A fuzzy decision model for color reproduction

Cecilia Temponia.*, Farid Derisavi Fardb, H.W. CorleyCa Southwest Texas State University. San Marcos. TX 78666. USA

b Xerox Corporation. Dallas. TX, USAc The University of Texas at Arlington. Arlington, Tx' USA

Received 4 June 1997; accepted 29 December 1997

Abstract

Color quality appraisal refers to a visual evaluation of an original color image and an assessment of the degree towhich that image is matched in photomechanical or electronic reproduction. The evaluation of reproduction quality isquite subjective since it relies on the individual judgments of color technicians who have different perceptions of colorand its component qualities. To reduce the amount of rework due to subjective evaluation of color reproduction, anobjective model is proposed here. In this model, a fuzzy set of quality factors is introduced and a numerical quality valuecalculated. This measure is then compared with an acceptable quality value experimentally established for particularcustomers and printing processes. Finally, the installed model is tested in actual production, and compared with theconventional evaluation method in a cost analysis. ~ 1999 Elsevier Science B.V. All rights reserved.

Keywords: Printing process; Color appraisal; Production; Fuzzy model; Cost analysis

I. Introduction tive and varies from one technician to another. Thisfact is the primary cause of production rework in

In color quality appraisal, the original color the prepress industry today when the color repro-image (color transparency, color print, or other duction does not meet the customer's expectation.reflection copy [I]) and the color reproduction are Quantitative decision techniques have not beencompared for hue, lightness, and saturation. They heretofore used, but the situation seems a naturalare also appraised in terms of tonal range, contrast, application of fuzzy logic.and texture, which affect the overall appearance of In this paper we construct a model for the nu-the color image. However, such analysis is subjec- merical evaluation of the quality level of a color

reproduction by introducing a set of factors fora color technician to use in their evaluation anda measure of their respective importance. This ap-proach enables the technician to judge the degree of

.Corresponding author. Tel.: + 1 512245 3189; fax: color match acceptable to a particular customer for+ 1 5122453089; e-mail: ct01@swt.edu. a specified printing process.

0925-5273/99/$ -see front matter I:!::) 1999 Elsevier Science B.V. All rights reservedPII: S 0925 -5 27 3 ( 98 ) 000 78.4

2. Preliminaries 1-1 (x)

2.1. Fuzzy set theory andfuzzy numbers I Introduced by Zadeh [2], the theory of fuzzy sets

concerns a subset A of a universe of discourse X ,where the transition between full membership andno membership is gradual rather than abrupt.A fuzzy subset has no well-defined boundaries ascompared with the universe X that consists of defi-nite objects. Most classes of objects encountered inthe real world are of a fuzzy nature, without a pre- 0cise criteria for membership. In such cases it is not 5 7 12 Daysnecessary for an object to belong or not belong toa particular class. The object may have an inter- Fig. I. Number of days required to train a new hire.

mediate class of membership [3,4].Fuzzy numbers provide a way to represent one's f on X with respect to the fuzzy measure Jl by

knowledge about real-world numbers. Informationor know!edge ab?ut a number is often based on

I f d = su [a I\ (X nF J] (1)someones experIence. An easy way to define Jl p Jl ,

Ae[O.cx»a fuzzy number is to determine its lowest value, itshighest value, and the most likely value. For where I\ is the minimum operator and F a = { x:

example, from the authors' experience in the pre- f(x) ~ a}. Now consider an object to be evaluated,press industry, it takes from 5 to 12 days to train and let X = {xl' X2' ..., Xn} denote the factor space

someone for a particular position, depending on of the object, that is, the set of all its quality factorsthe trainee's background. However, most people of interest. The goal is to obtain a numerical qualitycomplete the training program in seven days. evaluation of the given object from the evaluationsA fuzzy number representing the number of days for each individual factor.required to train a new hire for this particular Assume that each subset A of the factor spaceposition is shown in Fig. 1. This fuzzy number is X is associated with a real number X(A) betweenreferred to as a triangular fuzzy number (TFN). 0 and 1 that indicates the importance of A. Such

a real number should be the maximum possiblescore that the object can gain relying only on the

