.d

=

min[pA(d

h?(d17

(3)

pA\h)

=

midpA(

1 -

=

1 -

l

l

In effect, the particular index function we employ provides a realization of the abstract definition of erosion in terms of the index function; other fuzzy morphological algebras result from different index functions, As stated previously, the one we utilize achieves our dual ends of modeling fuzzy fitting and having an erosion operation that serves as a fuzzy marker. Of course, given our formulation of erosion in terms of a particular index function, it will have to be demonstrated that there exist fitting characterizations of both erosion and opening. Because there are fitting characterizations of the par-

b+(x)

(4)

p&)1,

pA(-d.

(5)

One can also rephrase the subset relations: A

c

B =

hB

=

maxhA,

pB1

e

PA

=

min[,uA,

,‘d.

(6)

The support of a (crisp or fuzzy) set A, denoted as Y(A), is a crisp set of those elements of % which belong to A with some certainty: Y(A) = {x : /Q(X) > O}.

(7)

SINHAANDDOUGHERTY

288

When we use sets to model images, we say that an image A is afinite image iff card 9(A) < m; otherwise, the image is said to be infinite. The translation of a (crisp or fuzzy) set A by vector u E %, denoted as T(A; u), is defined in terms of vector subtraction: /-%A;&)

=

pA(x

-

@I

u).

The reflection of a (crisp or fuzzy) set A, denoted as -A, is defined as /--A(x)

=

set1

set2,

where set1 denotes the image under investigation, and set2 denotes the probe or structuring element being employed. In this framework, all sets are crisp. There are four basic morphological operations: erosion (e), dilation (@), opening @), and closing (.), of which erosion is fundamental. Given two sets A and B, erosion of image A by structuring element B is defined by

(9)

PAt-x).

operation

A 8 B = 67 T(A; x). XE-B

(13)

The scalar addition of a fuzzy set A and constant CY, denoted as A 0 (Y, is defined as Other elementary morphological operations can be expressed in terms of erosion: dilution, opening, and clospAOh) = min(l, mado, pA(X) + aI). (10) ing are given by For example, for any arbitrary set A, we have A 0 1 = Q, and A 0 (-1) = 0. It is important to contrast the definition of 0 and 9 operations. The former transforms the range of membership functions, while the later transforms the domains of the membership functions. The effect of performing A 0 (Ycan be viewed pictorially as the lifting of the entire graph of PA upward (if (Y 2 0) or downward (if (Y < 0) by amount Ial. Of course, a clipping of the values may be needed at the two extremes of the valuation space. The effect of performing T(A; (-Y)on a I-D set A can be viewed pictorially as the shifting of the entire graph of @A to the right (if CC2 0) or to the left (if ac< 0) by amount llyl. Finally, let us introduce two operations that are vital to our present study. They were proposed by Giles [I81 in the context of fuzzy set theory. The bold union of two sets X and Y, denoted as X a Y, is defined as PXAY(Z)

Similarly,

= min[l, hdz) + PLY(Z

one defines the bold intersection PXVY(Z)

(11)

X V Y as

= maxLO,kdz) + PY(Z) - 11.

(12)

Zf X and Y are crisp sets, then X a Y = X U Y and X V Y = X n Y. The bold union and intersection share many properties of the usual union and intersection operations; these are listed in Appendix A. In Appendix A we also state a few properties of translation, reflection, and scalar addition operations. These properties are needed later on. 2.2.

Binary Morphological

All the morphological infix form:

Operations

operations are of the following

A $ B = (AC 8 -B)“,

(14)

A 0 B = (A 8 B) CDB,

(15)

A*B=(A$-8)0-B.

(16)

Equation (14) represents the duality formulation of dilation in terms of erosion. An equivalent formulation of dilation is A 69 B = u T(A; x). XEB

(17)

Like erosion and dilation, opening and closing are dual of each other: A

l

B = (A” 0 B)‘.

(18)

Since the morphological operations are set theoretic, one is naturally interested in the various interrelationships between the morphological operations and the fundamental set operations, as well as interrelationships among themselves. The totality of these relations is known as the Minkowski algebra. Among the basic relations are various associativity, commutativity, and distributivity properties (see Giardina and Dougherty [5]). Basic distributivity properties are given in Appendix A. For a fixed structuring element B, erosion, dilation, opening, and closing can each be viewed as an operator 4: % + % and, in this context, certain operator properties are important. For any operator $, 4 is said to be (1) (2) (3) (4) (5)

translation invariant if 4[FT(A; u)] = T-I+(A); v]; increasing if Al C_A2 implies $(Al) C $(A*); antiextensive if $(A) C A; extensive if $(A) _> A; idempotent if $[$(A)] = $(A).

FUZZY MATHEMATICAL

Erosion, dilation, opening, and closing are both translation invariant and increasing; opening (closing) is antiextensive (extensive); and opening and closing are idempotent. According to Matheron [l], a mapping that is increasing, idempotent, and antiextensive (extensive) is called an algebraic opening (algebraic closing). If it is also translation invariant, then it is called a T-opening (T-closing). Some of the other properties of morphological operations are stated in the next section. For further details, the reader is referred to [I] or [5].

289

MORPHOLOGY

3.1.2.

An Index-Function-Based

Approach

It is helpful to rewrite the above expressions more explicitly in a fitting paradigm. To this end, let us define an index .function 9: 2Q x 2”u + (0, I} such that $(A, B) =

1 ifAcB 0 otherwise.

3.

MORPHOLOGY

AND

FITTING

The fundamental representation theorem due to Matheron [I] states that every translation-invariant, monotonically increasing set-to-set mapping can be represented by a union of erosions. See [l] or [S] for details. Thus, the manner in which we treat erosions is crucial to extending binary morphology. 3.1.

Traditional

Approach

A 0 B = {x : Y(B; x) C A}.

A CDB = {x : 5(-B;

x) n A f 0).

G3N

Like erosion and dilation, opening and closing can also be given by fitting formulations: T(B;x),

(21)

J(B:x)GA

A

l

XEA

inf 1I

(24)

&A

The last relation follows because for crisp sets the bold union has the following properties:

x E A + PEA.&) = /-Q(X) xFfA+~m&) = 1. Other similar expressions are also possible. In terms of the index function, erosion takes the form

~~ed-4 = 4(T(B; 4, A).

(25)

(19)

According to the above equation, A 6 B can be found by taking all points x such that the translate of B by vector x is a subimage of A. Corresponding to the structuring element fitting characterization of erosion is a dual property concerning dilation: A $ B is the set of all points x such that the translate of -B by vector x intersects A. Formally,

u

$(A, B) = inf P&X) = min i inf Jo, XEA

Key to both morphological image processing and the current study is the fitting characterization of erosion:

AoB=

The above equation can be rewritten so as to express the index function directly in terms of characteristic functions:

Binary Morphology

Let us first consider the notion of fitting within the original formulation of mathematical morphology. We present two equivalent views on the “fitting” characterizations of the various operations: the traditional approach (which can be found in most books and papers on mathematical morphology) and a new approach (which is suitable for the subsequent generalizations proposed in this paper). 3.1 .I.

