MATHEMATICAL FOUNDATIONS OF AUTOMATED COLORIMETRIC QUALITY CONTROL OF BAST FIBER RAW MATERIALS

Abstract

The article is devoted to the development of mathematical foundations for automated colorimetric quality control of bast fiber raw materials, providing objective, reproducible and rapid determination of fiber color characteristics. It is emphasized that traditional organoleptic quality determination methods regulated by DSTU 4015:2001 are characterized by high subjectivity (error up to 15-20%), labor intensity (30-60 minutes per sample) and low productivity, making operational quality control impossible under industrial production conditions. The feasibility of using the sequence of color transformations RGB→XYZ→CIE L*a*b* to ensure perceptual uniformity of colorimetric measurements is theoretically substantiated. It is shown that the CIE L*a*b* space provides significantly better correspondence of geometric distances to visual color differences compared to the device-dependent RGB space. Mathematical models of nonlinear transformations between color spaces have been developed taking into account gamma correction for the standard D50 illuminant. The transformation process through matrix operations and a piecewise-defined nonlinear function that models the psychophysiological features of human brightness perception has been formalized in detail. The choice of the Delta E CMC (l:c) color difference metric with parameters l:c = 2:1 for classification tasks of bast fiber raw materials is substantiated. An algorithm for calculating compensation factors SL, SC, SH, which account for the non-uniformity of color difference perception by the human eye in different areas of the color space, has been formalized. The CMC metric provides 30-40% higher accuracy compared to the basic Delta E*ab formula for textile materials. Statistical methods for processing colorimetric data from multiple image pixels have been developed. The minimum sample size nₘᵢₙ ≈ 139 pixels has been determined to ensure accuracy of ±0.5 L* units at a 95% confidence level, with samples of 500- 1000 pixels recommended to ensure high statistical reliability of results. A classification algorithm based on the nearest reference method for four quality groups of flax fiber according to DSTU 4015-2001 has been formalized based on the Delta E CMC minimization criterion. A mathematical model of measurement errors has been constructed with separation of systematic (ΔL* = ±3-5 for uncalibrated systems) and random components (σrand,L ≈ 0.5-1.5). Methods for error compensation through calibration and statistical averaging have been substantiated, allowing achievement of total Delta E CMC determination error at the level of σΔE ≈ 0.1-0.2 units. The developed mathematical foundations enable creation of automated quality control systems with accuracy of ±0.1-0.2 Delta E CMC, result reproducibility with coefficient of variation CV < 3%, and capability to classify samples within seconds, significantly exceeding the capabilities of traditional organoleptic methods.