ISSN 1061�9348, Journal of Analytical Chemistry, 2014, Vol. 69, No. 4, pp. 322–326. © Pleiades Publishing, Ltd., 2014.Original Russian Text © S.V. Khimchenko, L.P. Eksperiandova, 2014, published in Zhurnal Analiticheskoi Khimii, 2014, Vol. 69, No. 4, pp. 363–368.

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Visual colorimetric test analysis is widely used as arapid and simple method for the detection and semi�quantitative determination of traces as colored com�pounds [1–4]. Accurate determination and absoluteand relative errors, which depend on the discreteness(step) of the color test scale, are among the majorproblems in the metrological support of test analysis.The step (q) of a test scale, which is usually plottedexponentially based on general considerations, shouldbe selected in accordance with two main conditions.First, a too large step of the scale leads to an unneces�sarily large error of analysis, and second, on the con�trary, with a too small step one cannot confidently dis�tinguish colors (mainly, the change in lightness, ΔL) ina standard scale. It should also be considered that anaverage human eye can distinguish the lightness ofobjects of different color tints with different efficien�cies [5]. Thus, a unique optimal step should corre�spond to each color tint. From the above it followsthat, to establish an optimal step of a test scale, toneshould compare the eye’s ability to distinguish colorsin lightness in the test scales with different color tintsand discreteness.

The optimization of the construction of a referencecolor scale for visual colorimetric assay is paid con�stant attention [6–10]. The history of this problem anddifferent ways to constructing test scales were consid�ered in [9], where a new method for constructing testscales was proposed based on the Fibonacci sequence(the rate of progression Φ = 1.618), in which each suc�cessive term in the series is the sum of the two previous

terms (1, 1, 2, 3, 5, 8, 13…) and, from the 5th term ofthe series and on, the rate of progression Φ = 1.618.The use of this sequence takes into account the advan�tages and disadvantages of using geometric and arith�metic progressions in test analysis [9]. In the samepaper, using colorimetry and a probabilistic approachon the example of a blue test scale (ion associates ofperchlorates with thionine), it was first supposed that,in the creating test scales uniform in contrast, oneshould not always follow the criterion of ΔЕ ≥ 10,common in the practice of visual test analysis, in orderto change the overall color difference ΔЕ of consecu�tive scale elements, because “…it turned out thatpoints with q = 2, clearly distinguishable in the scale,were characterized by a general color difference ofΔЕ ≈ 6, rather than 10, and it is said that criterion ΔЕ =10 cannot be applied unconditionally as the univer�sal.” Later, the validity of this thesis was further con�firmed in [10]. However, the authors referred to [9]only in connection with that ΔЕ ≈ 6, and did not payattention to the second part of the quote, indicatingthe priority of the authors [9] with respect to a moreessential conclusion on the difference of ΔЕ from 10.The fact that a significant difference between colorsappears at ΔЕ > 6 was also indicated in [11].

The goal of this work was to study two methods ofevaluating the minimum concentration step in con�structing reference color test scales with differentcolor tints, colorimetric, based on measurements ofthe total color difference of the scale elements with

Colorimetric and Stochastic Assessment of the Visual Limitof Color Perception for Visual Colorimetric Analysis

S. V. Khimchenko and L. P. EksperiandovaScience and Technology Complex “Institute of Single Crystals”, National Academy of Sciences of Ukraine,

pr. Lenina 60, Kharkov, 61001 UkraineReceived March 28, 2013; in final form, September 11, 2013

Abstract—Methods for evaluating the visual limit of color perception are proposed to create common rulesfor constructing color test scales in visual colorimetric assay using a probabilistic approach and instrumentalmeasurements of the total color difference ΔE. Using a wealth of experimental data, we confirmed the previ�ous idea that the minimum value of ΔE discernible by an average eye in 90% of cases may differ from the con�ventional criterion ΔE = 10. It has been shown that ΔE is not constant throughout the range correspondingto the visible spectral region and depends on the color tint of the scale. Using the dependence of the proba�bility of color difference on the size of the scale step, a minimum step for each color tint is found. This makespossible the a priori selection of a step for constructing a calibration scale depending on its tint.