2.2. Fuzzy measure quality factors in A. Obviously the empty set 0,which includes no relevant quality factors, has the

The notion of fuzzy measure introduced by minimum importance 0; and the whole factor spaceSugeno [5] can be used to model uncertainty, in X has the maximum importance 1. Moreover, ifparticular, the vagueness and uncertainty inherent each factor in a factor set A belongs to anotherin natural language. In this paper it is used to factor set B, then A is at most as important as B. Itquantify subjective and vague terms like "too follows that X( 0) = 0, X(X) = 1, and X(A) ~ X(B)dark", "more contrast", "more punch", or "match whenever A c: B.the original" used by designers and print buyers in The number of quality factors for the given ob-describing reproduction quality to color experts. ject is finite, so the set function X is a regular fuzzySuch terms are difficult to translate into precise measure on the measurable space (X, p(X)) calledlanguage. For other applications of fuzzy measure the importance measure on X [5]. It representssee Refs. [4,6-9]. here a composite expert opinion. Given a particular

In this work we implement fuzzy measure as in object to be evaluated, a factor space of the 9bject,Wang and Klir [10] and define the fuzzy integral of and an importance measure, the object is evaluated

by experts for each individual quality factor xi, The degree of color match between an originalx2, , , , , Xn' Their subjective opinions are aggreg- and a reproduction can vary due to numerousated into a composite quality evaluation function factors or steps involved in the color reproduction/ to yield the scores /(xJ, /(X2)' , , , , /(Xn), where process. In our experience the three major factors/ may be regarded as a measurable function on affecting a color technician's decision are gray bal-(X, p(X)) for which/(xJe [0, I] for each Xi eX, ance (G), tone reproduction (T), and overall ap-

As noted by Wang and Klir [10] and Zhong [7], pearance (A), Therefore, let xi be the quality factorthe fuzzy integral J/dX on X of the scores/(xJ with G, x2 be T, X3 be A, and X = {XI' X2' X3}.

respect to the importance measure X yields a singlenumerical quality evaluation of the given object,The term synthetic evaluation [II] refers to thisprocess of reducing sets of scores assigned to factors 3,2, Importance measureinfluencing ajudgment to a single number. It gener-alizes the weighted average approach to evaluation The importance measure X of a color reproduc-since interactions between sets of factors can be ti on gives the contribution of each separate qualitymodeled using a non-additive fuzzy measure, X, x, factor and combination of factors to the total qual-and/will next be defined for a color reproduction ity level. To establish X, target transparencies wereprocess, selected and reproduced as mid-range offset color

reproductions, Fifteen color technicians weretrained to rate the importance to color reproduc-

3 Th d I tion quality of the individual quality factors and.emoe bo 0 f " 0

I f bcom Inatlon o lactors as tnangu ar uzzy num ero

0 0 0 0 Next they were asked to evaluate each reproduc-Color reproduction 1S class1fied as eIther offset or t o " th lOt ~

t f b I (G)0 0 1on lor e qua 1 y lac ors o gray a anceflexo because these two pnntlng processes are com- 0 ,

I t I dofii t d 0 d ofii t I 1° appearance (A), tone reproduction (T), and then top e e y 1 eren an requ1re 1 eren co or qua Ity 0 0

0 0 0 present the results In the form of a tnangular fuzzycons1deratlono Furthermore, the offset market 1S b b t I d 10 b of 0 .

