HUE ANGLE DETERMINATIONS AND STATISTICAL ANALYSIS FOR MULTIQUADRANT HUNTER L,a,b DATA ABSTRACT

Transcription

1 HUE ANGLE DETERMINATIONS AND STATISTICAL ANALYSIS FOR MULTIQUADRANT HUNTER L,a,b DATA M.R. McLELLAN', L.R. LIND and R.W. KIME Department of Food Science and Technology Institute of Food Science Cornell University Geneva, New York Accepted for Publication September 4, 1994 ABSTRACT Forpurposes of data analysis and data interpretation, it is important to have uniform representation of hue derived from the Hunter L,a, b color space. Using the standard calculation for hue [Arc tan(ba)], positive signed results are generated for the first quadrant [+a, +b] only. l2e other quadrants should be handled so that a 360" representation is accommodated and results are expressed as positive signed numbers. Second quadrant [-a, +b] and third quadrant [-a, -bj calculations should be: hue = I80+Arc tan(ba). Fourth quadrant [+a, -b] calculations should be: hue = 360+Arc tan(ba). Furthermore when planning statistical analysis of data which transverse the IV and I quadrant, vanformations of the data will continue increasing in hue angle from 361 " on up as high as appropriate. A spreadsheet macro for achieving these results can be easily derived. INTRODUCTION In 1975, A.C. Little published "Off on a Tangent", which clearly presented the concept of hue angle being derived from the arc tangent of ba (angle 0) in the Hunter L,a,b color space. The convention of using tan-' ba for hue was also described in detail elsewhere (Francis and Clydesdale 1975). In a similar vein, Francis (1975) discussed the origin of tan-' ab as a method of representing hue. In general, these publications served to help standardize the reporting of derived values for hue and chroma using the Hunter L,a,b system and settle on tan-' ba as the most desired method to report hue (Fig. 1). The hue angle method of reporting color remains commonly used when there is a need to present measurements of visual attributes. When performing first quadrant 'To whom correspondence should be sent. Journal of Food Quality 18 (1995) Aff Rights Reserved. Copyright I995 by Food & Nutrition Press, Inc., Trumbull. Connecticut. 235

2 236 M.R. McLELLAN, L.R. LIND and R.W KIME calculations the results of the hue angle derivation is highly reliable and easily managed in statistical analysis. However, when sample measurements cross quadrant boundaries, direct relationship between hue angle calculations based on the above noted methodologies are not easily relatable unless quadrant changes are accounted for such that a full 360" hue angle potential is used. By accounting for the full 360" hue angle, virtually any change in hue is easily interpreted and analyzable statistically. Appropriate management of quadrant information is otherwise confusing. Green 0= ArcTan(ba),.- Quadrant II Hue Angle I I 0 - Quadrant Ill Hue Angle 0= ArcTan(ba) I1 \ I11 \ \ \ \\ 'N --- Yellow (90") tb chroma 112 (a2+b2) 4 -, O=ArcTan(ba) I \\, # -b ie (270") IV Quadrant I Hue Angle I I I I- 0 0 Quadrant IV Hue Angle 0= ArcTan(ba) +a Red (O", 360") FIG. 1. THE HUNTER L,a,b SYSTEM DIAGRAM SHOWING ONLY THE a-b PLANE The objective of this work was to devise a simple and easy method to take the input of Hunter a and Hunter b values and derive a proper calculation of hue angle, including quadrant compensation for use in statistical analysis where linearity and consistency of scale are important.

3 HUE ANGLE ANALYSIS OF HUNTER L,a,b DATA 237 METHOD A macro language based on Microsoft Excel (Microsoft Corp. 1992) was programmed to derive the hue value with quadrant sensitive compensation. When constructed in this manner, the macro is considered a Function Macro. The ph of a cherry wine sample was adjusted with 15.3 ml of 1N NaOH followed by 2.12 ml of 0.5N NaOH. The color shift, as measured by the HunterLab Colorimeter (Model D25-PC2A, Hunter Associates Laboratory, Reston, VA), was noted periodically during the ph adjustment. Calculations of hue and chroma were derived (Table 1). The hue calculation was based on the macro as described above. The hue angle data were further transformed for regression analysis and the transformed data were then fitted to a three degree polynomial model. TABLE 1. CHERRY WINE SAMPLES, ADJUSTED BY ph, WITH HUE MEASUREMENTS MAKING A TRANSVERSE MOVE FROM QUADRANT I TO QUADRANT IV , 5.50 I 5.90 I [ i 8.01 : 5.00 I 4.70 I I I 4.88 RESULTS AND DISCUSSION The function macro was used in a data spread sheet and referenced to the associated Hunter a and b values. The result was the corrected hue angle based on the appropriate quadrant. The macro accepted two arguments, which were the Hunter a and Hunter b values, respectively. The Hunter a value was first

