IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. PAMI-4, NO. 2, MARCH 1982

Regular Correspondence

A Note on the Quantitative Measure of Image EnhancementThrough Fuzziness

SANKAR K. PAL

Abstract-The "index of fuzziness" and "entropy" of an image reflecta kind of quantitative measure of its enhancement quality. Their valuesare found to decrease with enhancement of an image when differentsets of S-type membership functions with appropriate crossover pointswere considered for extracting the fuzzy property plane from thespatial domain of the image.

Index Terms-Entropy, fuzzy set, image processing, index of fuzzi-ness, property plane.

I. INTRODUCTIONThe present correspondence illustrates an application o

theory of fuzzy sets in image processing problems. The Ilem is to provide a quantitative measure of enhancement (

ity of an image through the evaluation of its amount of fness. These are explained by the terms "index of fuzzirand "entropy" [1], [2] of a fuzzy set. Index of fuzzreflects the ambiguity present in an image by measurinldistance between its fuzzy property plane and nearest ordiplane. The term "entropy," on the other hand, uses Shanrfunction in the property plane but its meaning is quite dient from the one of classical entropy because no probabiconcept is needed to define it. These two terms which giidea of "indefiniteness" or fuzziness of a set may be regaas the measure of an average intrinsic information whitreceived when one has to make a decision (as in palanalysis) in order to classify the ensembles of patternsscribed by a fuzzy set. These quantities are found to deciwith enhancement of image.The fuzzy property plane has been extracted from the sp

domain using S-type membership function [31, [4] awith two fuzzifiers. The role of the fuzzifiers is to intro(different amount of ambiguity in a property plane by chanthe crossover point and slope of the transformation funciThe effectiveness of the algorithm with different valuethese fuzzifiers is demonstrated on a set of enhanced image

II. FUZZY SET AND IMAGE DEFINITIONA fuzzy set (A) with its finite number of supports xl,

.*. , xn in the universe of discourse U is defined as

A = {(g,A (xi), xi)}

or, in union form

A=U gi/xi, i=1,2, ,ni

where the membership function MA (xi) having positive value inthe interval [0, l] denotes the degree to which an event ximay be a member of A. If /IA (Xi) = 0.5, xi is said to be thecrossover point in A.Since a gray tone image possesses some ambiguity within the

pixels due to the possible multivalued levels of brightness,it is justified to apply the concept and logic of fuzzy set ratherthan ordinary set theory to an image processing problem.Keeping this in mind, an image X of M X N dimension and Llevels can be considered as an array of fuzzy singletons, eachwith a value of membership function denoting the degree ofhaving brightness relative to some brightness level 1, I = 0, 1,* * L - 1. In the notion of fuzzy set, we may therefore write

X =U U Pmn/xmn m = 1, 2, ,M;n = 1, 2, ,Nm n

(2)where Pmn/xmn (O . Pmn 6 l) represents the grade of pos-sessing some property Pmn by the (m, n)th pixel xmn. Thisfuzzy property Pmn may be defined in a number of wayswith respect to any brightness level depending on the problemsto hand. This is explained in Section IV.

III. EVALUATION OF FUZZINESS OF AN IMAGEA. Index of FuzzinessThe index of fuzziness of a set A having n supporting points

is defined as [11

I(A) =2kd(A, A) (3)

where d(A, A) denotes the distance between fuzzy set A andits nearest ordinary set A. The set A is such that ,uA (xi) = 0if gA (x1) < 0.5 and 1 for ,uA (xi) > 0.5. The number 2 and thepositive constant k appear in order to make y(A) lie between0 and 1. The value of k depends on the type of distancefunction used. For example, k = 1 for a generalized Hammingdistance whereas k = 0.5 for an Euclidean distance. The cor-responding indexes of fuzziness are called the "linear index offuzziness" yj (A) and the "quadratic index of fuzziness" yq(A ).Considering "d" to be a generalized Hamming distance we have

d(A, A ) = ; LA (X i) - MA (X i) PA n A(Xi) (4 )i i

and

(la) Y.(A) =-2 PA n (xi), i 2, nn i

(5)

(lb) where A nA is the intersection between fuzzy set A and itscomplement A. gA nX(xi) denotes the grade of membershipof xi to such a fuzzy set A fl A and is defined as

Manuscript received December 29, 1980; revised March 17, 1981.This work was supported in part by the Association of CommonwealthUniversities, England, and in part by the Indian Statistical Institute,Calcutta.The author is with the Electronics and Communication Sciences Unit,

Indian Statistical Institute, Calcutta, India, on leave at the Departmentof Electrical Engineering, Imperial College of Science and Technology,London, England.