(23)

B = {x : Vy, x E 9(B; y) + 9(B; y) II A f 0}. CT?‘,

At each point x we determine whether the translated structuring element 9(B; x) fits beneath the image A or not as mandated by Eq. (19). If it fits, then the eroded value should be 1; if it does not fit, then the eroded value should be 0. To express dilation in similar terms, we make use of Eq. (14): /-LA&X) = 1 - $(Y(-B; x), AC). (26) As is shown later (cf. Eq. (33)), we have 9ac(x, Y) = 1 - 9(X, Thus, we can write

Y) = sup j+&x). XE%.

SINHAANDDOUGHERTY

290 3.2.

Traditional

Gray-Scale Morphology

If we consider binary morphology from the perspective of binary-valued functions defined on the Euclidean plane or a Cartesian grid and having the range {-x, 0}, then, relative to fitting formulation of Eq. (19), the original set-theoretic erosion off by g can be written as t&f-e g)(x) =

sup

s(g;x)oysf

Y,

(28)

where we have followed the convention thatf(x) = 0 iff x lies in the set determined by f; that is, the background gray value is assumed to be ---co. 5(g; x) denotes the translation of domain of function g by amount x in a manner analogous to Eq. (8). Also, if h is an image, and Q is a constant, then we define the scalar addition h 0 cyin the usual way: (h 0 a>(x) = h(x) + CY. Given this representation, it is quite natural to extend erosion to the extended gray scale [-=, +a] by employing the same definition. Such an extension leaves the binary Minkowski algebra as a subalgebra within the gray-scale Minkowski algebra (see Dougherty and Giardina [ 191 or Dougherty [20] for an image-algebra interpretation of this subalgebra relation). The various gray-scale morphological operations are defined pointwise as E(f, g)(x) =

SUP Y

(2%

T(&X)OY~f

~b(f, g)(x) = ,(y;F co[SE

Y) 0 dY)lW

(30)

w, g>= Bb[%(f> gL gl

(31)

%e(f, L?) = -w-f,

(32)

-s>.

Almost all the properties of binary morphology have a counterpart in this framework. For example, erosion is still dual of dilation, and the fitting characterization of opening is

fu-, g)cd = T,gyip -f [T(g; , a<

Y)

0 ~IW.

In addition to the natural algebraic extension, there is also a fundamental transform, the umbra transform, which maps gray-scale relations into corresponding binary relations if the generalizations of Eq. (28) are employed. It is the umbra that plays the central role in the developments by Sternberg [2-41 and Haralick et al. [211. Although the umbra is employed in [5], the gray-scale theory of Giardina and Dougherty is based on Eq. (28),

and not the umbra. Although the lattice-based theory of Serra takes a more axiomatic approach, it too ultimately reaches a counterpart of Eq. (28). 3.3.

A Different Approach to Fittings

Is there a second fruitful extension of the binary theory, one that is not grounded on Eq. (28)? Indeed, there is a distinct approach, one that preserves the notion of “marking a fit” as the ground for erosion. In the new approach fitting is intrinsically fuzzy. In this subsection, we make use of the notion of an index function that was introduced earlier. Suppose that the images are fuzzy and we wish to directly generalize the notion of fitting. Specifically, at each point x we wish simply to determine whether the translated structuring element T(g; x) fits beneath the image f or not (cf. Eq. (19)). (In effect, we wish to determine whether or not the graph of !7(g; x) fits between the graph of f and the plane.) If it fits, then the eroded value should be 1; if it does not fit, then the eroded value should be 0. This would directly extend the binary erosion as fitting, and would yield a fuzzy-to-binary image transformation. This is essentially the approach taken in Goetcherian [I51 and Sinha [16]. Rather then proceeding in a strictly fuzzy-to-binary fashion according to Eq. (19), we can recognize that a fuzzy structuring element that does not fit fully beneath an image might fit “more or less.” For instance, a structuring element that has a constant value $ on its domain fits under a constant g-valued image “less” than the structuring element that has a constant value 1 on its domain, although both structuring elements clearly fit beneath an image having constant gray value of 1. To model this notion of fuzzy fitting, we need an erosion operation that outputs a 1 for a crisp fitting, and less than 1 for a partial fit: The closer the value to 1, the higher the degree of fit. To proceed, we must generalize the notion of an index function (Eq. (23)) so that 9(A, B) gives the degree to which A is a subset of B. The formulation given by Eq. (24) suffices for this purpose, and our notion of fuzzy fitting can be captured by Eq. (25). Equations (24) and (25) form the basis of a fuzzy-set-theory-based mathematical morphology. The result is an extension of the binary theory yielding fuzzy-to-fuzzy image mappings. As opposed to Eq. (28) which treats binary images as being {-m, 0}-valued, the above scheme treats binary images as being (0, 1}-valued. A very recent paper by Koskinen et al. [221 has attempted to define morphological operations that are less sensitive to (additive) noise and small variations in the shape by incorporating other order statistics besides maximum and minimum. Though this paper shares some of the goals with the current paper, it did not place much

FUZZY

MATHEMATICAL

emphasis on preserving the Minkowski algebra. In any case, the resulting opening and closing are neither idempotent nor anti-extensive and extensive, respectively.

291

MORPHOLOGY

More formally, 4(B U C, A) = min[JYB, A), %C, AlI. (This property is used in the binary morphology to show that erosion “distributes” over union.) The index function should preserve the following property: If A is a subset of B and C, then A is also a subset of B n C. The converse also holds: l

4.

AN INDEX

FOR SET INCLUSION

Since the index function is crucial to the present developments, we begin our investigation with a detailed technical study of this function. The properties of morphological operations will be induced by the properties of the index function. So, let us begin by proposing the properties that the generalized function 9 must satisfy. It is important to keep in mind that in our application we will be looking at ,$(A, B) for different values of set B, and the set A will be essentially fixed. Consider any two fuzzy subsets A, B C %. We ignore the possibility that A = 0. 4.1.