4 238 M.R. McLELLAN, L.R. LIND and R.W KIME checked for its sign (-I+). If it was positive the macro proceeded to check the sign of the Hunter b value. If it was positive then the macro calculated the arc tangent in radians, and then converted the resultant to degrees for the proper hue angle. If the check of the Hunter b value turned up a negative sign, then 360 was added to the arc tangent after being converted to degrees. If the program identified a negative Hunter a value, then calculation of the hue angle in degrees would be added to 180 and used for both quadrants I1 and 111. The only exception happens with a transverse set of data on the 0" (360") point (quadrants I and IV border). In this transverse data set, measurements change from the 359" to 0" or from 1" to 360". In Table 1, the sample of ph adjusted cherry wine moves from a starting hue angle of 16.14" to a final value, after adjustment, of ". Clearly in this case the change of quadrant is important in establishing the value of the final hue angle. This technique would be helpful with other examples of hue measurements as reported in the literature. In a report on blueberries as colorants, cultivars were measured for hue and reported values ranged from a blue hue of to a reddish hue of 3.6 (Francis 1985). If these values were treated with a quadrant sensitive derivation, the values would be reported as ranging from to 3.6, which properly identifies the values as measurements from quadrant IV (270" to 360") and from quadrant I (0" to 90"). Handling these data for statistical analysis requires adept interpretation of the data trend prior to analysis. The quadrant IV to quadrant I transverse move of the data was handled for statistical purposes by simply continuing the 360" hue angle to values greater than 360". In other words, sample measurements changing from quadrant IV into quadrant I would continue increasing in angle from 361" on up as high as appropriate. Later after appropriate statistical analysis, values greater that 360" could be transformed to the correct angle. In Table 1, this type of data transform has been conducted on the cherry wine sample in order to produce a continuous data set ready for analysis. A three degree polynomial model was fit to the transformed data with an R2 of based on the fitted 9 observations (Fig. 2). The cubic model predicting hue of the cherry wine based on adjusted ph was: Hue = (pH) (pH)' (PH)~ An analysis of variance conducted on the model indicated a high degree of significance with an F ratio of and significance beyond the 99.99% level. Application of the model to the hue data and subsequent analysis of variance would not be possible without quadrant compensated calculation of hue value and transformation of the hue to a linear change in degrees. Hunter L,a,b data are not independent variables, and their use in straight calculations of Analysis of Variance and regression should be cautioned because

5 HUE ANGLE ANALYSIS OF HUNTER L,a,b DATA (20) c PH FIG. 2. PLOT OF POLYNOMIAL MODEL FIT TO HUE DATA EXPRESSED AS CONTINUOUS DEGREE MEASUREMENTS of this fact (Gill 1978; Francis and Clydesdale 1975; Francis 1994). Handling the hue angle data using this quadrant compensation approach with a further transformation of the data to a continuously linear scale during a quadrant IV to quadrant I transverse ensures the proper analysis of data from the Hunter color space based on independent variables of lightness, hue and chroma. SUMMARY The calculation of a hue angle from the Hunter L,a,b, color space should be done by compensating for the quadrant in which the data appears. Further transformation of the data during a transverse of the fourth and first quadrant is required to arrive at a continuous data set appropriate for further statistical analysis where linearity and consistency of scale are important. REFERENCES FRANCIS, F.J The origin of tan-' ab. J. Food Sci. 40, 412. FRANCIS, F.J Blueberries as a colorant ingredient in food products. J. Food Sci. 50,

6 240 M.R. McLELLAN, L.R. LIND and R.W KIME FRANCIS, F.J Personal Communication. Dept. Food Science, Univ. of Massachusetts, Amherst, MA. FRANCIS, F.J. and CLYDESDALE, F.M Food Colorimetry, Theory and Applications, p. 208, Van Nostrand ReinholdAVI, New York. GILL, J.L Design and Analysis of Experiments. The Iowa State University Press, Ames, Iowa. LITTLE, A.C Off on a tangent. J. Food Sci. 40(2), Microsoft Corp User s Guide 2: Worksheet Analysis, Exchanging Data, Customizing and Automating. Microsoft Corp., Redmond, WA.