Keywords: colorimetry, visual limit of color perception, minimum step of color test scale, a priori selection ofthe scale step

DOI: 10.1134/S1061934814040042

ARTICLES

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COLORIMETRIC AND STOCHASTIC ASSESSMENT OF THE VISUAL LIMIT 323

different discreteness, and stochastic using probabilityanalysis involving a large number of observers.

EXPERIMENTAL

Reagents and materials. Twice�distilled water andreagents of cp grade were used to prepare solutions. Asolution of CoSO4 ⋅ 7H2O (without nickel), preparedaccording to GOST (State Standard) 4212�76 andstandardized by GOST 10398�76 was used as a stocksolution of Co(II) (1.05 g/L). A solution of Fe(III)(0.97 g/L) was prepared by dissolving 1.43 g of Fe2O3

in HCl 1 : 1 followed by dilution with water to 1 L andstandardization according to GOST 10398�76. Work�ing solutions of Co(II) and Fe(III) were prepared bydiluting stock solutions with 0.001 M H2SO4 and0.001 M HCl, respectively. A solution of NaNO2

(0.30 g/L) was prepared by dissolving a weighed por�tion of anhydrous salt in water. We also used a 1.5 Msolution of KSCN, a 0.1 M solution of NaF, 2 MH2SO4, and 5 M HCl. Solutions of the following dyeswere used: a 0.1% solution of Methyl Violet (MV) anda 0.1% solution of Malachite Green (MG) in ethanoland a 0.1% solution of Metanil Yellow (MY) and a0.1% solution of Congo Red (CR) in water. To createmagenta color, a mixture was prepared consisting of0.1% solutions of MV in ethanol and Neutral Red(NR) in 50% ethanol (1 : 1, vol). Polyurethane foam(PUF) 22�3.0 based on ethers was used as an adsor�bent and a substrate. PUF tablets, 16 mm in diameterand ~0.025 g in weight, were cut from a 5�mm com�mercial polymer sheet with a metal punch. The tabletswere purified by soaking in 1 M H2SO4 for 1 h, thenwashed with water to neutral pH, and dried in air.

Preparation of color test scales. To construct colorscales, we used a set of colored complex compounds ordyes whose color tints covered a wide range of visiblespectrum. The test scales were constructed accordingto a geometric progression (step 1.5 or 2) or theFibonacci series (step 1.618).

Test scales for the determination of iron(III) andcobalt(II) were prepared according to procedures [1,2]. A sample solution containing a test component andthe reagents necessary for obtaining colored analyticalforms were placed in a 50�mL glass beaker. A PUFtablet, purified as described above, was placed in eachglass and pressed with a glass rod to remove air bub�bles. The contents of the beaker were stirred with amagnetic stirrer for 30 min. The tablet was thenremoved from the solution and squeezed with glassrod, the residue solution was removed with filter paper,and the tablet was left in air to complete dryness.

Test scales for the determination of perchlorateswere prepared according to the procedure [12]. Asolution of thionine and a formate buffer solution wereadded to a volumetric flask with a solution containingperchlorate, the mixture was diluted to the mark withwater, and the resulting solution was transferred into a

beaker into which a PUF tablet was then placed. Thecontents of the beaker were mixed with a magneticstirrer, the tablet was then squeezed with a glass rod,the residue solution was removed with filter paper, andthe tablet was left in air to complete dryness.

Test scales containing dyes MV, MG, MY, and CRand a mixture of MV and NR were prepared by theuniform impregnation of PUF tablets with solutions ofthese substances. For this purpose, a tablet was placedon a Teflon plate; a portion of a dye solution 0–32 μLin volume according to the selected step of the testscale and 100 μL of ethanol were applied; the tabletwas repeatedly pressed on top with a similar plate foruniformly impregnating the support and left in air tocomplete dryness.

Procedure for assessing probability by a group ofobservers. The test scales obtained using PUF tabletswere placed at an achromatic white background in anascending concentration order of colored impuritiesfor each of the studied sequences (q = 1.5; 1.618,and 2). Examinations of the test scales and visualassessments were conducted by observers in diffusedsunlight with an average illumination of ~1000 lux anda color temperature (Tc) of approximately 6500 K;such an illumination corresponded to the averagecolor temperature of a cloudy sky. A large number ofindependent observers (more than 100 students ofKharkov National University) were asked the ques�tion: “Do the neighboring tablets in a scale differ fromeach other by the intensity of the blue color and whichone is more intense?” By analyzing the results a con�clusion about the probability of “hits” was made; itwas evaluated as the ratio of the number of correctresponses to the total number of observers.