I...0 num er e ween an y spec1 ylng Its owest,further d1v1ded Into the three separate categones: hO h t d t l o k I I Th d 0 0

.19 es , an mos 1 e y va ue, e ec1s1on to use

good-enough color (GE), mid-range color (MR), t 0I f b b t I d 100 .0 0 nangu ar uzzy num ers e ween an was

and h1gh-end color (HE). We limit our presentation b d 0 0 th 0 tofi Id d th0 0 .ase on expenence In e pnn Ing e an e

here to mid-range offset color reproduction. S1m- , 0 .,'I It h b bt ' d " h h authors belief that color techmc1ans best relate to1 ar resu s ave een o alne lor t e ot er cases. th O

al1S sc e.The data collection forms for the 15 experts were

analyzed using a fuzzy spread sheet application3.1. Qua/ity factors package called FuzzyCalcTM for WindowsTM to

yield a composite importance measure for mid-Acceptance or rejection of a color reproduction range offset color reproduction. The resulting

is presently based on subjective evaluation by values of X are presented below in Table I.a color technician. A trained color technician candistinguish about 10 million colors. However, ver- Table 1bal systems can describe only a few hundred colors,physical sample systems a few thousand, and in- Importance measure

strumental systems hundreds of thousands [12]. X<A) = 0.569Color technicians monitor the color quality-con- X<T) = 0.342trol process. They accept or reject a reproduction x(G) = 0.397based on the degree of color match between the X<T, A) = 0.819original and its color reproduction, as well as their x(G, A) = 00908

0 f .. h d x(G, T) = 0.673perception O 1tS meetIng t e customer's stan ardso

3.3. Quality evaluation functions ity level of factors G, T, A with triangular fuzzynumbers between 1 and 10 for any reproduction

For a particular color reproduction a similar deemed acceptable for customer review. Theseapproach can be used to obtain values for f(xl), numbers were then entered on a quality factorf(x2)'f(x3). Each expert evaluates the reproduction evaluation log form. When a customer acceptedfor gray balance, tonal reproduction, and overall a color reproduction, FuzzyCalcTM averaged theappearance in terms of a triangular fuzzy number. three fuzzy numbers using fuzzy set theory and thenThese are then defuzzified and averaged using defuzzified the result. In this manner, the minimumFuzzyCalcTM to yield a crisp value forf(x;). Accord- quality value required for acceptance by a particu-ing to Wang and Klir [10], this average is very lar customer could be obtained. For example,close to the true quality level if the number of color magazine DM required a synthetic quality level ofexperts evaluating the color is large enough. 0.75 for acceptance.

3.4. Example4 R I.esu ts

Given a color reproduction, technicians evaluate ,th ' d ' .d 1 l .t so t G T A t ' Id th Twenty-five mId-range offset color reproduc-

e m IVI ua qua I y lac ors , , o yle e ,.t f(G) f(T) f(A) Th . 1 tIons were selected to test the performance of the

composl e scores, , .e numerIca d 1 . 1 d ..E hth t . l t . E f th I . f h 1 mo e m a co or repro uctIon envIronment. ac

syn e IC eva ua Ion o e qua Ity o t e co or. ,reproductIon was revIewed by three color experts

reproductIon IS then obtamed from the fuzzy mte- d 1 t d so h I .sof G T A ThJ ., an eva ua e lor t e qua Ity lactors o , , .e

gral fdx of the scoresf(x;) wIth respect to Import- h . T bl 2 F h d ,scores are s own m a e .or eac repro uctlon

ance measure X. h h . I . 1 1 1 d Th .

bLet f(G) = 0.6 j(T) = 0.8 f(A) = 0.9 for the t e synt etIc. qua Ity va ue was ca cu ate. .e jO .s

'd ~ t ' , t ' , .were then rejected or accepted after theIr synthetIcml -range ollse Impor ance measure gIven m ..

...qualIty value was compared wIth the acceptableTable 1. For thIs color reproductIon the numerIcal l .t 1 . T bl 3 h . h hh 20 ' b...qua I y va ues m a e w IC s ows t at jO s

synthetIc evaluatIon E IS then calculated as follows,dd fi ' .

d..were accepte an ve rejecte .where v IS the maxImum operator. ..