PA n A(xi) = min {PA (xi), 9.9(xi)}, for all i (6a)

= minm {A(xi), (1 - MA (Xi))}, for all i. (6b)

From (5) it is seen that

(7a)(7l)min = ° for pi= 0 or I

and

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(7Yl)max =1 for Al = 2= .=Afin=0.5.Furthermore, it follows that

(7b) Snf(PA (xi)) = -PA (xi) In PA (xi)

- (1 - PA(Xi))ln (1 - PA(Xi))-(7c) Therefore, like the index of fuzziness, this entropy, (14) is

also dependent on the absolute values of p and satisfies theproperties

(7d)

where A * is a "sharpened" version of A such that p, (xi)>PA (xi) for PA (Xi) > 0.5 and p1 (xi) < PA (xi) for PA (xi) < 0.5.Extending (5) in two-dimensional image plane we may write

7Y1 (X) = ZZE PXn (Xmn)MN m n

(8)Equation (8) defines the amount of fuzziness present in theproperty plane of an image X. pl corresponds to Pmn. X n Xis the intersection between fuzzy image planes X = {Pmn/xmn}and X = {(1 - Pmn)/xmn}, the complement of X. Pxnx(Xmn)denotes the degree of membership of (m, n)th pixel xmn tosuch a fuzzy property plane X n x so that

AxnFc(Xmn) = Pmn Pmnn mmin {Pmn, (1 - Pmn)},

Hmin = 0 for pi=Oorl

Hmax=l for iP2 =P..=An=0.5

H(A) > H(A *) (16c)

and

H(A) = H(A). (16d)

In fact, these conditions (7) or (16) may be regarded as thecriteria to be satisfied by a function in order to measure fuzzi-ness in a set.With this notion, we define the entropy of an M X N dimen-

sional image plane X as

IH()MNlIn 2 2:1 Snl(PX(Xmn))

m n

(9)for all (m, n).

Similarly, for an Euclidean distance we have

Tq(X) = [E [E (PX(Xmn) PIX(Xmn))2

m = 1 , 2, ,M;n = 1 , 2, ,N (1 0)where X is the nearest ordinary image plane of fuzzy plane X.

B. Fuzziness through EntropyThe entropy of a fuzzy set A having n supporting points is

defined as [ 1]

H(ipl,f2~~ ~~~>in= n pA(x ) ' ln (s,pA (xi)) (1In (n)

where

Z/A (xi)

The entropy is then seen to lie between 0 and I in a way

Hmin 0 for (pj= 1, jE{1, 2, **n}

fOi= , ij (13a)

Hmax for ePi = 1P2 /PnI ln. (13b)

It is to be mentioned here that this method (unlike the previousone) does not depend on the absolute values of p but theirrelative values. In other words, a fuzzy set with a single non-zero pA-value would have zero entropy and a set having a

constant ,u-value for all the elements would have H = 1. There-fore, an image X with IA(Xmn) = 1 or 0, xmn EX (i.e., fullybright or dark) according to this definition would be possess-ing maximum entropy, but this is intuitively unappealing.De Luca and Termini [21 defined entropy of a fuzzy set A

in analogy with information theoretic entropy, although quitedifferent conceptually, as

H(A)= Sn(pA(xi)), i = 1, 2, ,n (14)n ln22n s (xion

with the Shannon's function

with

Sn(gx(xmn)) = -PX(Xmn) ln PX(Xmn) - (1 Ix(Xmn))-In (1 lX(Xmn))

(18)

C. Interpretation of y(X) and H(X) for Image EnhancementIn the previous section, we have described y(X) and H(X)

for providing a measure of the fuzziness present in two-dimen-sional image plane X. y(X) measures the distance betweenfuzzy property plane of X and its nearest ordinary plane.H(X), on the other hand, is based on the well-known propertyof Shannon's function Sn(p) (1 5)-monotonically increasingin the interval [0, 0.51 and monotonically decreasing in [0.5,11 with a maximum (= In 2) at p = 0.5-in the fuzzy propertyplane of X.For gray tone image processing problem, an image pattern

X looks ambiguous to a people or device which knows onlyblack and white gray levels. The nature of this ambiguity(fuzziness) in X therefore arises from the "incertitude" presentwhen one has to decide whether the (m, n)th pixel intensityxmn has to be considered white or black. We may measurethis incertitude or uncertainty by Pxng(xmn) or Sn(pX(Xmn))which is O ifPx(xmn) =O or 1 and is maximum forpx(xmn) =

