Australan Journal of Basc and Appled Scences 4(10): 4601-4608 010 ISSN 1991-8178 Color Renderng Uncertanty 1 A.el Bally M.M. El-Ganany 3 A. Al-amel 1 Physcs Department Photometry department- NIS Abstract: In ths wor the method of the ISO Gude to the Expresson of uncertanty n measurement are appled to derve analytcal expressons for uncertanty n u v (CIE 1960 unform chromatcty scale coordnates ); wuv (CIE 1964 unform chromatcty color space coordnates); color shft specal color renderng (R ) and at last the general color renderng (R a ). The results of calculaton show that the uncertanty n (u v) s the lowest one whch the hghest s that for R One example s gven for calbraton of ncandescent lamp (100 watt). An uncertanty n general color renderng ndex for whch has been obtaned as a fracton of unt. The spectral rradance of the lamp has been measured usng spectroradometer the maxmum error of whch wthn the spectral range (380-780) s 47 %. Key word: Chromatcty Color space Color renderng Chromatc adaptaton. INTRODUCTION Experence has shown that the value of the color renderng ndex (CRI) may be nfluenced by the followng factors: The spectrum range (e.g. 400-700nm 380-780nm 360-830nm). A spectrum range 380-780 nm wll provde adequate results Spectral nterval of 10 nm have been shown to be satsfactory for fluorescent lamps but ntervals of almost 5nm are recommended for use n the case of metal halde lamps The most sgnfcant factor whch affects the value of (CRI) s the precse determnaton of the spectral power dstrbuton of the lght sources In ths wor attenton has been pad to determne the uncertanty of (CRI) accordng to such factor. The ISO Gude (Internatonal Organzaton for Standard 1993) descrbes consstent method by whch uncertantes of measurement are to be calculated. Color renderng ndex usually estmated accordng to the CIE method (CIE 1964). Here we determne the uncertanty of (CRI) from spectral rradance measurements by calculaton of the value of the color renderng ndex and hence ts standard uncertanty whch should also be calculated by the ISO method. The calculaton s complex as the relaton between the color renderng and chromatcty coordnates s a complex one and the coordnates themselves are based on expermentally determned response functons and whch are also correlated. Followng a bref outlne of color measurement ths paper derves expresson from whch the standard uncertanty of (CRI) may be determned. One example of source s consdered calculatng the uncertanty n (CRI) n terms of a constant relatve uncertanty n relatve spectral rradance. The dervaton of the specal and hence the general color renderng ndces shall be based on a general comparson of length of color dfference vectors n the CIE 1964 unform color space ( CIE1964 ) MATERIALS AND METHODS Sequence of dervatons must be carred out to derve the uncertanty n color renderng measurement accordng to C.I.E publcaton (Gardner000) one must carry out the dervaton of the uncertantes n (u v); (unform chromatcty scale coordnate ) (u v w ) (C.I.E 1964 unform color space coordnate); E (the resultant color shft) R specal color renderng and fnally R a (the general color renderng). -CIE 1960 (u v) coordnates for any lght sources or colored objects are defned as ( C.I.E1960). U = 4X / ( X + 15Y + 3Z ) (1) Correspondng Author: M.M.EL-Ganany Photometry department- NIS E-mal: alamel_ns@yahoo.com 4601
Aust. J. Basc & Appl. Sc. 4(10): 4601-4608 010 V = 6X / ( X + 15Y + 3Z ) () Where X Y and Z are the trstmulus response values founded by convolvng the color-matchng functons (Casmer1997) showng n Fg.1. wth spectral rradance dstrbuton of the source or the spectral dstrbuton of the lght reflected from the object. Accordng to the ISO gude the standard uncertanty n U c (u) and U c (v) are gven n terms of the uncertanty n rradance as follows(gardner and Frenel1999) and (Gardner000). Fg. 1: the CIE color matchng functons U c u u4 Uc E x u 5 Uc E y 9 Uc E z 30 4 c 6 4 c 90 c u u U E x y u u U E x z u U E y z 15 3 E x E y E z And smlarly U c v v Uc E y v Uc E x Uc E z c c c 95 9 6v 5v U E xy 6v U E xz 18v 5v U E yz 15 3 E x E y E z 1 1 / / (3) (4) Where E s the reflected rradance of the sample () and xy and z are the color matchng functon at wavelength. In ths wor we use the suffces r and to ndcate that the parameters XY Z u and v are related to the reference llumnant and test lamp respectvely. Also the suffces (r) and () to ndcate that such parameters related to the th Munsell sample llumnated by the reference and tested sources respectvely. 