SOP: Determination of Tristimulus Values by the Selected Ordinate Method

GIFSQ — Global Institute for Food Safety & Quality

Principle

Reduction of spectrophotometric data to the tristimulus values X, Y and Z is based on Grassmann’s laws. In the weighted ordinate method, the spectrum wavelengths are chosen at uniform intervals. In the selected ordinate method, instead, the intervals of spectrum selected correspond to uniform increments of each primary (X, Y or Z) in the spectrum of the source. The spectral reflectance of the specimen in one such interval is as important as that in another, so in computing an approximate value of a trichromatic coefficient for the whole spectrum no further weighting of each band is needed — merely adding together the reflectance values at different wavelengths for each tristimulus is sufficient. The curve is thus sampled more frequently in the wavelength regions of greater importance and is sparsely sampled near the ends. Depending on the accuracy needed and the shape of the spectrophotometric curve, up to 300 separate readings are required; ten selected ordinates can give sufficient information, but 30 selected ordinates are recommended for most work.

Apparatus required

• Spectrophotometer or spectrophotometer/colorimeter capable of measuring the amount of light reflected at wavelength intervals of 5 or 10 nm between 380 and 770 nm
• Large-size chromaticity diagram

Sample preparation

Measure the amount of light reflected by a representative sample at wavelength intervals of 5 or 10 nm between 380 and 770 nm.

Procedure

Measure the amount of light reflected at wavelength intervals of 5 or 10 nm between 380 and 770 nm

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Plot the % reflectance recorded against each wavelength to obtain the reflectance curve

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From the reflectance curve, read the reflectance for each specified wavelength ordinate from the selected ordinates table below and enter as shown in the worked example (in the example only 10 selected ordinates are taken; ninety separate entries are required if the thirty selected ordinates method is used)

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Add the entries in the X column and multiply the total by the factor indicated in the table; this is the X value; obtain Y and Z similarly

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Using the values of X, Y and Z, calculate x and y (x = X/(X + Y + Z); y = Y/(X + Y + Z))

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Using the values of x and y, determine the dominant wavelength and purity from the chromaticity diagram

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The brightness (visual efficiency) is equal to Y

Calculation

X = (factor for X) × (sum of reflectances at the X ordinates); Y = (factor for Y) × (sum of reflectances at the Y ordinates); Z = (factor for Z) × (sum of reflectances at the Z ordinates).

x = X / (X + Y + Z); y = Y / (X + Y + Z); z = Z / (X + Y + Z).

Where,

• X, Y, Z = tristimulus values
• x, y, z = chromaticity coordinates
• The selected ordinates are the wavelengths at which the reflectance curve is sampled; the factor is the constant given in the table for the chosen illuminant and the number of ordinates used (10 or 30)

Worked example — ten selected ordinates for Illuminant C

X = 0.09806 × 4.114 = 0.4033; Y = 0.10000 × 4.853 = 0.4853; Z = 0.11814 × 5.843 = 0.6903. Therefore x = 0.2554; y = 0.3074; z = 0.4372. Dominant wavelength = 489.0 nm; purity = 21.2%; visual efficiency = 48.5%.

Note: the available copy prints the Illuminant C 10-ordinate Y factor as 0.10800 (Table 17-6) and 0.1080. (Table 17-7); the worked example computes with 0.10000 (Y = 0.10000 × 4.853 = 0.4853), which the 30-ordinate factor 0.03332 also supports — 0.10000 is adopted here; confirm against a clean print.

* Table 17-7 prints this ordinate as 551.8; normalized to 551.7 per Table 17-6 — confirm against a clean print.

Reference — Selected ordinates for Illuminants A, B and C

The table gives the wavelengths to be chosen for 30 selected ordinates. If only 10 selected ordinates are required, use the wavelengths in rows 2, 5, 8, 11, 14, 17, 20, 23, 26 and 29 (shown in bold type in the book) and the corresponding 10-ordinate factor.

Note on the table: in the available copy of the book, many wavelength values are OCR-damaged in which “$” and “§” stand for the digit “5” (for example “$16.9” for 516.9); these have been restored as printed, and the restorations are confirmed by cross-reference to the worked ten-ordinate example above, whose ordinates match rows 2, 5, 8, 11, 14, 17, 20, 23, 26 and 29 of the Illuminant C columns. “—” marks cells that cannot be read in the available copy and must be checked against a clean print.

Precautions

• Depending on the accuracy needed and the shape of the spectrophotometric curve, up to 300 separate readings may be required.
• Ten selected ordinates can give sufficient information, but 30 selected ordinates are recommended for most work.
• The spectrophotometric curve characterizes the colour of the product; additional calculations are required to make this data meaningful.

References

Ranganna, S., Handbook of Analysis and Quality Control for Fruit and Vegetable Products, 2nd edition.

Hardy, A.C., Handbook of Colorimetry, The Technology Press MIT, Cambridge, Mass. (1936) — cited in the book.