Procedure of colorimetric measurements. The testscales were scanned using an Epson V350 Photo flat�bed scanner in a 24�bit color mode with the resolution300 dpi. A cold�cathode xenon lamp with a correlatedcolor temperature of approximately 6300 K served asthe radiation source in the scanner. The images wereprocessed in an Adobe Photoshop 8.0 graphics editor.To do this, an image of each tablet was recovered usinga Gaussian Blur filter with a radius of 30 pixels. Thecenter of the resulting image was isolated and, using aHistogram function, the average values of color coor�dinates L, A, and B in the CIE LAB system were found.The obtained values of color coordinates were statisti�cally processed in accordance with recommendations[13, 14].

RESULTS AND DISCUSSION

Colorimetric method. Based on the determinationof color coordinates using the scanner and the graphiceditor, the total color difference ΔE calculated by thefollowing equation [5, p. 94] was selected as an analyt�ical signal:

2 2 2 1 2( ) ,E L A BΔ = Δ + Δ + Δ

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where ΔL, ΔA, and ΔB are changes in the color coor�dinates in the CIE LAB system (1976).

The study of the test scales by colorimetry showedthat the change in the mean total color difference δΔEbetween two neighboring samples in the scale dependson the progression step and on color of samples. Themeasured values of δΔE for different colors in the vis�ible spectral region are given in the table, and thedependences of δΔE on the color tint corresponding tothe visible spectrum (λ = 380–780 nm), including the“nonspectral” magenta color, which is a combinationof red and blue, are presented in Fig. 1. It can be seenthat the smallest number of color discrimination

thresholds δΔE corresponds to dark blue, blue, andgreen, that is, colors with wavelengths of reflectedcolor of 430–550 nm, whereas the largest number, toyellow, red, and violet (580–780 nm). We took thevalue of δΔE (change in ΔE) between the nearestbenchmark elements in the scale at which 90% of aver�age observers felt visual difference in the of colorintensities, for a minimum number of thresholds.Thus, based on the distribution pattern of δΔE over thespectral region and the data of the mass experimentinvolving a large number of independent observers, wecan conclude for the color scales with the steps q = 1.5,1.618, and 2 that the minimum number of distinguish�able thresholds is 4–6 for the middle and 10–15 for theedges of the spectral region. Therefore, the color scalesfor dark blue, blue, and green colors should be betterconstructed exponentially with the step q ≤ 1.618, as alarger step is “excessive” and unnecessarily increasethe error of analysis; for yellow, red, and violet colors,the scale step q should be 1.618 ≤ q ≤ 2–3.

Note that the observed distribution pattern for δΔEover the spectral region (color tint) is almost exactlythe same as the dependence of the color discrimina�tion threshold on the wavelength of monochromaticradiation [5, p. 64] (Fig. 1). This coincidence can beexplained as follows. First, the spectral sensitivity ofthe charge coupled devise (CCD) sensor of the scan�ner is close to the spectral sensitivity of the human eye;therefore, the three minimum and maximum valuesobserved in the curve correspond to three types oflight�sensitive elements in the sensor matrix (RGB).Second, in obedience to the Weber–Fechner law, thechange in lightness is visually better noticeable in darkcolors than its change by the same amount in light col�ors, as the lightness increment in the former case willbe higher than that in the latter [5, 15]. In addition,according to the known Purkinje effect, violet, dark

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ΔE

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Fig. 1. Dependence of δΔE on color tint in the visible spec�tral region (380–780 nm) for the scales with q = (I) 1.5,(II) 1.618, and (III) 2; on the abscissa axis: (1) violet,(2) dark blue, (3) blue�green, (4) green, (5) yellow, (6) yel�low�orange, (7) red, (8) red�brown, and (9) magenta(nonspectral color); the dashed line shows the conven�tional value for δΔE = 10.