Properties That Binary Morphology Index Function

Requires of the

After surveying the proofs of the various results of binary mathematical morphology, we have found certain properties of the index function that are essential for these proofs to work. For the proposed generalization, we choose to include these properties as necessary criteria for the selection of a generalized index for set inclusion. In any case, we postulate the following “reasonable” properties for the index function 4: $(A, B) E [O, I]. If A and B are crisp sets, then $(A, B) E (0, l}. A c B w $(A, B) = 1. If B c C, then $(A, B) I $(A, C). (This property is used in the binary morphology to show that dilation and erosion are increasing operations .) If B C C, then sl(C, A) % Si(B, A). 4 should preserve the translation-invariant nature of subset relations: A C B e T(A; T) C T(B; T); that is, we must have $(A, B) = Si[T(A; 7), T(B; T)]. (This property is used in the binary morphology to show that dilation and erosion are translation invariant.) si should preserve the identity A C B G Bc c AC; that is, we must have $(A, B) = .9(BC, A”). (This property is used in the binary morphology to establish duality.) S should be invariant under reflection: $(A, B) = 41(-A, -B). (This property is used in the binary morphology to establish duality.) The index function should preserve the following relationship: If B and C are subsets of A, then so is B u C. The converse also holds: l l l l

(A c B) A (A C C) e A c (B f’ C). More formally, $(A, B f’ C) = min[4i(A, B), $(A, C)]. (This property is used in the binary morphology to show that erosion distributes over intersection.) Alternately stated, we require that the relationship

be preserved, that is, .!?“(A, B n C) = max[P(A, B), P(A, C)]. (This property is used in the binary morphology to show that dilation distributes over union.) 4.2.

Proposed Index Function and Its Properties

As stated in the Introduction, one of our goals is to provide a framework for extensions of binary mathematical morphology relative to generalized index functions. The vehicle for this extension is Eq. (25), which defines traditional binary morphology in terms of crisp-set inclusion by means of the index function of Eq. (23). Subsequently, we define fuzzy erosion by means of Eq. (25), namely, by 4[T(B; x), Al; however, this is a generalized formulation, the specific meaning of which lies in the defining index function 9. As mentioned in the last section, herein we choose to define the index function (see Eq. (24)) by

l l

l

This definition will eventually yield the desired fuzzy extension of binary morphology, Even though the above index function has been mentioned in Dubois and Prade [23], we have not been able to find a reference where its properties have been formally studied.

l

l

(B c A) A (C L A) e (B U C) c A.

EXAMPLE 4.1. Let us consider iliO.1, 211.0, 310.71, B = (ljO.5, {2lO.9,3lO.21,at~m ={1~0.6,2~1.0, % = (1, 2, 3). We have

B’AA

the fuzzy sets A = 2/O.& 3/0.5}, c = 310.8}.Assumethat

= (110.5, 210.4, 310.5) A {l/0.1, = (110.6, 211.0, 3/1.0},

211.0, 310.7)

292

SINHA

C”AA

FAA

= (ljl.0,

210.1, 3~0.8}A{l~O.l,

= (ljl.0,

211.0, 3/1.0},

= (110.4, 210.0, 3~0.2}A{l~O.l,

AND

211.0, 310.7)

DOUGHERTY

set-theoretic characterization of this complement? The answer is yes, and it can be expressed in terms of the dual of bold union (that is, the bold intersection operation):

211.0, 310.7)

$“(A, B) = 1 - $(A, B)

= (1~0.5, 211.0, 310.9).

= 1 - ,‘i; p/p&x)

Therefore, 9(B, A) = 0.6, 9(C, A) = 1.0, and41(D, A) = 0.5.

= 1+ su &

-~AWb)

The next example illustrates a property of the index function that has important consequences for the fuzzy morphological operations.

=

:zg

1 -

~AcA&)

=

“,F$

&A%E,.(x)*

EXAMPLE 4.2. Let us consider two fuzzy subsets A = {1~0.8,2~0.9}andB={1~0.8,2/1.0}ofth e set q = (1, 2). We have

Note that we have made use of Properties A. l(8), and the following relations:

A.1(7) and

x + syp aj = svp (x + aJ A” A 0 = (110.2, 210.1) A {l/0.0,

210.0) = (110.2, 210.1)

syp (-ai)

B’ A 0 = (110.2, 210.0) A (110.0, 210.0) = (110.2, 210.0).

Thus, $(A, 0) = 0.1 and 4(B, 0) = 0. Note that $(A, 0) > 0 even though A g 0. Let us mention that this phenomenon arises only when x~$~J

PA(x)

<

1.

This phenomenon can be avoided; however, avoidance leads to rather nonintuitive and much more involved discussions and derivations. In any case, most of the similar indices that have been introduced in the literature (see Dubois and Prade [23]) have this as well as many other similar idiosyncrasies. Let us now formally establish the properties of the index function. The proofs of various results in this section are stated in Appendix B. THEOREM 4.1.

(Basic Properties of Index Function). For any two fuzzy sets A and B, we have 1. 2. 3. 4. 5. 6. 7. 8.

$(A, B) E [0, 11. Zf A and B are crisp sets, then $(A, B) E (0, l}.

A c B (Ij $(A, B) = 1. Zf B C C, then $(A, B) ‘= $(A, C). IfB C C, then $(C, A) 5 4(B, A). $(A, B) = .9(-A, -B). $(A, B) = Si(B”, A”). min[l, (Y + /3] 2 $(A, B) B p, where p = 1 -

Therefore,

= -iTf ai.

we obtain the identity

$“(A, B) = 1 - $(A, B) = Wl

PA&X).

(33)

Proceeding along the lines of Theorem 4.1, we can show that the function $c checks for nonempty differences: For crisp sets it is well known that AcB@9’c(A,B)=

lGA\B#OGAABc#O. (34)

THEOREM 4.2 (Properties of Index Function). three fuzzy sets A, B, and C, we have

For any

1. Zf7 E (Il., then $(A, B) = 4i[T(A; T), 5(B; T)]. 2. $(B U C, A) = min[.9(B, A), 9(C, A)]. 3. .9jc(A, B n C) = max[Jjc(A, B),.Y(A, C)]. 4. $(A, B n C) = min[$(A, B), $(A, C)]. To end this subsection, let us note that the properties of the index function stated in the previous subsection do not guarantee a unique realization of the index function. Choosing a different realization will at least yield different fitting characterizations of various fuzzy morphological operations (these are discussed in Section 6). 4.3.

Fitting Characterization

of the Index Function

Having established the basic algebraic properties of the selected index function, we now turn to the manner in which it (as possibly opposed to others) possesses a fitting characterization. Given our desire to model fuzzy suP@U~A(x> and CY = WXE~/-&>. fitting, such a characterization is crucial. We give two results in this direction. In eq. (26), we encountered a term of the form 1 To obtain an intuitive feel for the following results, it is 41(* * e), which represents the complement of 4. Is there a

FUZZY

MATHEMATICAL

helpful to recall the “pictorial” definition of the scalar addition operation (cf. discussions immediately following the definition (10)). When A C B, the graph of ,.,&Afits beneath the graph of PE and we have $(A, B) = 1. When A p B, we must “lower down” the graph of j+, (this will be accomplished via the scalar addition operation) in order to make it fit beneath the graph of pg. The results of this subsection (especially Corollary 4.1) essentially state that the value of 1 - $(A, B) is the smallest amount by which the graph of PA must be lowered down so that it fits beneath the graph of pB. THEOREM 4.3. If $(A, B) = r, then r is the largest number for which the following holds for every x E (3.:

or alternately, /-b(x) + PA4X) 2 7. Let us mention yet another restatement of the above result. This also shows a strong connection between the bold union and scalar addition operations that was hitherto unknown, COROLLARY 4.1. If $(A, B) = r, then r is the largest number in [0, l] for which the relation [A 0 (r - l)] c B holds. Equivalently stated, if K is the smallest number in [0, l] for which the relation [A 0 (-K)] C B holds, then $(A, B) = 1 - K. All subsequent fuzzy morphological fitting characterizations follow from the next theorem. Thus, in a technical sense, it is a key proposition in the paper.