0.5; the average (normalized) amount of incertitude is mea-sured by the terms y(X) or H(X).Now through processing, if we can remove partially the un-

certainty on the gray levels of X, we say that we have obtainedan average amount of information given by 67= 7(X) - y(X')or 6H = H(X) - H(X') where X' is the processed (sharpened)version of X. The criteria y(X') <7(X) and H(X'))H(X)in order to have positive 67y and 5H-values are followed from(7c) and (16c), respectively. If the uncertainty is completelyremoved, then y(X ) = H(X') = 0. In other words, y(X) andH(X) can be regarded as the measure of average amount ofinformation (about the gray levels of pixels) which has beenlost for transforming the classical pattern (two tone) into a

fuzzy pattern X.

IV. PROPERTY PLANE AND FUZZIFIERSThe operations described in Section III are restricted in

fuzzy property plane. To enter this domain from the spatialimage plane, we need a membership function which will

and

71(A) =71(A) (16a)

(16b)

(17)

205

(15)

M=1,2,...,M; n=112,...,N.

1) 2, M; n = 1, 2, N.

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. PAMI-4, NO. 2, MARCH 1982

transform each xmn in the spatial domain to its correspondingPmn-value in the property domain. This function may beeither S-type or r-type or their complements depending onthe problem in hand. The S-function defines the compatibilityfunction corresponding to fuzzy plane "xmn is Xmax" whereasthe r-function corresponds to a plane "xmn is 1," 0 <1 <xmax. The corresponding fuzzy pmn-values denote the degreeof possessing maximum brightness level xmax and some otherlevel I by the (m, n)th pixel Xmn.Now in our problem of measuring fuzziness of an image we

are interested in a monotonic increasing/decreasing functionas represented by S/(l - S)-function which will result in anone-to-one mapping of the elements in the x-plane rangingfrom 0 to xmax to the p-plane in the interval [0, 1]/[1, 01.To represent such an S-function, we define a simple expression

P = GS(Xmn) = [+ (Xmax Xmn)]d

(19)

where Fe and Fd are the exponential and denominationalfuzzifiers, respectively. These two positive constants have theeffect of altering the ambiguity (fuzziness) in the fuzzyproperty plane by changing the crossover point and slope ofthe S-function. Their effect on the y and H-values has beenstudied in the next section. The function Gs is symmetric inthe interval [0, Xmax] if it leads to the crossover point atxmax/2. Otherwise, it is said to be nonsymmetric.

If for example, we use (xmax/2) - Xmn instead of (xmax -Xmn), (19) would represent a 7T-function (G,r) symmetric overXmax /2. Since such a function would result in the samep-value for any two pixel intensities located symmetrically onopposite sides of xmax/2, the conditions (7a), (7b), and (16a),(16b) (except for ,ui = 0) would not convey the appropriateinterpretation of fuzziness of an image. For example, animage X with xmn = Xmax/2, xmn EX would have Pmn =G,r(xmn) = 1 and hence 'y(X) = H(X) = 0. Similarly, theimage X with xkl = Xmax/4 and xmn = 3xmax/4, (k, 1) # (m, n),Xkl, xmn E X would have (for symmetrical G7-function)Pmn = 0.5 and hence y(X) = H(X) = 1. Both of the cases arenot intuitively appealing. Only the case when xmn = 0 orxmax for which Pmn = 0 and y(X) = H(X) = 0, conveys anappropriate information regarding ambiguity in X.Again, it is to be mentioned here that the above Gs-function

results in an et-level property plane where a is the value of Pmnfor xmn = 0. Since this violates the condition (7a) or (16a)for xmn = 0, the algorithm includes provision for constrainingall the zero xmn-values to zero Pmn-value. However, theresults without using this constraint are also reported forcomparison.

V. EXAMPLE AND DISCUSSIONFig. 1 shows a 96 X 99, 32-level image of handwritten script

("Shu"). Fig. 2(a), (b), and (c) are its different enhancedversions as obtained by histogram equalization technique,contrast intensification technique, and contrast intensificationalong with smoothing, respectively [5]. Table I illustratesthe values of 'Y7 (X), ^Yq(X), and H(X) for different slopes ofthe symmetrical Gs-function (crossover point at 1 5.5). Resultsfor different nonsymmetrical Gs-functions (for Fe = 2) areexplained in Table II. Here we have considered the values ofFd to be 70, 60, 50, 45, 35, 30, 25, and 15 so that the cross-over point of the nonsymmetrical Gs-function can lie betweenthe gray levels 2 and 3, 6 and 7, 10 and 11, 12 and 13, 14 and15, 16 and 17, 18 and 19, 20 and 21, and 24 and 25,respectively.From Table I it is seen that the quadratic distance when used

in (3) results in higher effective values of y as compared tothose of linear distance. The absolute 7I-values and H-valuesfor a fixed corssover point are decreased as the curve tends tobe steeper (with increase in the values of Fe and Fd) resulting

