Systems and methods for approximating a 5-angle color difference model

A method of approximating a 5-angle color difference model with 3-angle measurements data, the method comprising the steps of:

• receiving a 3-angle standard color measurement of a standard color at each of 3 color measurement angles;
• receiving a 3-angle test color measurement of a test color at each of the 3 color measurement angles;
• entering one of:
• (i) the 3-angle standard color measurement;
• or (ii) a geometric mean of the lightness L* and chroma C* of the standard color measurement and the test color measurement;
• into a neural network empirical model comprising a plurality of input nodes, a plurality of hidden nodes connected to the plurality of input nodes, and a plurality of output nodes connected to the plurality of hidden nodes;
• outputting 3-angle tolerance values S3L, S3c, and S3H at each of the 3 color measurement angles from the neural network empirical model;
• and
• calculate a 3-angle color difference value for at least one of the 3 color measurement angles using a 5-angle equation (A) with the 3-angle standard color measurements, the 3-angle test color measurements, and the 3-angle tolerance values S3L, S3c, and S3H, in place of 5-angle tolerance values S5L, S5c, and S5H, wherein the 5-angle equation (A) is;
• =the square root of ((d/kL·)2+(d/kc·)2+(d/kH·)2), (A)
• wherein d is a 5-angle color difference value, dL* is a difference in a lightness between a standard color−test color pair, is a measurement angle, i is an iteration of the measurement angle, dC* is a difference in a chroma between the standard color−test color pair, dH* is a difference in a hue between the standard color−test color pair, KL, Kc, and KH are acceptance limits, S5L, S5c, and S5H are 5-angle tolerance values for the lightness, the chroma, and the hue, respectively.

A method of producing a 3-angle color difference model that approximates a 5-angle color difference model, the method comprising the steps of:

• receiving a 5-angle standard color measurement of a standard color at each of 5 color measurement angles;
• calculating a plurality of 5-angle tolerance values S5L, S5c, and S5H using equations (B), (C), and (D),
• =1.000·((|−|)/(i+1−i))2/3+0.002·C*45+0.33;
• (B)
• =1.478·((|−|)/(i+1−i))+0.014·C*45+0.27;
• (C)
• and
• =0.800·((|−|)/(i+1−i))+0.004·C*45+0.30;
• (D)
• wherein where L*, C* and H* are a lightness, a chroma, and a hue, is a measurement angle, i is an iteration of the measurement angle, and where C*45 is the chroma of the standard color measured at 45 degrees;
• receiving a 3-angle standard color measurement of the standard color at each of 3 color measurement angles, wherein the 3 color measurement angles are a subset of the 5 color measurement angles;
• producing a neural network empirical model comprising a plurality of input nodes, a plurality of hidden nodes connected to the plurality of input nodes, and a plurality of output nodes connected to the plurality of hidden nodes, wherein the neural network empirical model is configured to output 3-angle tolerance values S3L, S3c, and S3H at each of the 3 color measurement angles;
• training the neural network empirical model with (1) the 5-angle standard color measurement, (2) the 3-angle standard color measurement, and (3) the 5-angle tolerance values such that the 3-angle tolerance values output by the neural network empirical model are about the same as the 5-angle tolerance values at each of the 3 color measurement angles.