293

MORPHOLOGY

higher the value, the more that pixel belongs to the image and vice versa. These comments do not differ from the traditional way in which researchers have employed fuzzy set theory in image processing. To arrive at a fuzzy version of the various imaging operations, researchers have essentially replaced the ordinary set operations by the fuzzy set operations . We wish to introduce fuzziness in the operations in more subtle ways. The fuzziness is introduced by considering the degree to which a structuring element fits inside the given image. the index function 9, plays a crucial role. Our aim is to arrive at a framework possessing a fuzzy Minkowski algebra in which the key properties of the binary morphological operations are preserved. We begin with the generalization of the erosion operation and then use Eq. (14) et seq. to define the other operations. The actual definition of erosion does not mention bold union, but is stated merely in terms of translation (Y) and index function (9). Of course, since we have a particular realization for 9,) it is this realization that will be operationally employed. In what follows, we use different symbols to denote the fuzzy operations. Furthermore, we use the prefix form to denote them: operator(set1,

where set1 and set2 denote the image and structuring element, respectively. DEFINITION 5.1. The erosion of a set A by another set B, denoted as %(A, B), is defined by h(A,B)b)

THEOREM 4.4 (Fitting Function).

Characterization

=

x),

A).

(36)

DEFINITION 5.2. The dilation of a set A by another set B, denoted as %(A, B), is dejined by (35) /-b(A,B)(X)

where a! 2 -supZEQ ~~(2). FUZZY

%T(B;

of the Index

Wb B) = rAygE1 + min[O,al,

5.

set2),

Alternatively, MORPHOLOGY

We assume that images are fuzzy subsets of the Cartesian grid % x % or the Euclidean plane S? x ?&. Note that % and 9? denote the set of all integers and reals, respectively. In the examples we always consider images to be defined on the Cartesian grid. Consider any image A and a point (pixel) (i, j). The Cpntity ,-&A(i, j) iS the VZdUe of image A at pixel (i, j}. On the ordinal scale [0, 11, 0 represents “white” or background and 1 represents “black” or foreground. The usual fuzzy set-theoretic interpretation is also valid: the

/-%(A.B)(x)

=

1 -

=

/%(Ac,-L&X)*

(37)

we have (cf. Eqs. (12) and (33)) ,%(Ac,-B)(x)

= 4”(3(-B;

x), A”) (38)

Before defining the opening and closing, let us illustrate the dilation and erosion operations. The following example also illustrates the fuzzy notion of “soft fitting”: Although the structuring element does not actually fit, the eroded value gives the degree to which it fits.

SINHA AND DOUGHERTY

294 EXAMPLE 5.1. Consider the following (given in bound matrix notation’): 0.2

1

two images

0 .8 0 . 1 O.O \

I A = \ 0.3

0.9

0.9

0.2 I

0.1

0.9

1.0

0.3 @,O)

and

B = (0.8

0.9)$yo,

Since the structuring element Jts entirely under the image when it is translated by (I, - 1) (that is, Y(B; 1, - 1) C A), we have %(A, B)(l, - 1) = 1.0. Similarly, we can also show that %(A, B)(l, -2) = 1.0. Next, consider the vector (1,O). Even though T(B; 1,O) c A, the subset relationship almost holds: A n Y(B; 1, 0) = (0.8

0.8)(,,,, = T(B; 1, 0).

Therefore, we expect a relatively high value of %(A, B)(l) 0). Indeed, 4[5(B;

1, O), A] = min{min[l, min[l,

By appropriate

thresholding,

1 - s;ps, /+(x)

= 1 - 0.9 = 0.1

Let us formally establish the background erty of the erosion operation. PROPERTY 5.1 (Background

color prop-

Color of Eroded Image).

For have any two images A and B and any vector x E Q, we

minC1,7 + 4 2 MW&) 2-7, where r = 1 - supigq &z)

and (Y = supzEou am.

Proof. It follows immediately erosion and Theorem 4.1(8). s

from the definition

of

1 + 0.21,

0.8 + 0.11) = 0.9.

Thus, %(A, B)(l, 0) = 0.9. Proceeding obtain an injinite eroded image

Erosion of a jinite image has resulted in an infinite image. This is because the background has been painted by the value 0.1. This value for buckground color is not arbitrary:

along this line, we

we can obtain the image

According to Property 5.1, when we erode an image with a structuring element which may not be a crisp set, then the background of the eroded image does not necessarily have the value 0: the background’s value is 7. However, this phenomenon does not exist if the structuring element has value of 1 .O at one or more pixels. This phenomenon does not exist in binary morphology as we are then dealing with crisp structuring elements. It is one of the characteristics of fuzziness, and is related to the fact that we can have $(A, 0) > 0 even though A f 0. In binary morphology, if (0, 0) E B, then A 8 B c A. Analogous to this result, in the fuzzy morphology we have ,d(O,

\l.dc1.0:

0)) = 1 3 %(A, B) C A.

(39)

In the binary case, for A 8 B = A, A finite, we must have in the fuzzy case, the structuring element need not be a singleton set. For notational simplicity, let us assume that the structuring element B is a 3 x 3 mask centered at origin and that ~~((0, 0)) = 1. Let y be the largest difference between the value of a pixel in A and any one its eight neighbors, that is,

B = ((0, O)}. However, Intuitively speaking, we have a noisy image of a 3 X 2 box which we are eroding with a 2-unit-long horizontal straight line. If this were binary morphology, we would expect to obtain a vertical straight line of height 3 units. This is essentially what we have obtained.

VX E Q VY E ’ In this representation, we enclose the area of interest in a given image by a (smallest) rectangle. The membership values within this rectangular region are specified in a matrix format. The membership values outside this region are assumed to be fixed and this value is specified as a superscript of the matrix. To complete the specification, we state the coordinates of the topmost-leftmost element of the matrix as a subscript. For example, in the matrix A the unspecified membership values are 0, and the coordinates of the topmost-leftmost non-0 element in A is (0, 0), that is, ~~((0, 0)) = 0.2.

Jvx

I/-k&)

-

k&)l

~2 Y>

where sir, denotes the 3 by 3 neighborhood around point x. Now, if the smallest difference between the value of pixel (0, 0) in B and any one its eight neighbors is at least y, that is, v x E N(O,O)

IPBm

0)) - PB(X)I

2 Y,

FUZZY

MATHEMATICAL

then %(A, B) = A. This result has practical significance as it points out one of the criteria for the selection of values for the mask. Let us now illustrate the dilation operation. It is apparent that computing the dilation of two finite images via expression (37) implies that we must deal with infinite images. Expression (38), on the other hand, leads to a more practical algorithm.