Fig. 1. Original image.

in a decrease in ambiguity in p-plane. These values are mini-mum for Fig. 2(c) and maximum for Fig. 1. This relativeorder (as in Table I) for different enhanced images is seen tobe maintained as long as the crossover point is restricted in theleft half [0, 15.51 of the gray scale. As we keep the crossoverpoint moving from 15.5 towards xmax = 31, the amount offuzziness in the equalized image [Fig. 2(a)] tends to be greater(after the crossover points 16.5, 16.5, and 18.6 are reachedfor yl, 'Yq, and H, respectively) than that of input image(Fig. 1). It is revealed under investigation that Fig. 2(a),since it possesses an almost uniform histogram, contains ascompared to Fig. 1, a large number of levels around the cross-over points (as selected in the right half of gray scale) and itis these levels which cause an increase in (Pmn n imn)-valueof y(X) and Sn(pmn)-value for H(X).As mentioned in the previous section, the above results were

obtained using the constraint a = 0 in (19). For comparisonof these results, the parameters y and H were also computed1) without using this constraint and 2) using an ideal S-func-tion which is defined as [31

Pmn = 2(xmn/Xmax)2 for Xmn -< (20a)

1- 2((Xmn - Xmax)/Xmax)

for ,B.Xmn S Xmax (20b)with ,B (crossover point) = xmax/2 and 0 < Pmn < 1. Theresults are shown only for yi (Table I) as a typical case ofillustration. The use of (19) alone (i.e., with oa # 0) results inan increase in the absolute values of fuzziness (especially forFig. 2(b) and (c) having a large number of zero gray levels)but does not change the relative order of fuzziness for theseimages, whereas (20) does change. It is also to be noted that(20) is symmetric across ,B and there is no control over thecrossover point in order to make it nonsymmetric.

In a part of the experiment, the entropy under Kaufmann'sdefinition (11) was also computed for these images. Sincethese images are neither fully bright nor fully dark (for whichit leads to an unappealing concept of H = 1), the order of theirH-values is not changed from that obtained with (14) (Table I).Experiments were also conducted using G,7-functions, but theresults as explained in Section IV, did not reflect the appro-priate measure of enhancement-quality for different values ofthe fuzzifiers.

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(a)

(b) (c)Fig. 2. Different enhanced images.

TABLE I"INDEXES OF FuzzINESS" AND "ENTROPY" OF IMAGES FOR DIFFERENTSYMMETRICAL S-FUNCTIONS (CROSSOVER POINT AT X =12= 15.5)

IMAGE y1(X) Yq(X) H(X)

(1) (2) (2) (3) ISF (1) (2) (3) (1) (2) (2) (3)Fig. 1 0.800 0.760 0.760 0.744 0.286 0.804 0.767 0.752 0.966 0.951 0.999 0.944

Fig. 2a 0.711 0.681 0.684 0.670 0.367 0.753 0.727 0.717 0.880 0.863 0.994 0.857

Fig. 2b 0.196 0.189 0.641 o.186 0.103 0.403 0.389 0.384 0.233 0.230 0.842 0.229

Fig. 2c 0.096 0.091 0.578 o.o89 0.043 0.261 0.250 0.246 0.134 0.129 0.812 0.128

(1): Fe= 1 and Fd= 15.5; (2): Fe= 2 and Fd= 37.42; (3): Fe= 3 and Fd= 59.63* without the constraint a = 0 in equation (19)

ISF using Ideal S-Function (equation 20)** using equation (11), Entropy under Kaufmann's definition

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TABLE II"INDEXES OF FUZZINESS" AND 'ENTROPY" OF IMAGES FOR DIFFERENTNONSYMMETRICAL S-FUNCTIONS (CORRESPONDING CROSSOVER POINTS,

CP FOR Fe = 2 ARE ALSO MENTIONED)

IMAGE Fd = 70 F = 60 Fd = 50 Fd= 45 F = 40 Fd= 35 Fd= 30 Fd= 25 Fd= 15CP = 2.01 CP = 6.15 CP = 10.28 CP = 12.36 CP = 14.43 CP = 16.5 CP = 18.57 CP = 20.64 CP = 24.79