Results of stochastic (Fig. 2) and colorimetric (Fig. 1) approaches to assessing the minimum step of the test scale; numberof observations N = 100–150*

Color tint (color�forming agent)δΔE Probability of positive responses, % Recommended

step of the visual test scaleq = 1.5 q = 1.618 q = 2.0 q = 1.5 q = 1.618 q = 2.0

Violet (MV) 10.4 15.8 17.2 74.1 97.3 98.0 ≥1.618

Dark blue (Cl with thionine) 2.6 5.2 6.4 80.0 90.0 97.9 ≥1.618

Blue�green (Co2+ thiocyanate) 5.0 6.2 6.6 89.8 95.9 98.9 ≥1.5

Green (MG) 2.8 4.0 6.0 74.2 94.0 97.1 ≥1.618

Yellow (MY) 7.0 9.4 11.4 72.6 77.2 80.3 >2

Yellow�orange (ions (N ) 6.0 6.4 8.2 80.9 89.8 93.4 ≥1.618

Red (CR) 7.6 10.4 16.0 68.7 95.9 97.6 ≥1.618

Red�brown (Fe3+ thiocyanate) 10.2 11.8 11.2 85.4 90.5 91.1 ≥1.618

Magenta (MV + NR) 9.4 10.6 9.4 80.9 93.9 94.8 ≥1.618

* The values of δΔE corresponding to the minimum allowable probability of positive responses and, simultaneously, to the minimum stepof the visual test scale are highlighted in italics.

O4–

O2–

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blue, and blue fields always seem brighter than red,yellow, or orange fields at an equal brightness of theilluminated (photometric) fields.

One should pay particular attention to the fact that,in the case of scales similar to some scales preparedusing PUF tablets studied in this paper, namely, blue(Co2+ thiocyanate), yellow (nitrites), and red (Fe3+

thiocyanate) scales [10, p. 192], rather small color dif�ferences (δΔE = 4) were obtained for the red and bluecolors and improbably low values (δΔE = 3) for theyellow color. Such an unnaturally low estimate for theyellow color can be explained only by an incorrectexecution of the experiment. Apparently, in measuringcolor coordinates by diffuse reflection spectra, theauthors did not take into account the type of equip�ment used in the standard illuminant, which probablyrepresented a halogen incandescent lamp with colortemperature of Tc = 2800–3200 K. Then, these resultswere incorrectly compared with the results of proba�bility estimates made by independent observers, whoconducted their observations in diffuse daylight withTc = 6500 K. This assumption is supported by the cal�culations of ΔE for different types of standard sourcesrelative to a standard source with Tc = 3250 K,reported in [11, p. 82]. These calculations clearly showlarge differences in ΔE measured using standardsources with different Tc. It is likely that erroneousconclusions were made in [10] for this reason. For acorrect comparison of the results, the lighting condi�tions in the visual evaluations and instrumental mea�surements must be as close as possible.

Stochastic method of evaluation. To estimate theminimum step of the scale, we used a probabilisticapproach used to evaluate the detection limit [10]. Forall scales with q = 1.5, 1.618, and 2, the colors of twoneighboring samples in all scales were compared, con�sidering each interval as a separate area of an unreli�able reaction. The authors of [10] considered theprobability of P = 0.95 a criterion for reliable detec�tion. We have shown previously [9] that it is moreappropriate to use P = 0.90 as a criterion of reliabledetection, as it was known that approximately 8% ofmen and 0.5% of women suffer from color blindnessand can not give an objective assessment [5, 15]. Giventhis and the fact that, according to statistics, theUkrainian population has 46.3% of men and 53.7% ofwomen, the widely accepted probability value of 0.95should be reduced to its real level, by ~0.04–0.05.Thus, concentration matching the difference betweenadjacent scale concentrations determined with a prob�ability of P = 0.90 was considered the minimum step ofthe scale. It should be noted that the authors of [10]also used P = 0.90, but with no reference made to theearlier selection of this value of Р [9] and without otherargumentation.

Note that, in determining phosphates [8], the min�imum step of the geometric scale was evaluated alsostatistically by involving a large number of “naive”

observers, but the authors used a slightly differentapproach. They studied the variation of the resultsaround the true value, which a priori increases theerror due to colorblind participants in the experiment.