3-Consderaton of Adaptve Color Shft(Meshove1981): To account for the adaptve color shft due to the dfferent state of chromatc adaptaton under the lamp to be tested and the reference llumnant r there are formula whch can be appled to obtan the values of the chromatcty coordnates U ' V ' of a test color th sample after consderaton of the adaptve color shft obtaned by movng the lght sources to be tested to the reference llumnant e U' = U r V' =V r. However f the chromatcty dfference between the lamp to be tested and the reference llumnant ( c) 1.10-3 > 54 10-3 then accordng to the table (5)(C.I.E1964) the dference between (u' u ) and (v' v ) can be neglected. 460
Aust. J. Basc & Appl. Sc. 4(10): 4601-4608 010 4- Transformaton nto C..e 1964 Unform Color Space Coordnates(C.I.E1964): Colormetrc data must be transformed nto C.I.E unform color space coordnates by usng the followng formula: w 5 1/3 Yr 17; w 1/3 5 Y. 17 13 U 13w U U 13 V 13w V Vr r U w U U r r r r U w V V r r r r The values Y r and Y must be normalzed so that Y r =Y =100 Accordng to the ISO method the uncertanty of the three varable W U and V can be gven by the followng formula w c c Y U w U Y As Y E y E y (7) Consderng the concept of relatve uncertanty Then we have ( ) U Y U E y E y Y c c U ( E ) E c ) U E y U E y E y Y c c 5 3 c U w Y 3 U u 13w U U 13u u U w c c c u u r Ucw U w cu u w u Where (Where s a constant). (5) (6) (8) (9) (10) u w u u 13wu 13w u w u u u r U E U E U E w u c c c E E E E (11) w E 5 3 Y 3 y 4603
Aust. J. Basc & Appl. Sc. 4(10): 4601-4608 010 u E 4 x u t / E t Where t x 15y 3z Smlarly 13 13 U V w U V v v U w c c c v v rw v U c w Uc v w v Where (1) v w 13wv v v v 13w w v v v r U E U E U E w v c c c E E E E (13) v E 6 y vt / E t 6-the Determnaton of the Resultant Color Shft: To calculate the dfference between the perceved color of a test color sample llumnated by the lght source to be tested and that of the same sample llumnated by the reference llumnant r and consderng the above mentoned condton c =1.110-3 the C.I.E 1964 color dfference formula [1] shall be used. u u v v w w r r r (14) Where w r =5(Y r ) 1/3-17.and..w I =5(Y ) 1/3-17 7-calculaton of the Color Renderng Indces: The specal color renderng ndex R based on E obtaned from the formula (15) for any ndvdual test color sample s to be derved by the use of the followng formula: R = 100-4.6 E (15) The general color renderng ndex R a s to be derved as the arthmetc mean of the eght specal color renderng ndces R for the C.I.E /Tc 3 test color samples No. 1.8 4604
Aust. J. Basc & Appl. Sc. 4(10): 4601-4608 010 R a 1 8 8 1 R (16) 8-uncertanty n the Determnaton of Color Renderng Indces: From equaton (15) & (16) the uncertanty n determnaton of R a equals the arthmetc means of the uncertantes of the 8 uncertantes of R for the Munsell samples. From equaton (15) R depends on E u v whch n ts turn depends on the three varables and. w Then applyng the ISO method the standard uncertanty of E can be gven by E Uc E Uc u u E E Uc v Uc w v w E E ruv.. U cu Ucv u v E E ru w.. U cu Uc w u w E E rv w.. U cv Ucw u w r r r / t u u v v w w Puttng (18) Then we have E u r u u / t ' (7) E v r v v / t ' (19) E w r w w / t ' The correlaton factors can be gven by the followng formulae r U E U E U E u v E E E E u v u v c c c 4605
Aust. J. Basc & Appl. Sc. 4(10): 4601-4608 010 r U E U E U E u w E E E E u w u v c c c r U E U E U E u w E E E E v w v w c c c From equatons ( 5 ) we have. u w uu 13 ( ) u w u u E E E E 13u u 13w (0) (1) Smlarly v w v v E E E E 13v v 13w w w w w E E E E 13w w 13w From equaton (15) the uncertanty n specal color renderng U R 4.6 U ( E ) c c Then the uncertanty n general color renderng U ( R ) 1 U ( R ) 8 c a c 1 8 () (3) (4) (5) 9-Sequence of Calculaton: 1- Calc\ulaton of X.Y.Z and hence uv coordnates for the tested source - Calculate the XYand Z for Munsell sample 3- Calculatng the chromatcty coordnates (u v ) for the Munsell samples llumnants by the tested lamp 4- The chromatcty coordnates (u r v r ) for the Munsell sample llumnated by the reference llumnant can be gven from table ( 4 ) n [ 1 ] 5-Calculate the uncertanty of u v usng equatons ( 1 ) and ( ) 6- Calculatng u v and w for the Munsell samples usng Equaton ( 5 ) 7-Calculate the uncertantes of u v and w 8- Calculatng the uncertantes u v and W usng equatons ( 3 ) ( 4 ) and ( 6 ). Then accordng to equatons (17) and (4) calculatng the U c ( E ) and hence U c (R. ) from equaton (4). The uncertanty of R a from equaton (5) s the arthmetc mean of R for the eght Munsell samples. RESULTS AND DISCUSSION -In ths wor we apply the above mentoned method to obtan the uncertanty n the general color renderng ndex for an ncandescent lamp (100 watt) usng Spectroradometer for whch the maxmum error n rradance measurement wth the spectral range (380-780) s 4.7 %. 4606