EXAMPLE5.2.

Consider the following

two images

We choose to employ expression (38). From this expression it is clear that P(%(-B;

x), A”) > 0 3 Y(A) n Y[Y(-B;

x)] # 0.

295

MORPHOLOGY

Proof. (37). n

It follows directly from Property 5.1 and Eq.

According to Property 5.2, when we dilate an image with a structuring element which is not a crisp set, then the highest value of the dilated image is not necessarily 1: the highest value is K. In certain situations (when K + CY 2 I), the background may also be painted with the constant value K + CY2 1, thereby making the image infinite. However, this phenomenon does not exist if the structuring element has value of 1.0 at one or more pixels. Hence, this phenomenon does not exist in binary morphology. Let us now illustrate some consequences of Property 5.2 with a few examples.

EXAMPLE5.3.

Let us dejine two images

A=

and

B = (0.9

0.6)&$,.

In any case, we obtain We have ‘0.5

if x = (0, 0)

0.6 9’(!T(-B;

if x = (1, 0), (0, -2)

x), A”) = { 0.7

if x = (0, -l),

0.8

ifx = (1, -1)

(0.0

%(A, B) =

(1, -2) Here, (Y = 0.1,

otherwise

i !

EXAMPLE5.4.

0 . 5 0 .6 ‘.’

Let us define two images

.

0.7

0.8

0.6

0.7 (o,o)

Intuitively speaking, we have a noisy vertical height 3 which is being dilated by a 2-unit-long horizontal line. As is the case with crisp sets, here obtain a 3 x 2 rectangular image. Finally, note that the highest value in the dilated is 0.8, which is smaller than the highest value structuring element B. Analogous to Property result for dilation:

color is 0.

The following example shows that dilation of an infinite image need not be finite.

Hence,

Eb(A, B) =

= 0.9. Thus, the background

K

and line of noisy too we

PROPERTY 5.2 (Background

Color of Dilated Image). For any two images A and B, we have

where K = supZE%pB(z) and CY= inf,,qpcL,(z). Zf K + CY5 1, then %(A, B) is ajnite image. In particular, if A isfinite (so that CY= 0), then so is %(A, B).

0.6)&.

We have

image in the

5.1, we have the following

B = (0.9

!SJ(A, B) = Here, CY= 0.2, 0.01.

K

= 0.9. Thus, the background

color is

Finally, let us define the other two morphological operations: opening and closing. We make use of Eqs. (15) and (16) to define these operations:

fJYA> BQ= 9B(A, Z-3,Bl

(40)

%(A, B3 = %[%(A, - B), -B].

(41)

Let us now illustrate fuzzy opening.

296

SINHA

AND

EXAMPLE 5.5. Let us consider the two images of Example 5.1 again. We had

DOUGHERTY

Proof.

5.2.

It follows immediately

from Properties 5.1 and

n

6. FUZZY MINKOWSKI ALGEBRA

P = %(A, B) =

. In this section we formally study various algebraic properties of the fuzzy morphological operations. Note that Theorems 6.1 through 6.4, which are strictly algebraic and do not involve fitting, derive directly from abstract algebraic properties of the index function, and not from particularities of its definition in terms of bold union (insofar as those particularities concern fitting).

Therefore,

6.1.

Hence,

Erosion and Dilation

We first show that the dilation operation tive. B(A, B) = %(Pc, -B)C =

0.3

0.8

0.9

0.2 !

.

0.1

0.8

0.9

0.3 (-1.0)

THEOREM 6.1 (Commutativity of Dilation). For any two nonempty images A and B, %(A, B) = S(B, A). Proof.

Upon appropriate

thresholding,

is commuta-

All we need to establish is that

we can obtain the image Si”(S(-B,

x), A”) = 9c(3(-A;

x), Bc).

To this end, we make repeated use of Theorems 4.1 and

4.2. Zn the binary case, if we open a 3 X 2 box with a horizontal line of width 2, we get the same box back. This is essentially what we have obtained here. It is also of interest to note that, like the binary case, we have 6(A, B) c A. It is of interest to contrast the output of the fuzzy opening operation with that of the traditional gray-scale opening, which yields the image

0.2 0.7 0.8 0.3 0.8 0.9 0.2 \O.l

0.9

1.0

.

0.3/c-l,oj

Analogous to Properties 5.1 and 5.2, we have the following results for opening and closing: PROPERTY5.3.

For any two images A and B, we have

$[Y(-B. x) A”] = $[A, &B; x)“]

(cf. Theorem 4-l(7))

= $[A, 3(-B”;

(cf. Properties A.2(3) and A.4(5))

= 41[S(A; -x), - B”]

(cf. Theorem 4.2(l))

= $[-T(A;

-x), B”]

(cf. Theorem 4.1(6))

= .!+[Y(-A;

x), B”]

(cf. Property A.2(1)).

The next two theorems creasing r-mappings. ”

5.2.

It follows immediately

from Properties 5.1 and

n

enable us to talk about “in-

THEOREM 6.2 (Increasing). Consider any three images A, B, and C such that A C B. Then a(A, C) C B(B, C) and %(A, C) c %(B, C). Proof. Since Bc c AC, from Theorem 4.1(4) we obtain the inequality VT E Q,

Proof.

x)]

9[3-(--C;

T), BCl 5 9[Y(-C;

T), A"1

from which we may conclude that (cf. Eq. (38))

n v?-

E

q11,

/-%A,C)(d

s

k%B,dd.

PROPERTY5.4. For any two images A and B, we have 1 z

~%(A,&)

z

1 -

SUP ZPU

!-dd.

Consequently, %(A, C) c S(B, C). Similarly, one can show the result for erosion.

n

FUZZYMATHEMATICALMORPHOLOGY THEOREM 6.3 (Translation Invariance). Consider any two images A and B, and a vector r E W Then 9[Y(A;

T), B] = WS(A,

Proof.

We only establish that erosion is invariant under translation. To this end, we have, for any x E %, Proof.

=

&(A,&

= $[Y(B;

-

7)

x - T), A]

= .9[Y(B; x), 3(A; G-)] (cf. Theorem 4.2(l)) =

/%[Y(A;&B](X).

’

There is a crucial difference between Theorem 6.3 and ordinary gray-scale translation invariance, which would in our setting take the form %[T(A;

7) 0 (Y, B[ = T[%(A,

B); 71 0 (Y.

In general, the preceding equality does not hold in fuzzy morphology. THEOREM 6.4 (Distributive Laws). three images A, B, and C. Then

S[A,

Consider

B U C] = S[B U C, A] = %(A, B) U %(A, C),

%[A n B, C] = %(A, C) n %(B, C). Proof. The proofs of these results follow immediately from Theorem 4.2. For example, =

/-%[B”C,A](d

(cf.