To clarify the minimum value of δΔЕ for its use asa criterion for constructing visual test scales, weobtained the dependence of the probability P of thedetection of color difference on the scale step size q forthe entire region of the visible spectrum. The tableshows the results of observations, P(q), for differentcolors in the visible spectral region; the dependenceP(q) for scales with q = 1.5, 1.618, and 2 on color tintis presented in Fig. 2. To facilitate understanding,these dependences are superimposed on a spectrum inthe visible region, including the “nonspectral”magenta color. It can be seen that the scale with q =1.618 and 2 for all colors, except for yellow, and thescale with q = 1.5 for blue color (cobalt thiocyanatecomplex) correspond to P(q) = 0.9. Thus, a scale stepof q = 1.618 can be considered the minimum possiblefor most color tints of the studied scales, except foryellow.

Recall that it was found in [10] that the yellow coloris perceived by observers better than red and blue. Asexplained above, this statement is inaccurate. The fal�lacy of this conclusion is confirmed by the fact that,according to the empirical dependence of the colordiscrimination threshold for color tint on wavelengthand the Weber–Fechner law, yellow color has an evenlarger value of color discrimination threshold than thered and violet colors perceived more difficult [5] andthus cannot be easily distinguished visually. Thus, itfollows from the theory of color perception and theconsistent theory resulting from our experiment(Fig. 2) that yellow color with λ = 550 ± 20 nm is dis�cernible by the eye worse than all other colors, includ�ing red and dark blue colors. It is also surprising that,

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P(q), %

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Fig. 2. Dependences P(q) for the scales with q = (I) 1.5,(II) 1.618, and (III) 2 on the color tint. The dashed lineindicates the probability P = 0.9 adopted for visual testanalysis; designations on the abscissa axis are as in Fig. 1.

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as described in [10], over 90% of the observers wereable to confidently distinguish elements of the scale onthe films, while the value of δΔE = 0, reported bythe authors, indicate that this discrimination is impos�sible.

Limit of color perception and the Fibonacci series.As already noted, the geometric progression of the col�orimetric scale is more preferable than the arithmetic.In fact, in the low concentration region, it would bebetter to build the scale in an arithmetic progressionwith its small step, because the geometric progressionmakes the initial separation between the adjacentpoints of the scale too large. On contrast, for distantmembers of the scale, it would be wiser to use a geo�metric progression, as an arithmetic one in this casewould be inappropriate (the relative error analysiswould be 100%). Namely the scale based on theFibonacci series meets these requirements [9]. It isassumed that, apparently, the eyes, being a part ofnature, enable the optimal perception of the color dif�ferences of two objects of observation, by analogy withmany others proportions of nature [16] correspondingexactly to the number of Φ = 1.618 [9].

Indeed, in the present study, the construction of thetest scale satisfies in the most cases the condition ofP(q) ≥ 0.9, at which the analyte concentration changesaccording to the Fibonacci sequence; so its use in thetest assay is justified (table). It can be seen from thetable that the scales of the blue�green color can beconstructed even with the step q = 1.5, the value of qfor the yellow scales must be greater than 2, and thescales of all other colors can be used with q = 1.618. Itis also clear that, in some cases highlighted by italics inthe table, the value of δΔE corresponding to the mini�mum step of the scale can vary from 4 to 6. It followsfrom the table in general that the step of the test scale,if possible, should be the minimum number of thecolor discrimination thresholds (from 4 to >10) anddepends on the scale color. Thus, the previousassumption that the condition δΔE = 10 is optional isconfirmed once again.

It is important that in the construction of visual testscales, the use of ratios between analyte concentra�tions in the samples corresponding to the Fibonaccisequence (see above) is convenient in practice for themajority of the color tints. This is due to the combina�tion of a minimum discreteness of the scale (and,therefore, the minimum error of the test determina�tion) and the ease of dosing an analyte by introducinginteger�valued volumes of test solution. It should benoted that, to increase the sensitivity of test analysis, itwas previously proposed to arrange the scale elementsin the background whose color is strictly complemen�tary to the color of the scale elements [9]. The use ofthis technique, based on the phenomenon of chro�matic contrast, improves visual color perception.

Thus, in the present study, the previously statedassumption was confirmed that the minimum value of

ΔЕ discernible by an average eye in 90% of cases maydiffer from the conventional criterion ΔЕ = 10. Thevalidity of the conclusion is confirmed by a large set ofexperimental data. It is shown that ΔЕ is not constantthroughout the visible spectrum region and dependson the color tint of the scale. The minimal step of thereference visual test scale prepared using PUF tabletswas found and proved for each color tint, whichenables a priori selection of the step in constructingsuch a scale.

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Translated by O. Zhukova

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