Aust. J. Basc & Appl. Sc. 4(10): 4601-4608 010 The reference llumnant s selected to be the absolute radator at color temperature 650 K from C.I.E publcaton ts chromatcty coordnates U r = 0.671 V r = 0.35. Then the chromatcty dfference between the reference and test llumnant s found to be 310-4. Then we can apply the approxmaton mentoned before.e. U' = U and V ' = V. Also the values of U r V r and W r have been obtaned from reference [1] The results of calculatons are represented as follows Table (1) llustrates the trstmuls values of tested source Table () llustrates the (X Y Z) of Munsell samples Table (3) llustrates the uncertanty n (u v) Table (4) Chromatcty coordnates (u v w ) of Munsell's samples and ther uncertantes under tested source. Table (5) uncertantes of color shfts specal color renderng and general color renderng (1964 system) of Munsell's samples and ts uncertantes under tested source. -Expresson has been gven for the uncertanty n measurng general color renderng Index measurement accordng to method of the ISO gude. Acheve such expresson a sequence of dervaton the uncertantes n ( u v ) ( u v w ) coordnates the color shft E and hence the specal color renderng R for each Munsell's samples have been obtaned. One cane notce that the values of the uncertantes rse from step to the other when transfer. The uncertanty of R a "the general color renderng" s the arthmetc mean of the uncertantes n specal color renderng ndces for the eght Munsell's samples. The result of uncertanty of Ra s a fracton of the unt of measurng R a Table 1: Trastmulus values of tested source Source X Y Z Incandescent 100 Watt 11.9715 100.000 30.37 Table : Trastmulus values of Munsell's samples under tested source Sample X Y Z 1 43.965 33.141 6.759 36.58 30.764 4.4 3 30.46 30.39 3.161 4 3.138 6.595 6.636 5 3.138 7.764 11.471 6 8.10 6.99 15.804 7 38.106 9.93 14.004 8 48. 34.337 1.04 Table.3: Chromatcty coordnates of Munsell's samples and ts uncertantes under tested source. Sample U V U c (u ) U c (v ) 1 0.313 0.354 0.0009 0.00008 0.86 0.361 0.00009 0.00096 3 0.46 0.368 0.00016 0.00009 4 0.09 0.361 0.00004 0.0008 5 0.09 0.349 0.0001 0.0000 6 0.35 0.337 0.00111 0.0001 7 0.88 0.339 0.00007 0.0000 8 0.3 0.344 0.00008 0.00015 Table 4: Chromatcty coordnates (1964 system) of Munsell's samples and ts uncertantes under tested source. Sample U v w U c (U ) U c (v ) U c (w ) 1 35.817 1.046 63.30 0.59 0.06 1.077 1.8 6.713 61.335 0.731 0.774 1.044 3-19.773 1.77 61.018 0.141 0.110 1.015 4-46.0 5.9 57.64 0.197 0.63 0.995 5-46.0-3.531 58.701 0.187 0.03 1.009 6-6.697-13.45 57.934 0.831 0.141 0.998 7 14.374-11.699 60.6 0.095 0.055 1.034 8 43.580-8.363 64.57 0.6 0.134 1.08 Table 5: Uncertanty of color shfts specal color renderng and general color renderng (1964 system) of munsell's samples and ts uncertantes under tested source. Sample 1 3 4 5 6 7 8 U (E) 0.046 0.774 0.018-0.385 0.09 0.558 0.003 0.011 U (R) 0.1 3.559 0.084-1.77 0.134.569 0.016 0.051 U(Ra) 0.607 4607
Aust. J. Basc & Appl. Sc. 4(10): 4601-4608 010 Concluson: The lowest value of the uncertantes s that of the (uv). Whle the hghest one s the uncertanty of R. The uncertanty of R a s a fracton of unt whch s reasonable snce the value of R a for ncandescent lamp near 100. REFERENCES C.I.E publcaton color renderng of lght sources 1964. C.I.E publcaton color renderng of lght sources 1960. Casmer De Cusats 1997. Optcal Socety of Amerca Handboo of Appled Photometry pp: 354-356. Internatonal Orgnzaton for Standard Geneva 1993. Gude to the expresson of uncertanty n measurement. Gardner 000. correlated color temperature uncertanty and estmaton Metrologa 37: 381-384. Gardner J.L. R.B. Frenel 1999. Metrologa 36: 477-480. Gardner J.L. 000. Color Res Appl 5: 349-355. Meshove V.V. 1981. Fundamentals of llumnaton engneerng pp: 80-8. 4608