Theorem

= 9”(9[-A;

xl, (B U Cy)

= .9”(S[-A;

x], BC fl Cc)

= max[$c(T[-A; =

maXh-G(A,B)(~)~

=

/%(A,B)uwA&)

It follows directly from Theorem 4.4.

/-%(A,&)

6.1)

=

1

%(A>

(43)

BKd,

where % denotes the traditional gray-scale erosion and A denotes the image that is obtained from A by clipping it at the maximum gray value in B (call it p): if PA(z)

P‘?(Z) = p 1 PA(z)

k

p

otherwise.

Not only is this nonffuzzy formulation specific to the particular bold union based extension, it reflects neither the key role of the index function as the basis of extension nor the goal of modeling fuzzy fitting. Regarding the general notion of an index function, whereas Theorems 6.1 through 6.4 employ only abstract algebraic properties of sl, Theorem 6.5 (and hence the specific relation between fuzzy and traditional gray-scale morphology as given by Eq. (43)) depends upon the fitting characterization of 9, which in turn derives from the bold union realization. It is also of interest to contrast the following result with that of the definition of gray-scale dilation (cf. Eq. (30)). THEOREM 6.6 (Fitting

xl, BC), 4C(3[-A; xl, Cc)] (cf. Theorem 4.2(3))

1 +

PFo,(A,B)(X) =

Characterization inf

of Dilation).

1 +min[O, (~1,

T[(-B)Oa;x]cAC

(44)

!-%(A,&]

where /3 = supZEq pB(z) and (Y 2 -/3.

The other two results similarly follow.

Characterization

q

Proof. It follows directly from Eq. (37), Theorem 6.5, and the identities

n

We end this subsection with another structuring element fitting characterization of eroded images. It is interesting to note the similarity of the following result on erosion with that of the traditional definition of gray-scale erosion (cf. Eq. (29)). THEOREM 6.5 (Fitting

(42)

Let (0, 0) E Y(B). (Recall that Y(B) denotes the support of fuzzy set B.) Then, crudely speaking, to find the erosion of A by B at point x, we shift B so that it is centered at X, find the maximum value of (Ythat will leave 3(B; X) 0 ac“beneath” image A, and then compute 1 + min[O, a]. (Note that 3[B; x] 0 (Y = Y[(B) 0 (Y; xl). An immediate consequence of Theorem 6.5 (and the one that was alluded to in the Introduction) is that our specific fuzzy extension of binary morphology can be framed without reference to bold union or fuzzy fitting, Indeed, --

any

%[A, B U C] = %(A, B) II %(A, C).

/-h(A,B”C)(d

SUP 1 + min[O, (.w], P%(A,E&) = S(BOa;x)cA where Q! I -suP,~%~B(z).

B); 71 and

%[FF(A; Q-),B] = !Y[%(A, B); ~-1.

kT[CR(A,B);&)

297

of Erosion).

max[p, min[p,

ql = -min[-p,

q] + r = min[p

6.2

-q]

and

+ r, q + r].

W

Opening and Closing

In this subsection, we formally of opening and closing.

establish the properties

298

SINHA

THEOREM 6.7 (Increasing).

(1)

W‘L

(2)

%(A, C> C WB, Cl.

Cl L W,

AND

DOUGHERTY

Zf A c B, then

0,

Proof. We only establish (1). Since erosion and dilation are increasing operations (cf. Theorem 6.2), we have the following sequence of implications:

= sup max[O, ~g(-~:Jz) ZGQ = sup max[O,

zPU =

Hence, the result.

n

41

+ min(O,

sup

max[O, ~T(-B&)

+ min(0, cr>l

TU3Oa:z)CA

sup

max[O, p~(-~;~)(z) + n-MO, ~11

sup

maxP, PY(B&)

FT(BOu;&A

THEOREM 6.8 (Translation T(B; T)] = 6(A, B),

Invariance).

(1)

6[A,

zz

+ minP-4 ~11.

(47)

TT(BOLU;Z)~A

(2)

QW-(A; ~1,Bl = W’(A, B); ~1,

(3) (4)

%[A, T@; 711= %(A,B),

Consider any a and z such that !T(B 0 a; z) c A. It can be easily shown that

%]%(A; T), B] = S-[%(A, B); ~1.

Proof. These properties follow from the translation invariance of erosion and dilation operations (cf. Theorem 6.3). n The following result states that opening and closing operations are dual of each other. THEOREM 6.9 (Duality

I + min(O, or)]

T(BOor;z)CA

= sup

sb[z(A,Cl, Cl C 9[WB, C), Cl.

sup T(BOcu;z)CA

SUP M-BJZ)

ZPU

A C B 3 %(A, C> C z(B, Cl Zs

- 1 +

Principle).

B(A, B) = %(Ac,

a 2 0 3

max[O, p 9(B;&

+

a < 03

max[O, PT(B&)

+ min(0, a>1 =

Minsk

a>]

=

p~(B00;&) ~.LT(~o~;~)(x>.

Equation (46) now follows directly from Eq. (47). According

q

to Theorem 6.10,

B)C. Proof.

O(A, B) C

We have

%(A”, B)C = %[%(A’,

-B),

= ‘3[G3(Ac, -B),

-B]’ B]

= 9b%(A,B), Bl = B(A, B).

(cf. Eq. (37))

sup zPU

n

6(A, B) =

Characterization

U

T(BOa:i)CA

T[BOa;

of Opening). z],

=

zl,

(48)

%(A, B) =

where 1y 2 -sup,,% p*(p). Note that in Eqs. (45) and (46), the supremum is over all values of z and (Y that satisfy the constraints T(B 0 (Y; z) C_A and (Y5 0. Proof. By making use of Eq. (38) and Theorem 6.5, we obtain

Characterization

of Closing).

F3-[Boo; z]“,

n T(BOa;z)CAC

(49)

and asO

or in terms of membership functions,

(45)

sup

CY5 0.

sup

J(BOa:z)iA

THEOREM 6.11 (Fitting

/-%(A,B,(~)

=

inf TT(BOa;z)GAC rind a50

md a50

or in terms of membership functions, /-%A,B)(x)

TWa;

the later union giving B(A, B) in binary morphology. In fuzzy morphology, equality holds in Eq. (48) if and only if

(cf. Eq. (37))

Like the binary case, here too we have a structuring element fitting characterization of opening. However, the form is slightly different compared to the binary case (cf. Eq. (21)). THEOREM 6.10 (Fitting

u

P 9[BOa;7] (x)3

(50)

where CY2 -sup,,% pB( p). Again note that the injmum is over all values of z and (Y that satisfy the constraints T(B 0 CY;z) C A’ and QI 5 0. Proof. 6.10. n

It follows immediately

THEOREM

6.12

have (1) (2)

fYA, B) C A, %(A, 9 2 A.

from Theorems 6.9 and

(Antiextensivity/Extensivity).