0.869 0.917 0.882 0.845 0.794 0.725 0.643 0.547 0.319

Fig. 1 0.876 0.921 0.884 0.849 0.799 0.733 0.652 0.557 0.3300.979 0.989 0.987 0.979 0.963 0.937 0.896 0.836 0.624

0.569 0.619 0.658 0.674 0.678 0.679 0.666 0.637 0.508

Fig. 2a 0.636 0.683 0.715 0.727 0.725 0.723 0.707 0.677 0.5580.780 0.81o 0.838 0.850 0.860 0.865 0.864 0.852 0.763

0.166 0.180 0.188 0. ^91 0.189 0.187 0.179 0.166 0.120

Fig. 2b 0.352 0.380 0.392 0.395 0.391 0.385 0.371 0.346 0.2640.215 0.222 0.228 0.230 0.231 0.230 0.227 0.220 o.186

o.o68 0.075 0.082 o.o86 o.o89 0.092 0.095 0.098 0.097Fig. 2c 0.202 0.219 0.235 0.242 0.247 0.251 0.256 0.258 0.253

0.107 0.113 0.120 0.124 0.127 0.131 0.134 0.137 0.138

Upper Score: y1(X); Middle Score: yq(X) and Lower Score: H(X) of equation (14)

ACKNOWLEDGMENTThe author wishes to thank Dr. R. A. King for his interest

in this work.

REFERENCES[1] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets-

Fundamental Theoretical Elements, vol. 1. New York: Academic,1975.

[2] A. De Luca and S. Termini, "A definition of a nonprobabilisticentropy in the setting of fuzzy set theory," Inform. Contr., vol.20, pp. 301-312, May 1972.

[3] L. A. Zadeh, K. S. Fu, K. Tanaka, and M. Shimura, Eds., FuzzySets and their Applications to Cognitive and Decision Processes.London: Academic, 1975.

[4] P. P. Wang and S. K. Chang, Eds., Fuzzy Sets-Theory and Appli-cations to Policy Analysis and Information Systems. New York:Plenum, 1980.

[5] S. K. Pal and R. A. King, "Image enhancement using smoothingwith fuzzy set," IEEE Trans. Syst., Man, Cybern., vol. SMC-1 1,pp. 494-501, July 1981.

Efficient Spiral Search in Bounded Spaces

RICHARD W. HALL

Abstract-This correspondence defines approaches for the efficientgeneration of a spiral-like search pattern within bounded rectangularlytessellated regions. The defined spiral-like search pattern grows out-ward from a given source in a two-dimensional space, thus tending tominimize search time in many sequential tracking tasks. Efficientspiral generation is achieved by minimizing the number of operationsrequired for interaction with boundaries. Algorithms are developed

Manuscript received April 29, 1980; revised September 29, 1981.This work was supported in part by the National Eye Institute of theNational Institutes of Health under Grant 5-R23-EY02457-02.The author is with the Department of Electrical Engineering, Uni-

versity of Pittsburgh, Pittsburgh, PA 15261.

for both rectangular search regions and for arbitrary convex searchregions.

Index Terms-Convex and rectangular search regions, image track-ing component, iterative growth techniques, search pattern generation,spiral search pattern.

I. INTRODUCTIONAn important facet of many image processing applications

is the search in a given image space for a particular object orobjects followed by some form of subsequent analysis (e.g.,see [1] -[3]). Although many current applications utilizeglobal processing with exhaustive search strategies, the order-ing of a given search can reduce image processing time (e.g.,see [41). In particular, we have been developing an imageprocessing approach to the eye tracking problem [51 whichhas generated a particular interest in efficient search strategiesfor the detection of small (-4-5 pixels) moving objects in abinary image space. In one approach we are considering theuse of a spiral-like search pattern which proceeds outwardfrom the previous known position of the object being tracked.This allows us to economize on the size of the search spacewhen objects to be tracked are not numerous and have rela-tively small velocities. Spiral-like search patterns found earlyuse in analog character recognition approaches [61 and in lightpen tracking for random access CRT display technologies [ 71.The definition of a spiral-like search pattern is quite straight-

forward except when one must interact efficiently with bound-aries as in our application. Consequently, characterizationshave been developed for spiral-like search which are efficientin use of computation time by minimizing interactions witharbitrary boundaries in both rectangular and convex searchregions. These characterizations are the subject of this note.

II. THE SPIRAL SEARCH CHARACTERIZATION INRECTANGULAR SEARCH REGIONS

A useful class of spiral-like search over a rectangularlytessellated space is illustrated and defined in Fig. 1. Prior toreaching any boundaries, we may characterize the clockwise

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