To compare these results wIth the conventIonal

E - I f d subjective method, a fourth color expert evaluated-X the 25 reproductions and rejected eight. Regardless

of a job's acceptance or rejection using either ap-= [0.61\ X(X nF 0,6)] v [0.81\ X(X nF 0.8)] proach, each job was sent to the customer.

v [0.91\ X(X nF )] An evaluation of the model's performance re-0,9 quires that costs be associated with the four pos-

= [0.61\ X(X)] v [0.81\ X(T, A)] v [0.91\ X(A)] sible combinations of rejecting or accepting a job

= [0.61\ 1] v [0.81\ 0.819] v [0.91\ 0.569] by color expert and customer. Ajob here is definedas an 8.5" by 11 " color reproduction, and every

= 0.6 v 0.8 v 0.569 round of color modification involves the following

= 0.8. steps (1-6) and factors (7-10):

1. locating the digital file,3.5. Quantifjling customer quality expectation 2. loading the file on the color workstation,

3. performing the color modification,Once the importance measure is established, 4. archiving the new image,

a decision (accept or reject) on a particular color 5. manufacturing a set of film,reproduction requires that the customer's quality 6. manufacturing a prepress color proof,expectation be quantified. For nine months three 7. disrupting the production schedule,color 1echnicians were instructed to grade the qual- 8. increase in overtime on misplaced jobs,

Table 3 Table 4Accepted or rejected targeted mid-range color reproductions Crisp cost matrix C

Job Quality value Result Cust(A) Cust(R)

01 0.818595 A CX(A) $0 $17502 0.766667 A CX(R) $350 $15003 0.85 A04 0.85 A05 0.716667 R06 0.75 A07 0.75 A that the expert rejected. These results may be repre-

08 0.75 A sented by the matrices09 0.716667 R10 0.766667 A C(A) C(R) C(A) C(R)11 0.783333 A12 0.7 R CX(A) [18 2] CX(A) [16 1]13 0.766667 A CX(R) 1 4 and CX(R) 2 6 .

14 0.7 R15 0.783333 A~~ ~:~:~~~~ ~ The cost CM associated with the model is then given

18 0.8 A by19 0.8 A20 0.75 A [18 2] [ $0 $175

]C --130021 0.783333 A M- 1 4 <8) $350 $150 -$ ,

22 0.816667 A23 0.8 A h /0\ d h . 1 . I ..24 0.783333 A were IC/ enotes t e matrIx mu tIp lcanon25 0.8 A (18)(0) + (2)(175) + (1)(350) + (4)(150). Similarly Cs

for the standard or conventional approach is

[16 1] [ $0 $175]Cs = <8) = $1775.

crIncal one ID the Industry. Being too good is ex- 2 6 $350 $150

pensive in color reproduction. From such dis-cussions we concluded that a reasonable cost These numbers show a performance by themodel was a triangular probability distribution be- model superior to that for the usual approach. Thetween $300 and $400 with a mean, median, and main reason is that the model rejected fewer accept-mode being $350. able jobs than the usual approach. Based on this

Since there is no cost associated with acceptance small sample and cost estimates, our model seemsof a job by both the color expert and the customer, well-suite~ to make decisions for mid-range color

the previous information suggests the following reproductIon.matrix C of mean costs (also the median and modalcosts). In Table 4, CX(A) and CX(R) denote thecolor expert accepting and rejecting a job, respec- 5. Conclusiontively, while C(A) and C(R) refer to the customer'sacceptance and rejection. The fuzzy approach to color quality evaluation

Recall that all 25 jobs were sent to the customer. proposed here produced better acceptance/rejec-Of the 20 accepted by the model, two were rejected tion results on a small sample of mid-range colorby the customer, who also accepted one of five that reproductions than the conventional method. Thethe model rejected. Of the 17 accepted by the color limitations of the work are the cost estimates, whichexpert using the usual approach, one was rejected are difficult to obtain, and the size of the sample.by the customer, who also accepted two of eight Unfortunately, both are difficult to obtain, and one

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