We

FUZZY

Proof. 6.11. n THEOREM

It follows

directly

from Theorems

6.13 (Idempotence).

MATHEMATICAL

6.10 and

We have

299

MORPHOLOGY

mappings and T-openings applies, the development of a fuzzy granulometric theory, and the investigation of other morphological algebras resulting from relaxing the constraints on the index function.

(1) fJFX% B), Bl = fWL B), (2)

%[%(A,B), Bl = %(A,B).

APPENDIX

Proof. We only establish part (1); part (2) follows by duality (cf. Theorem 6.9). If 5[B 0 a; z] is an element of the union forming 6(A, B), then it is also in the union forming 6[6(A, B), B], so that B(A, B) L 6(A, B), Bl. Idempotence follows from the anti-extensive property of opening (cf. Theorem 6.12). n Since f%L

B)(x)

sup zE%

Thus, by idempotence, tion B(A, B) =

5

SUP /-dp)v

pEC

sup

CY5 0.

~i(BOa;:)CC(A,B)

we have the recursive formula-

U

T[BOa;

z].

(51)

b(BOa;;)CC(A,B)

This is completely analogous to the traditional gray-scale morphology, since there is no constraint CY5 0 governing the union, as there is in the fitting characterization of Eq. (45). 7.

CONCLUSION

By employing fuzzy set modeling for images, a mathematical morphology results which is different from the one typically studied. Extension of the binary theory into a fuzzy theory is accomplished by generalizing the crispsubset index function, and the specific resulting fuzzy theory depends upon the particular form of the generalization. A major objective of the present paper has been to model fuzzy fitting, and in line with this objective a fundamental aspect of the specifically developed fuzzy theory is a fuzzy fitting paradigm that preserves the original marker notion inherent in binary erosion. Consequently, rather than being forced to make a binary decision as to whether a structuring element does or does not fit, a fuzzy decision can be made. Each of the elementary erosion, dilation, opening, and closing operations possesses a fuzzy counterpart, as do the basic algebraic properties of increasing monotonicity, translation invariance, anti-extensivity, and idempotence. Relative to these and the customary fuzzy set algebra, there is a fuzzy extension of the Minkowski algebra. Future work will center on the manner in which the Matheron filter theory pertaining to increasing, translation-invariant

A

Properties of Set-Theoretic Operations In this appendix we state the properties of various settheoretic operations that we defined in Section 2. We also mention a few properties of the binary morphological operations. Property A.1. The bold union and intersection operations satisfy the following properties: (1) A A (B A C) = (A A B) A C, (2) A A 0 = A, (3) A A Q = %, (4) A V (B V C) = (A V B) V C, (5) A V 0 = 0, (6) A V (3 = A, (7) (A A B)C = A” V BC, (8) (A V B)C = A” A BC, (9)AcBeAcAB=%, (IO) A c B 3 A A C c B A C. However, the absorption, idempotence, and distributive properties do not hold. For a proof of the above results and for further details, see Giles [IS] or Dubois and Prade [23]. Property A.2. Translation distributes ouer reflection, scalar addition, and all set operations: (1) -5(A; 7) = Y(-A; T) (2) 5(A; G-)0 (Y = %(A 0 c-w;Q-) (3) T(A; Q-)~= T(Ac; 7) (4) T(A U B; 7) = Y(A; 7) U T(B; 7) (5) T(A fI B; 7) = %(A; 7) f’~ 5(B; 7) (6) Y(A A B; 7) = ?7(A; 7) A %(B; 7) (7) Y(A V B; 7) = 5(A; 7) V %(B; 7). Property A.3. Scalar addition distributes tion, and all set operations: (1) (-A) 0 7 = -(A 0 7) (2) (A U B) 0 r = (A 0 7) U (B 0 7) (3) (A n B) 0 T = (A 0 Q-)n (B 0 7) (4) (A A B) 0 7 = (A 0 7) A (B 0 Q-) (5) (A V B) 0 Q-= (A 0 7) V (B 0 7) (6) A” 0 7 = [A 0 (--7)lc Property ations: (1) -(A (2) -(A (3) -(A (4) -(A (5) (-AC

A.4.

Reflection distributes

U B) = (-A)

U (-B) n B) = (-A) /I (-B) A B) = (-A) A (-B) V B) = (-A) V (-B) = -AC.

over rejlec-

over all set oper-

300

SINHA

Finally, let us state the various distributive of the binary morphological operations: Property AS. We (1) A @ (B U C) = (2) (A U B) @3C = (3) A @ (B f7 C) C (4) (A f- B) G3 C 5 (5) A 9 (B U C) = (6) (A n B) 6 C =

have (A G3B) (A 43 C) (A G3B) (A @ C) (A 8 B) (A 0 C)

APPENDIX

AND

DOUGHERTY

properties

=

fF,f

PACAB

= $(A, B). U U n I-I f-I n

(A (B (A (B (A (B

$ C), 69 C), EE C), 83 C), 0 C), 8 C).

2. We have 9i[B U C, A] = It:,f P(BUC)dX)

B

Proofs of Displayed Results in Section 4 In this appendix, we provide the proofs of various displayed results of Section 4. Proof of Theorem 4.1. All the statements follow from the properties of bold union: 1. Bold union of two fuzzy sets is a fuzzy set. 2. If A and B are crisp sets, then A A B = A U B. The result now follows from the definition of union. 3. From the definition it is clear that 9(A, B) = 1 iff AC A B = % iff A c B (cf. Property A.1(9)). 4. IfB c C, then AC A B c AC A C (cf. Property A.1(7)). The result now follows from eq. (6). 5. If B c C, then Cc c BC. Consequently, Cc A A c BC A A (cf. Property A.l(lO)). The result now follows from Eq. (6). 6. Follows immediately from the definitions. 7. Follows immediately from Eq. (24). 8. We have

PXAY@ 2 PXAD(~

= min[S(B, A), 4(C, A)]. 3. We have from eq. (33)

JYA, B n Cl = ;y

(cf. Property A. 1(lo)) (cf. Property A. l(2)).

= PA4 Hence, $(A, B) 2 j$

/.+(x)

Proof of Theorem 4.2.

= 1 - ;F$ /LA(X).

n

The proofs of these properties

are more involved. 1. We have Si[T(A; T), 5(B; T)] =

~AV(B”c+)

gi

= fb min[l, PUA&)

+ PW&)I

= &i min[l, PCL~(A&)+ k~&)l p,&

;Fg

=

max[;Fg

max[pAV&>,

pAV&)l

~AVB’(X)~

= max[$c(A,

;ig

pAVCc(x)l

B), $“(A, C)l.

4. Follows immediately

from part (3).

Proof of Theorem 4.3.

From the definition

q

of 9, we

obtain

/-~T(A;T)~A~(B;&)

= in{ min[l,

=

- 7) + j.~~(x - 7)l

= $$ miniI, ~~4z) + PB~)I

7 - 1 = i$

min[O, j-&r)

- pA(x

Hence, for all x E %, min[O, j@(x) - PA(x)] 2 T - 1. Therefore, pB(x) L r - 1 + ,.&A(x). That r is the largest such number easily follows. n

FUZZY MATHEMATICAL

PXO(,-Ml

with CI 2 -p. On the other hand, from Corollary 4.1, we know that if 9(A, B) = 7, then A 0 (T - 1) C B. All we need to show then is that (Y = 7 - 1 satisfies the constraints 1 + min[O, a] = 7 and (Y 2 -/3. From Theorem 4.2, we know that r 2 1 - /I. Hence, CY2 -p. Consequently, we obtain the desired result: 1 + min[O, 7 - l] = 1 + (7 - 1) = 7. n

= max[O,py(d 7 - 1 + p,d(.u)I

ACKNOWLEDGMENTS

Since, r I 1, we have

4.1.

Proof of Corollary

= max[O,7 - 1 + PxWI.

PXO(~-~)(~

Hence, P[xo(~-~)I~Y@)

= maxb.44,

= max[En,y(a), 7 - 1 + pAa)]

D. Sinha was partially supported via a “summer from the College of Staten Island.

= PA(Y) (cf. Theorem 4.3). Hence, [X 0 (Q- follows. n

l)] U Y = Y, and the result now

In order to prove Theorem 4.4, we need to establish a technical lemma. If [A 0 cx] c B, then

B.l.

LEMMA

$(A, B) z 1 + min[O, max(a,

-/3)],

(52)

where p = SUP,~% PA(Z). Since

Proof.

we obtain /-b(z)

-

@A(z)

z

PAOak)

=

mid1

-

2 min[l

PA(z) pA(zh

m&a,

-pA(

- p, max(a, -p)].

(53)

Therefore, $(A, B) - 1 =

jg$

min[l,

1 +

pdz)

=

$i

mid%

h&(z)

-

-

pA(

-

1

pA(

I I’ll min[O, 1 - /3, max(c-u, -p)]

(cf. Eq. (53))

= in& min[O, max(or, -p)] = min[O, max(a, -@)I. Hence, the result.

301

MORPHOLOGY

n

Proof of Theorem 4.4.

From Lemma B.l, it follows

that

JW B) 2 rafts 1 + minKAal

research award”

REFERENCES 1. G. Matheron, Random Sets and Integral Geometry, Wiley, New York, 1973. 2, S. Sternberg, Morphology for grey tone functions, J. Comput. Vision Graphics Image Process. 35, 1986. 3. S. Sternberg, Image algebra, course notes, 1983. 4. S. Sternberg, Cellular computers and biomedical image processing, in BiomedicalImages and Computers (J. Sklansky and J. Bisconte, Eds.), Springer-Verlag, Berlin, 1982. 5. C. R. Giardina and E. R. Dougherty, Morphological Methods in Image and Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1988. 6. J. Serra, Mathematical morphology for complete lattices, in Image Analysis and Mathematical Morphology (J. Serra, Ed.), Vol. 2, Academic Press, New York, 1988. 7. J. Serra, Mathematical morphology for Bollean lattices, in Image Analysis and Mathematical Morphology (J. Serra, Ed.), Vol. 2, Academic Press, New York, 1988. 8. J. Serra, Introduction to morphological filters, in Image Analysis and Mathematical Morphology, (J. Serra, Ed.), Vol. 2, Academic Press, New York, 1988. 9. G. Matheron, Filters and lattices, in Image Analysis and Mathematical Morphology (J. Serra, Ed.), Vol. 2, Academic Press, New York, 1988. 10. H. J. Heijmans and C. Ronse, The algebraic basis of mathematical morphology. 1. Dilations and erosions, J. Comput. Vision Graphics Image Process. 50, 1990. 11. H. J. Heijmans and (2~Ronse, The Algebraic Basis ofMathematical Morphology. II. Openings and Closings, .I. Comput. Vision Graphics Image Proc. 54, 1991. 12. H. Hadwiger, Vorslesungen Uber Inhalt, Oberfache and Isoperimetrie, Springer-Verlag, Berlin, 1957. 13. L. A. Zadeh, Fuzzy sets, Inform. Control 8, 1965. 14. L. A. Zadeh, Theory of fuzzy sets, in Encyclopedia of Computer Science and Technology (J. Belzer, A. Holzman, and A. Kent, Eds.), Dekker, New York, 1977. 15. V. Goetcherian, From binary to grey level tone image processing using fuzzy logic concepts, Pattern Recognition, 12, 1980. 16. D. Sinha, Fuzzy Sets, Possibility Distributions, and Their Application to Image Processing, Ph.D. thesis, Stevens Institute of Technology, 1987. 17. C. R. Giardina, and D. Sinha, Pointed fuzzy sets, Proc. SPIE 1192, 1989, 659-668. 18. R. Giles, Luckasiewicz logic and fuzzy theory, Internat. J. ManMach. Stud. 8, 1976.

302

SINHA AND DOUGHERTY

19. E. R. Dougherty and C. R. Giardina, Image algebra-Induced operators and induced subalgebras, Proc. SPIE 845, 1987. 20. E. R. Dougherty, A homogeneous unification of image algebra. Part I. The homogeneous algebra, J. Imaging Sci. 33, JulyiAug. 1989. R. Haralick, S. Sternberg, and X. Zhuang, Image analysis using mathematical morphology, IEEE Trans. Pattern Anal. Mach. Zntell. PAMI-9, 4, 1987. 22. L. Koskinen, J. Astola, and Y. Neuvo, Soft Morphological Filters, Proc. SPIE 1568, July 1991, 262-269. 23. D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and App/ications, Academic Press, San Diego, 1980.

DIVYERDU SINHA obtained a B.E. (Honors) in computer science and engineering from the Indian Institute of Technology in 1982 and a Ph.D. in computer science from Stevens Institute of Technology in 1987. From 1987 to 1989 he was on the faculty of electrical engineering and computer science at Stevens Institute of Technology. Since January 1990. he has been on the faculty of the College of Staten Island, CUNY, and on the doctoral faculty of computer science at the Graduate

Center of CUNY. Dr. Sinha has wide ranging research interests and has authored numerous papers on fuzzy set theory, mathematical morphology, pattern recognition, distributed decision making, nonsupervised learning, and network performance management.

EDWARD DOUGHERTY is an associate professor at the Center for Imaging Science of the Rochester Institute of Technology. He holds an M.S. in computer science from Stevens Institute of Technology and a Ph.D. in mathematics from Rutgers University. He has written numerous papers in the areas of mathematical morphology and image algebra, and has authored/coauthored six books in the areas of image processing, mathematical morphology, artificial intelligence, and statistics. His current research is centered on the design of statistically optimal morphological filters, development of model-based approaches to morphological image analysis, characterization of the statistical distributions of granulometric moments, and development of intrinsically fuzzy approaches to mathematical morphology.